Using numbers and handling data
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Contents
Introduction
Learning Outcomes
1 Decimals
1.1 Introducing the decimal system of numbers
1.2 Decimal points
1.3 Marking decimals on a scale
1.4 Decimal places
1.5 Rounding to decimal places
1.6 Multiplication and division by factors of ten
1.7 SI units and conversions
1.8 Adding and subtracting decimal numbers
1.9 Addition of decimal numbers
1.10 Subtraction of decimal numbers
1.11 Addition and subtraction in practice - fluid balance
2 Accuracy, precision and common errors
2.1
2.2
2.3
2.4
2.5
Differences between accuracy and precision
Checking accuracy and precision
Common maths problems and errors in the workplace
Sources of errors
What is a sensible dose?
3 Handling data
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
Graphs
The anatomy of a graph
Types of graphs and their uses
Bar graphs
Line graphs
Graphs with unusual scales - graphing exponentials
Descriptive statistics
Descriptive statistics
4
5
6
6
7
8
8
11
12
15
19
19
21
22
24
24
24
25
27
29
29
29
30
31
32
33
38
45
46
Conclusion
Keep on learning
Acknowledgements
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Introduction
Introduction
This course is designed for those who are employed in the health services, perhaps as a
paramedic or as operating theatre staff. If you are a student, you will have a tutor to help
you, and perhaps a work-based mentor supplied by the employer - normally the NHS. The
aim is to use the workplace as a teaching arena that helps provide relevance and
meaning to the activities you undertake, and it is especially designed to be relevant to
students' current or future employment in health areas.
This OpenLearn course provides a sample of level 1 study in Health & Wellbeing
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Learning Outcomes
After studying this course, you should be able to:
l
understand the decimal system of numbering (hundreds, tens, units)
l
explain the best way to write down decimal numbers and associated units of measurement in the healthcare
workplace, in a manner that avoids confusion
l
understand the concepts of discrete and continuous variables and the best types of graphs used to represent
these data
l
analyse, construct and extract information from graphs.
1 Decimals
1 Decimals
1.1 Introducing the decimal system of numbers
Many different systems for writing numbers have been developed over the history of
humankind.
The easiest way of counting small numbers is to use your fingers, and for this reason
many numerical systems, such as the decimal system, are based around the number ten.
But what happens when you run out of fingers to count on?
Numbering systems get round this problem by using a system of scale in which many
small units are represented by a single larger unit, and many of these larger units are
represented by a single even bigger unit, and so on.
For instance, the number five can be represented as five fingers, or by a single hand. In
the same way, if you were writing numbers, you might want to use one symbol to
represent single units and another to represent a larger collection of these units. How
about a line to represent an individual unit and a star
to represent a collection of five
individual units? Using this system, the number eleven could be represented as two stars
and a line
(i.e. five plus five plus one).
The decimal system uses a similar concept of scale, but it is arranged such that ten of any
particular unit makes one unit of the next size up. This also works the other way round:
each larger unit consists of ten units of the next size down.
For instance, ten pennies (ten 'units') are the same as one 10p piece (one 'ten'). Similarly,
ten 10p pieces (in other words one hundred pennies) are the same as one pound.
These different groups differ by a factor of ten (each group is ten times larger or smaller
than the one that precedes or follows it), which is also known as an order of magnitude
(see Figure 1).
Figure 1 Three orders of magnitude: 100, 10 and 1
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1 Decimals
Groups of decimal numbers start from zero, so within a group of ten, any particular unit
may be numbered between 0 and 9. Unlike many earlier systems of numbering, the
decimal system doesn't rely on having to use different symbols to indicate larger and
larger groupings of numbers. Instead, the order in which the symbols 0 to 9 are written
provides the information that tells you the size of a number. In other words the decimal
system is a positional system of numbering - the numbers are read and written from left
to right, and the order in which they are written or spoken is used to tell you how many
different factors of ten are present in the number. For example, the number 125 (one
hundred and twenty five) contains three different orders of magnitude. On the left are the
numbers of hundreds (one hundred), then the numbers of tens (two tens), and finally the
numbers of units (five units).
The name of the unit of measurement is usually written to the right of the number, after a
space, and this information tells you what individual objects are being counted (people,
grams, metres, etc.). As such, a distance of one hundred and twenty five metres would be
written as 125 m, where 'm' represents the unit of measurement.
1.2 Decimal points
Suppose you have less than one of any particular unit: how would you represent that
using the decimal system?
Well, we've already seen that decimal numbers rely on a positional system, in which
values get smaller by factors of ten as you read from left to right. If we continue doing this,
then the number to the right of a single unit represents tenths of that unit. A decimal point
is then used to mark the boundary between the whole units and tenths of that unit.
For instance, I look in my pocket and find I have one pound ten pence. If my units of
measurement were pounds, then I would write this amount as £1.1 (one and one tenth of
a pound). However, if my units of measurement were pence, then I would write this as
110p (one hundred and ten pence).
In Table 1, you can see how the position of each digit in the sequence is used to tell you
the overall size of the number.
Table 1 Decimal numbers described in terms of their orders of magnitude
Description
Tens
Units
Decimal
point
Tenths
Hundredths
4.5 is 4 units and 5 tenths
0
4
•
5
0
6.87 is 6 units, 8 tenths and 7
hundredths
0
6
•
8
7
98.04 is 9 tens, 8 units, 0 tenths and
4 hundredths
9
8
•
0
4
0.06 is 6 hundredths
0
0
•
0
6
Now try placing decimal points appropriately and reading the values of decimal numbers
for yourself with practice questions 1, 2 and 3.
Right click and open the practice questions in a separate window, then you can switch
easily between the course text and the questions.
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1 Decimals
1.2.1 Study Note 1
Simple rules for dealing with orders of magnitude and decimal points in decimal numbers:
values ten times bigger than the order of magnitude you are looking at go to the left, ten
times smaller go to the right, and less than 1 to the right of the decimal point.
Note: in many European countries, a comma is used instead of a decimal point. For
instance in France and Germany two and a half (in other words 2.5) can be written as 2,5.
This is important to bear in mind, for example, if the drug you are measuring out comes
from a European supplier, or you are following a laboratory protocol that was developed
abroad.
1.3 Marking decimals on a scale
Figure 2 shows a picture of a ruler. The major units are marked in centimetres (1 to 11
cm), whilst the intervals between the centimetres have each been split into ten equal,
smaller units. These minor units are therefore tenths of a centimetre, commonly known as
'millimetres'. (There are 10 millimetres in 1 centimetre.) A similar type of decimal scale is
used on many devices that you will work with, such as syringes, gas pressure gauges and
pH meters, even though the physical quantities they measure, and therefore their units of
measurement, are different.
In Figure 2 several numbers have been highlighted to allow you to relate the way a
decimal number is written to the quantity that number represents. For instance, if our units
of measurement are centimetres, then we can see that 0.3 cm is less than 1 cm (it is
actually 3 millimetres). Similarly, 2 cm is less than 3.5 cm. However, 70 mm is less than
10.2 cm, because 70 mm means the same as 7 cm.
Figure 2 Units and tenths of units represented on a ruler
Now practise placing decimal values on a similar scale for yourself with practice questions
4 to 8.
Right click and open the practice questions in a separate window, then you can switch
easily between the course text and the questions.
1.4 Decimal places
If you have less than one unit you should put a zero before the decimal point to make it
easier for yourself and others to read the value (e.g. you should write 0.4 rather than just
.4, as will be explained later in this course). However, how many zeros should you put
after the last whole number in the series? For instance, is 0.4 the same as 0.40?
The short answer is that on one level, it is. However, by writing 0.40 we are saying that
there are four tenths and zero hundredths, and importantly we are saying that we can
actually measure to an accuracy of an individual hundredth of a unit; in other words to two
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1 Decimals
decimal places. In contrast, by writing 0.4 we are only claiming an accuracy to the level
of individual tenths of a unit, or to one decimal place.
One way of getting a more accurate measurement is to use an instrument with a more
finely divided scale.
Figure 3 shows a close-up of two thermometers, labelled A and B, that were placed side
by side to record the air temperature in a room.
Figure 3 Thermometers A and B measuring the same temperature in a room. The finer the
scale, the more certain you can be of the reading, and the more decimal places you can
quote with confidence
In terms of accuracy the scale on thermometer A is quite coarse, as the markings
represent individual degrees Celsius (°?C). Using this scale, we can see that the room
temperature is somewhere between 21 °?C and 22 °?C. On closer inspection, someone
might estimate it at 21.7 °?C, but someone else could easily record it as 21.6 °?C or
21.8 °?C. There is some uncertainty in the first decimal place, and there is certainly no
way we could accurately state the temperature to two decimal places using this
thermometer.
In order to give someone an idea of how confident we are about the measurements we
make with thermometer A we should quote the range of possible values (i.e. the highest
and lowest values) that the actual temperature could be. Since we estimate the reading to
be between 21.6 °?C and 21.8 °?C we would choose the mid-point of these values, and
say that the temperature was within 0.1 °?C of 21.7 °?C. As we will see later, another way
to write this would be 21.7 °?C 'plus or minus' 0.1 °?C, or 21.7 ± 0.1 °?C. This gives us a
measure of the uncertainty of thermometer A.
Now look at thermometer B. This thermometer has a finer scale, with divisions marked
every 0.1 °?C. Now we can clearly see that the room temperature is between 21.6 °?C
and 21.7 °?C. This is within the range of possible values that we estimated from
thermometer A, but the finer scale of thermometer B allows us to be more certain of the
temperature. Nevertheless, even though thermometer B allows us to read the
temperature to within 0.1 °?C we cannot be so sure about the second decimal place;
someone might read it as 21.63 °?C, whilst another person might read it as 21.61 °?C or
21.65 °?C. With this scale, we can be sure of the first decimal place, but not the second.
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1 Decimals
1.4.1 Study Note 2
An important point to remember when writing down measurements from a scale is never
to quote more decimal places than you can reliably read from the measuring device you
are using.
Figure 4 The right tools for the job? A 5 ml syringe and a 20 ml syringe. Match the syringe
size to the volume of liquid to be administered
SAQ 1
Figure 4 shows a photo of a 5 ml syringe and a 20 ml syringe. Which is the most
appropriate syringe for measuring to an accuracy of 0.5 ml?
Answer
Looking at the scales given on the sides of the syringes the markings on the 5 ml
syringe go up in 0.5 ml jumps, whilst the marking on the 20 ml syringe go up in 2 ml
jumps. Therefore, you could only be confident about measuring to an accuracy of 0.5
ml using the 5 ml syringe.
SAQ 2
What would you do if you needed to inject 0.6 ml?
Answer
You would need to select a smaller syringe than either of those shown in Figure 4 in
order to be confident of measuring and administering the correct volume of liquid. A 1
ml syringe, marked in 0.1 ml increments, would be ideal.
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1 Decimals
In general, the smaller the syringe size the more finely the scale is marked and therefore
the easier it is to distinguish small changes in volume. For this reason (as well as reasons
of cost; large syringes are generally more expensive than small ones), it's generally easier
to read and more accurate if you use the smallest syringe that can accommodate the
volume you need to inject.
1.5 Rounding to decimal places
Sometimes the result of a calculation gives a number with lots of decimal places - far more
than you need or could reliably measure. For instance, suppose a patient is required to
receive 5 ml of medicine a day, evenly spaced in three injections. How much medicine
should they be given in each dose?
To divide the 5 ml of medicine into three equal parts would mean measuring out 5 ÷ 3 =
1.6666 ml (where the 6s keep repeating, or recurring indefinitely). It's not realistic or
feasible to measure out the medicine to this kind of accuracy. Instead, you first need to
think about what level of accuracy is needed. An injection of this volume would be most
accurately dispensed using a 2 ml syringe, marked in 0.1 ml increments. In this case then,
the accuracy of your measurement would be limited to 0.1 ml, or in other words to one
decimal place. To administer the amount calculated above you would need to round the
figure to the nearest decimal place.
The rule to remember with rounding to a particular decimal place is that if the next number
to the right of that decimal place is 5 or more, you round the figure up to the next highest
number, and if it's 4 or less it remains the same. For instance, to correct 1.6666 ml to one
decimal place, find the first decimal place and then look at the next (smaller) decimal
place to its right, which we've highlighted here as 1.6666 ml. As this number is greater
than 5 we have to round up, and the amount becomes 1.7 ml corrected to one decimal
place (1 dp). If the original number had been 1.6466 ml then the value corrected to one
decimal place would be 1.6 ml (1 dp).
Here are some more worked examples for you to practise with:
Now try rounding numbers to specific decimal places for yourself with practice question 9.
Right click and open the practice questions in a separate window, then you can switch
easily between the course text and the questions.
Box 1 General rules for numbers in healthcare
l
Try to avoid the need for a decimal point
Use 500 mg not 0.5 g
Use 125 mcg not 0.125 mg
l
Never leave a decimal point 'naked'
Paracetamol 0.5 mg not Paracetamol .5 mg
l
Avoid using a terminal zero
Diazepam 2 mg not Diazepam 2.0 mg
l
Put a space between the drug name and dose
Apresoline 55 mg not Apresoline55 mg
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1 Decimals
Study Note 3 'Dos and don'ts' with decimals in the healthcare
workplace
As suggested in Box 1 above, there are a number of common 'dos and don'ts' that you
need to remember and apply whenever you are dealing with decimals in your workplace.
l
Look carefully! Because a decimal point is just a dot on the page it is sometimes
easy to miss when reading, especially on lined paper or in faxed documents. For this
reason if there are no whole units, always place a zero before the decimal point when
writing decimal numbers, e.g. seven tenths should be written as 0.7 and not as .7.
l
Similarly, don't add additional zeros after a decimal point as this may indicate a
degree of accuracy to which you are unable to measure, e.g. one and a half should
be written as 1.5 and not as 1.50 (unless, for a specific reason, you need to quote the
value to 2 decimal places).
l
In general, try to avoid the need for decimal places by changing the scale to use a
different unit of measurement. For example, half a gram can be written as 0.5 g. ,
This is the same as 500 mg (500 milligrams). Similarly, 0.125 mg can be rewritten as
125 micrograms. Note that although the accepted scientific symbol for 'micrograms'
is 'μg', when this is hand-written it can often be confused with 'mg', the symbol for
milligrams. To avoid confusion, and help to reduce the risk of error, many hospitals
prefer to use the symbol 'mcg' for micrograms. We will be covering scales and units
of measurement in the next section.
Listen to the audio track below. It contains information that reinforces what you have
just learned here.
Click to listen to the track [3 minutes 30 seconds, 4.02MB]
Audio content is not available in this format.
1.6 Multiplication and division by factors of ten
1.6.1 Getting comfortable with factors of ten
Moving a decimal point by one place changes the value of the number by a factor of ten.
For instance, to multiply a value by ten you can just move the decimal point one place to
the right:
Notice that if the starting number doesn't have a decimal point shown we can place an
imaginary decimal point after the last digit, and a zero to the right of this, in order to help
us see the change in the order of magnitude.
To multiply a number by 100, move the decimal point 2 places to the right:
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1 Decimals
The same principle applies to dividing by 10, 100, 1000 etc., except you move the decimal
place to the left instead, thus making the original number smaller. For instance:
Look at the animated examples in the second 'Info' box of your practice questions. These
animations illustrate how sequentially moving a decimal point by one place at a time
changes that number by a factor of ten each time the point is moved.
In Table 2, each number in the second column differs from the one in the row immediately
above or below it by a factor of ten, or in other words by one order of magnitude.
Table 2 Names, units and symbols for factors of ten found within the
decimal system of numbering
Name
Number
Order of
magnitude
Power
Unit
Symbol
Million
1 000 000
6
106
mega
M
5
kilo
k
hecto
h
Hundred thousand
100 000
5
10
Ten thousand
10 000
4
104
Thousand
1 000
3
103
2
Hundred
100
2
10
Ten
10
1
101
One
1
0
100
Tenth
0.1
−1
10−1
deci
d
−2
10
−2
centi
c
−3
milli
m
micro
μ
nano
n
Hundredth
0.01
Thousandth
0.001
−3
10
Ten thousandth
0.000 1
−4
10−4
Hundred
thousandth
0.000 01
−5
10−5
Millionth
0.000 001
−6
10−6
Ten millionth
0.000 000 1
−7
10−7
Hundred millionth
0.000 000 01
−8
10−8
Billionth
0.000
000 001
−9
10−9
The units shown in bold are those that you are likely to encounter on a daily basis in your work.
In the column marked 'Power', there is a 10 and a small, raised (superscript) number next
to it for each of the names of the factors of ten mentioned. This superscript number is
called the power or the exponent, and it indicates how many times the first number or
'base' (in this case a 10) is multiplied by itself in order to give the actual amount being
signified. For instance, 102 ('ten squared'), means 10 × 10 = 100. Similarly, 104 means 10
× 10 × 10 × 10 = 10 000.
Notice also that numbers less than zero can be shown by a negative exponent. For
instance, 10−1 means '1 divided by 10', where 1 ÷ 10 = 0.1. Similarly, 10−4 means '1
divided by 104', which is 1 ÷ 10 000, or 0.000 1. Note that any number to the power zero
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1 Decimals
means that it is divided by itself and is therefore 1 (e.g. 10 ÷ 10 = 1), which explains why
Table 2 states that 100 = 1.
An exponent will often have a number and a multiplication sign written before it, for
example 2.4 × 103, which also means 2.4 × 1000, or 2400. An easy way to work out the
value of numbers expressed as exponents is to move the decimal point the same number
of places as the exponent.
l
If the exponent is positive, then move the decimal point to the right by the same
number of places as the exponent.
l
If the exponent is negative, then move the decimal point to the left by the same
number of places as the exponent.
Common units of measurement
In practice, the most commonly used units differ from each other by a factor of 1000, and
the names of some of these have been highlighted previously in Table 2. As an example
of this, you can see in Figure 5 how the units of weight - tonne (t), kilogram (kg), gram (g),
milligram (mg), and microgram (μg) - differ from each other by a factor of 1000.
Figure 5: Common units of measurement differ by a factor of 1000
Interactive content is not available in this format.
It takes 1000 kilograms to make 1 tonne, so if you had a value written in kilograms you
could convert it to tonnes by dividing by 1000 and re-labelling it as tonnes (i.e. 500 kg =
0.5 t). Similarly, you could convert a value written in tonnes to kilograms by multiplying by
1000 (so 2 t = 2000 kg).
1.6.3 Litres and kilograms
The two physical units of measurement that you will probably come across most often in
your workplace concern volumes of liquids and weight measurements. It's important to get
a feeling for what various factors of ten look like, so that you can spot when there seems
to be a mistake in a value that you've calculated or have been given by someone else.
The litre is the main unit of measurement for liquid volumes (written as liter in America),
but what does a litre of fluid look like? What about a millilitre (ml; one thousandth of a litre)
or a microlitre (μl; one millionth of a litre)?
A litre is the volume of liquid or gas that would fit into a cube measuring 10 cm on each
side (10 cm × 10 cm × 10 cm = 1000 cubic centimetres (cm3) = 1 litre). A millilitre is the
volume of a cube measuring 1 cm on each side (1 cm × 1 cm × 1 cm = 1 cm3 = 1 ml). A
microlitre is the volume of a cube measuring 1 mm on each side (1 mm × 1 mm × 1 mm =
1mm3 = 1μl)
l
A typical carton of fruit juice has a capacity of a litre.
l
A teaspoon holds about 5 millilitres of liquid.
l
One raindrop is about thirty microlitres.
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Look around your workplace to find out the typical volumes used for various applications:
e.g. what's the capacity of a blood bag or a drip bag? How about the various syringes or
pipettes you might use? If your work is lab-based, then you will probably be measuring
volumes down to a smaller scale of microlitres. Again, get used to the volumes of typical
containers (e.g. 7 ml and 20 ml specimen tubes; 2 ml, 1.5 ml and 0.5 ml micro-centrifuge
tubes) and the equipment appropriate to measure out each of these various volumes
(measuring cylinders 1-1000 ml; pipettes 0.1-30 ml; micropipettes 0.1-1000 μl).
The kilogram is the basic unit of measurement for weight. Again, what does a kilogram
feel like? What about a gram (g; one thousandth of a kilogram)? What about a milligram
(mg; one thousandth of a gram)?
A litre of water weighs one kilogram.
l
l
A £20 note weighs about one gram, whilst a pound coin is almost 10 grams.
Originally, a 'pound' was the monetary value given to a pound weight of sterling silver
(an alloy of 92.5% silver and 7.5% copper) - hence 'pound sterling'.
l
About 12 grains of salt weigh one milligram.
Once more, look around your workplace and find the typical weights that you might deal
with. Get used to the different sizes of various tablets and medications and the typical
doses that patients are given. This will help you spot when a calculation might be wrong
and you need to double-check with someone.
1.7 SI units and conversions
The international system of units (Le Système International d'Unités: abbreviated to SI)
was developed in France during the 18th century in an effort to create a unified and
rational system of weights and measures. The SI system became adopted as the world
standard in 1960.
There are seven basic units (or base units) to the SI system and these are shown in
Table 3. All other units of measurement can be derived from combinations of these base
SI units.
Table 3 The seven base SI units
Quantity
Name
Symbol
Length
metre
m
Mass
kilogram
kg
Time
second
s
Temperature
kelvin
K
Electric current
ampere
A
Luminous
intensity
candela
cd
Amount of
substance
mole
mol
Many units of measurement arise from a combination of different base SI units, and some
of these are given in Table 4. For instance, you will see from the table that speed is
defined in 'metres per second'. This can be written as metres divided by seconds, i.e. m/s,
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where the slanted line indicates the act of division. Another way you may see this
expressed is as m s−1 where the negative exponent indicates the division.
Similarly, the newton is a SI-derived unit of force. It is defined as the amount of force
needed to accelerate a mass of 1 kg at a rate of 1 metre per second per second, and is
expressed in terms of the following SI units: m kg s−2.
Table 4 Commonly used units that are derived from
base SI units
Quantity
Derived unit
Symbol
−1
SI units
m s−1
Speed
metres per
second
ms
Force
newton
N
m kg s−2
Energy
joule
J
m2 kg s−2
Volume
cubic metre
m3
m3
Pressure
pascal
Pa
kg m −1 s−2
Absorbed dose (radiation)
gray
Gy
m2 s−2
Equivalent dose
(radiation)
sievert
Sv
m2 s−2
Radioactivity
becquerel
Bq
s−1
In addition, some non-standard SI units are in common usage, and a selection of these is
given in Table 5.
Table 5 Commonly used units that are not derived from base SI
units
Quantity
Symbol
Non-SI unit
Conversion factors
Time
min
minute
1 min = 60 s
Time
h
hour
1 h = 60 min = 3600 s
Time
d
day
1 d = 24 h = 1440 min = 86400 s
Volume
1
litre
1 l = 0.001 m3
Mass
t
tonne
1 t = 1000 kg
Energy
cal
calorie
1 cal = 4.18 J
Temperature
°?C
Celsius
1 °?C = 274.15 K
Pressure
mmHg
millimetres of
mercury
1 mmHg = 133.3 Pa
When you are dealing with particularly large or small quantities, the SI system is
combined with the decimal system, such that you write the decimal notation first then
follow it with the relevant SI unit.
For instance, 1000 metres is 1 kilometre and this is abbreviated to 1 km. One hundredth of
a metre is one centimetre, abbreviated as 1 cm.
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Box 2 A note about temperature
Although the basic SI unit of temperature is the kelvin, most people use and are familiar
with degrees Celsius (°?C). (Note that when using kelvins, the 'degrees' symbol '°' is not
used.) These two temperature scales are equivalent, so a temperature change of 1 °?C is
the same actual increase or decrease as a temperature change of 1 K. The only difference
is that these temperature scales don't start at the same place. The Celsius scale takes its
starting point to be the temperature at which water freezes (0 °?C), whilst the Kelvin scale
starts from absolute zero (0 K; the coldest temperature theoretically possible, where even
molecules stop moving). Zero kelvin is the same as −273.15 °?C.
It is vitally important that you are comfortable with how the SI and decimal systems
interact and how to make conversions within these units of measurement. In order to help
you achieve this, now test yourself with practice questions 10 and 11.
Right click and open the practice questions in a separate window, then you can switch
easily between the course text and the questions.
Activity 1 will allow you to practise comparing quantities (in this case weights and
temperatures) that are in different units of measurement. You can practise similar
calculations to those you will be asked to perform in Part 1 for yourself using many
common foodstuffs that list the quantities of vitamins, fats, fibre, etc. that they contain. The
time you should allow to complete the three parts of the activity is 45 minutes. Please try
working through the questions first before looking at the answers.
Activity 1.1 Comparing SI units
0 25
Part 1 Vitamin tablets
A multivitamin tablet contains various vitamins and minerals. Table 6 shows the
recommended daily allowance (RDA) for a selection of them, as shown on the packet.
Table 6 Label from packet of
vitamin tablets
Component
%
RDA
Amount per
tablet
Vitamin A
100
800 μg
Vitamin B6
100
2 mg
Vitamin C
100
60 mg
Vitamin D
100
5 μg
Vitamin E
100
10 mg
Magnesium
50
150 mg
Note: although denoting micrograms as 'μg' is scientifically correct, remember that in
the healthcare workplace it's generally better to write it as 'mcg' to avoid possible
confusion with milligrams (mg).
Using this list, can you order the components, from the highest to the lowest amount,
by the weight of the substance that equates to 100% RDA?
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Answer
The first thing to notice is that not all of the units of measurement are the same, some
are in milligrams (mg) and others are in micrograms (μg). It helps if you convert all the
units of measurement to be the same before attempting to put them in order.
Only Vitamin A and D are expressed in micrograms, so let's convert them to
milligrams. Because a microgram is 1000 times smaller than a milligram, to convert μg
into mg, divide by 1000. (Remember, to divide by 1000 you can move the decimal point
3 places to the left: 1 μg = 0.001 mg).
The amount of Vitamin A present is 800 μg, or 800 ÷ 1000 = 0.8 mg. This is the full
RDA (100%), so the RDA is 0.8 mg.
The amount of Vitamin D present is 5 μg, or 5 ÷ 1000 = 0.005 mg. Again, this is the full
RDA, so the RDA is 0.005 mg.
The amount of magnesium present is 150 mg. However, this is only half of the RDA
(50%), so the full RDA is twice this amount:
150 mg x 2 = 300 mg
Therefore, in order of decreasing RDA, the components should read: magnesium (300
mg), vitamin C (60 mg), vitamin E (10 mg), vitamin B6 (2 mg), vitamin A (0.8 mg), and
vitamin D (0.005 mg).
Activity 1.2 Comparing SI units
05
Part 2 Temperature
What is the freezing point of water on the Kelvin scale?
Answer
The freezing point of water is 0 °?C, which is +273.15 K.
Activity 1.3 Comparing SI units
0 15
Part 3 Temperature
What is the normal range of human body temperature in degrees Celsius? Why is
there a range of values?
Answer
Body temperature varies with age, time of the day, menstrual cycle and location in the
body: in a 6-year-old the temperature can vary by around 1 °?C per day; your
temperature is generally highest in the evening; ovulation can increase the
temperature by about 0.6 °?C; the body core (gut, liver) is generally warmer than the
periphery (arms, legs). Bearing all this in mind, the range 36.5-37.2 °?C is considered
normal for a healthy person. 38 °?C or above is a significant fever.
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1.8 Adding and subtracting decimal numbers
By now you should be familiar with decimal numbers, the different ways in which they can
be represented (written in full, or simplified by using an exponent), and understand that by
moving the decimal point to the left or right of its original position you can change the
value of a decimal number by various factors of ten. In this section, we're going to explore
how to add and subtract decimal numbers without using a calculator.
We use addition and subtraction all the time in our daily lives (for example, when
estimating if we've enough money to buy a round of drinks, or comparing prices whilst out
shopping). However, addition and subtraction are skills that you need to master, because
you cannot rely on there always being a calculator to hand. Although this may sound like a
trivial task that anyone can do, surprisingly few people are confident in these two skills,
especially when under pressure! (It has been estimated that in England there are 14.9
million people aged between 16-65 who lack the skills to pass a maths GCSE qualification
(DfES, 2003).)
1.8.1 Study Note 4
If you have difficulty with this section, you might find it helpful to investigate some of the
Government schemes aimed at improving maths skills. More information about such
schemes can be found at http://www.direct.gov.uk/en/EducationAndLearning/AdultLearning/ImprovingYourSkills/index.htm (accessed 5 March 2008).
Box 3 The basics
When adding or subtracting two or more decimal numbers you need to be sure that they are
lined up so that you can compare units with units, tens with tens, and hundreds with
hundreds, etc. Even if the numbers contain a decimal point, they can still be lined up into
columns of the same magnitude.
1.9 Addition of decimal numbers
If we add 109.8 ml of one liquid to 6.5 ml of another liquid, what would be the total volume
of liquid in ml?
To compare 109.8 with 6.5, you need to remember that
Place the two numbers in a grid on top of each other and make sure that columns
representing the same magnitude line up with each other, and add an extra line at the
bottom where you can put the result of the addition. You should end up with something
that looks like this:
To add these numbers start from the right, with the smallest sized group and compare the
numbers in the Tenths column.
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In the decimal system, ten of one group must be expressed as one of the next sized order
of magnitude, so 13 tenths is expressed as one unit and three tenths.
Put your addition into the results line at the bottom.
Then move up one order of magnitude to the next column to the left (Units) and add these
values together.
Add the five units here to the one that's already in the result line, giving six units, and put
the single ten into the Tens box in the results line.
Moving up one order of magnitude and adding the tens together we have
There are no tens to add to the result line, but you keep the 1 that you've already put there
from the addition of the units.
Move up to the final order of magnitude and add the hundreds together.
So, as you can see from your results line, the sum of 109.8 ml and 6.5 ml = 116.3 ml
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Now practise some additions for yourself with practice question 12.
Right click and open the practice questions in a separate window, then you can switch
easily between the course text and the questions.
1.10 Subtraction of decimal numbers
Subtraction of numbers can be used to answer questions such as 'what's the difference
between two values?' or 'if something has decreased by a certain amount, what's its new
value?' Subtraction can also be thought of as undoing the process of addition. For
instance, instead of saying '£10 take away £7.85 leaves how much?' you could say, 'what
do I have to add to £7.85 to get back to £10?'
Let's work through an example: how much is 25.18 - 16.87?
The way to write down decimal numbers for a subtraction is the same as for an addition in
terms of arranging them into columns that contain numbers of the same size category
(hundreds, tens, units, etc.). The only difference is that the number that you want to
subtract must go below the number you are subtracting it from. Just like addition, you
should work through your calculation from right to left.
The other important rule to remember with subtractions is that, for each column, if the
number you are subtracting from is smaller than the number you want to subtract, then
you need to make that number larger by 'giving it' one of the units from the next highest
order of magnitude.
Let's see how that works out in practice with our example of 25.18 - 16.87.
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In the first column from the right (Hundredths), we need to subtract 7 from 8. Eight take
away seven leaves us with one hundredth.
In the next column (Tenths), we need to subtract 8 from 1. In order to do this, remove one
unit from the next highest order of magnitude (here, units) and add this to the tenths. One
unit is the same as ten tenths, so we add ten to the Tenths column: 1 tenth plus 10 tenths,
gives 11 tenths in this column.
Now we can subtract 8 from the resulting 11 tenths, to leave a total of 3 tenths.
In the next column (Units) we need to subtract 6 from 4, so once again we need to give
this column a unit from the next highest order of magnitude (tens). 1 ten is 10 units and if
we add this to the 4 units we already have, this gives 14 units in this column. Now we can
subtract 6 from 14, to leave 8 units.
Finally, in the Tens column, we're left with 1 ten to subtract from 1 ten, which leaves 0 tens.
So, the result of 25.18 - 16.87 = 8.31.
You can double-check you're right by adding 8.31 to 16.87 in order to confirm you get
back to the original 25.18.
Now practise some subtractions for yourself with practice question 13.
Right click and open the practice questions in a separate window, then you can switch
easily between the course text and the questions.
1.11 Addition and subtraction in practice - fluid
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balance
A common healthcare example that uses addition and subtraction involves calculating the
fluid balance of a patient.
Fluid balance is a simple but very useful way to estimate whether a patient is either
becoming dehydrated or overfilled with liquids. It is calculated, on a daily basis, by adding
up the total volume of liquid that has gone into their body (drinks, oral liquid medicines,
intravenous drips, transfusions), then adding up the total volume of liquid that has come
out of their body (urine, wound drains, blood lost during surgery, vomit). The fluid balance
is then calculated by subtracting the total output from the total input, and is generally
quoted in millilitres.
Ideally, the total volume of liquids that goes into a person ought to balance the total
volume that eventually comes out of them, so the difference in the total input and output
should be almost zero. However, if the fluid balance is positive, then this indicates that
more liquid is going in than is coming out (i.e. they are swelling up: not necessarily a bad
thing if, for instance, they were admitted suffering from dehydration). A negative fluid
balance indicates that more fluid is coming out than is going in and the patient is at risk of
becoming dehydrated.
SAQ 3
Calculate the fluid balance for the patient described in the following scenario.
Over the course of a day, a patient who has just undergone chest surgery receives an
intravenous saline drip of 1 litre and another of 900 ml. They drink 550 ml of water and
200 ml of fruit juice. During the same day, they produce 2.5 litres of urine, and they lose
110 ml of fluid from a tube that is draining their chest wound.
Answer
The fluid balance is the difference between the total fluid input and the total fluid output
during a day.
To calculate this, first, check whether the units of measurement are in millilitres;
convert them to millilitres, if necessary.
Then add together all the fluid inputs in millilitres.
Then add together all the fluid outputs in millilitres.
The fluid balance is the daily total inputs minus the daily total outputs.
This is a positive fluid balance, indicating the patient is taking on more fluid than they
are losing, although only by a tiny amount (40 ml is about half a tea cup). If the output
had been larger than the input, then the fluid balance would be a negative number
(e.g. input 2000 ml - output 2500 ml = -500 ml fluid balance).
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2 Accuracy, precision and common errors
2.1 Differences between accuracy and precision
Accuracy is a measure of how close a result is to the true value. Precision is a measure of
how repeatable the result is. For instance, a group of three friends tried the shooting
gallery at a fair and their targets are shown in Figure 6. The first person was an expert
marksman, but they were using a rifle with sights that had not been set properly. Although
they aimed the sights at the bull's-eye they consistently hit the target off to the left side
instead. They were not accurate, but they were precise. The second person was also an
expert marksman, but noticed the incorrectly set sights and compensated by aiming to the
right of the bull's-eye. Consequently, all their shots hit the centre of the target - they were
both accurate and precise and their results were good. The third person was hopeless
with the rifle and their shots landed all over the target - they were neither accurate nor
precise.
Figure 6 Target practice analogy, demonstrating the difference between accuracy and
precision. Low accuracy but high precision = systematic error. High accuracy and high
precision = good results. Low accuracy and low precision = bad results
From the example given you can see how it is possible to be very precise, but not at all
accurate. This is called a systematic error (sometimes also called bias) and can normally
be corrected.
2.2 Checking accuracy and precision
2.1.1 Accuracy
The way to ensure that equipment is accurate is to use a series of known standards
against which to calibrate the equipment. Calibrating should be done at least each day
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and sometimes more frequently (such as before using the equipment to measure
unknown samples). Many types of measuring equipment go through an automatic
calibration when they are switched on, but others require the user to provide a series of
known calibration standards.
2.2.2 Precision
Measuring the same sample should give the same result every time if the equipment is
precise. In practice, the information displayed by a measuring device can depend on
several factors (such as temperature and humidity) and can drift slightly over time.
Nevertheless, during the time it takes to complete a measurement sequence, all
measurements ought to remain within a specified, small margin of error, often marked on
the equipment. We will see later on, in Section 3.5, how to quantify the precision of a
series of measurements.
2.3 Common maths problems and errors in the
workplace
In a busy, hospital environment mistakes with medicines and other treatments can
happen at any time. Some of these are caused by communication/administrative
problems, whilst others are due to mathematical errors (the news stories shown in
Figure 7 are sadly typical).
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Figure 7 Mistakes with placing a decimal point can cost lives; articles available from http://
news.bbc.co.uk and http://news.scotsman.com (websites accessed 6 March 2008)
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However, many of these types of errors can be avoided by pausing and thinking, 'is this
right'?
2.4 Sources of errors
The following is a list of common problems that can lead to medication errors. They fall
into three broad categories according to where they occur in the sequence from a drug
being prescribed to it being administered to a patient. As you can see, the same types of
mistake can occur in each category. Those errors that involve maths are highlighted in
italics:
Prescription errors
l
Wrong drug prescribed (contraindicated, or allergy, or interferes with existing drug
therapy).
l
Prescription illegible/misread (number, decimal point, units of measurement) or
unsigned.
l
Wrong dosage form selected (tablets vs. liquid).
l
Wrong administration route selected (oral, intravenous, etc.).
l
l
Wrong prescribed drug concentration, or drug quantity/volume, or drug rate of
administration.
Verbal (orally communicated) orders.
Dispensing errors
l
l
Drug not available.
Drug preparation error (wrong drug selected or misreading/miscalculation of drug
concentration, or drug quantity/volume).
l
Equipment failure/malfunction.
l
Labelling error (e.g. wrongly labelled syringe).
Administration/omission errors
l
Misreading/miscalculating drug quantity/volume or drug rate of administration, or
dose of radiation, or route of administration.
l
Wrong drug administration technique/equipment.
l
Equipment failure/malfunction.
l
Drug chart not kept up to date/needs re-writing.
l
Drug not given at correct time, or correct frequency, or not given at all.
l
Inadequate patient ID or drug given to wrong patient.
In addition, the following factors can all contribute to medication errors:
l
excessive workload;
l
lapses in individual performance;
l
inadequate training;
l
inadequate communication.
(Adapted from ASHP, 1993)
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Box 4 How can you help to minimise errors?
Take a pro-active role in minimising errors by keeping yourself up to date with the latest
alerts and safe medical practices.
The Medicines and Healthcare Products Regulatory Agency website (http://www.mhra.gov.
uk/) is run by the UK Government (website accessed 13 March 2014). It has up-to-date
safety information and alerts about problems with medicines, especially problems
concerned with the labelling and packaging of medicines that might lead to drug errors.
The Institute for Safe Medication Practices (in the USA) also has a good website (http://
www.ismp.org/) that you can browse for more information (website accessed 13
March 2014). Of particular interest are the newsletters that highlight potential risks to be
aware of and good practice to adopt. You may wish to look at some of the articles and
extracts that are available from the current issue and past issues of the newsletter, or
search for a particular topic of interest (note: for the purposes of this course it is not
necessary to subscribe to the newsletters). At present (2007), this website only covers
America and Spain, so whilst there's much useful and practical advice here, please be
aware that procedural differences may occur in the UK.
The four audio tracks linked below contain information that reinforces what you have
just learned here. Listen to them now.
Click to listen to the track [3 minutes 56 seconds, 4.50MB]
Audio content is not available in this format.
Click to listen to the track [4 minutes 38 seconds, 5.32MB]
Audio content is not available in this format.
Click to listen to the track [1 minute 25 seconds, 1.63MB]
Audio content is not available in this format.
Click to listen to the track [3 minutes 30 seconds, 4.02MB]
Audio content is not available in this format.
As you have just seen from the errors list above, it is not always safe to assume that a
prescription has been filled out correctly. The most common numerical mistakes are
differences by a factor of 10, 100 or 1000, either because of a mistake with a
calculation or because the wrong unit of measurement has been written down. For
instance, as you have seen, when handwritten, the symbols for 'milli' (m) and 'micro'
(μ) can often be confused, but are different by a factor of 1000, which is why 'mcg' is
routinely used in hospitals to indicate micrograms.
With experience, you will gain a common-sense knowledge of when particular
calculations are correct or not, and by developing a reflective sense of your working
practices you will quickly become better and more confident with these routine
calculations. In particular, you should get used to the recommended dose ranges of
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the drugs or reagents you routinely deal with, and check the dose range against your
calculations for all new drugs that you come across.
2.5 What is a sensible dose?
This will vary from drug to drug and patient to patient, but bear in mind that most drugs
need to be swallowed or injected, so the manufacturer has designed the dose sizes to
be as easy as possible for a patient to take and for the health worker to administer.
The following dose ranges are the most sensible and practical for adults:
Table 7 Typical drug doses
Drug formulation
Typical dose at any one time
Liquid
oral: 5-20 ml (1-4 teaspoons full)
injection: generally 0.25-2 ml
subcutaneous: 1 ml or less
intramuscular: adults - up to 3 ml in large muscles
children and elderly - up to 2 ml
infants - 1 ml or less
Solid
1-4 tablets
Gas
0.2-150 litres/min
Radiation dose
20-70 μSv for an X-ray, 100-200 Sv for radiotherapy
For each category, the doses for a baby or a child are normally much less than adult doses.
If you find that the dose you have calculated or the prescription you have been given is
outside of this range, especially if it is out by a factor of 10, 100 etc., then it's likely that
a mistake has happened somewhere. If it's your own calculation, then double-check it.
If it still doesn't look right, or was written that way on the prescription then check with a
senior colleague.
3 Handling data
3.1 Graphs
Information is everywhere these days - in the form of images, written records, tables
and graphs. In this part of the course we want you to realise how useful graphs can be
to analyse numerical information, and to show you some techniques that can help you
decide how reliable this numerical information is.
It's often difficult to spot a trend or a relationship in a long list of numbers. Because the
human mind is highly adapted to recognising visual patterns, it is often much easier to
understand a series of numbers or measurements by representing them visually as a
graph.
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3.2 The anatomy of a graph
A graph shows how two different types of data that can take on different values (known
as variables) are related, or change in relation to each other; for instance, how a
patient's temperature changes over time. Each measurement consists of two variable
values: the patient's temperature and the time at which the temperature was taken.
Table 8 shows what these measurements might look like.
Table 8 A patient's temperature,
measured throughout the day
Time of day
Patient temperature/°?C
13:00
38.2
14:00
37.9
15:00
37.9
16:00
37.5
18:00
37.1
19:00
36.9
Notice that a measurement was not made at 17:00.
As we shall see, this information can become easier to interrogate once it has been
assembled graphically.
3.2.1 Axes
A graph is made using two different scales or axes, forming a right angle. The
horizontal axis (x-axis) is used to represent the variable that changes in a consistent
way, such as time, or in a way that you can control. The vertical axis (y-axis) is used to
represent a variable that you measure but may not be able to control directly, such as a
patient's temperature.
Each axis should be carefully labelled to indicate what it represents. To plot a graph,
you put a mark at the point where the two variables in each measurement meet.
One way to remember which way round the axes go is to remember that X comes
before Y in the alphabet and then picture entering someone's house; you go along the
corridor (horizontal, x-axis) before you can go up the stairs (vertical, y-axis).
3.2.2 Choice of scale
It's important to choose a scale that covers the range of values you have recorded for
that particular axis. If the scale is too big, then all of your measurements will be
bunched up at one end of the graph, making it difficult to read. It is also very important
to keep the scale consistent all along the axis, i.e. don't suddenly change the spacing
between the units of measurement on an axis.
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3.3 Types of graphs and their uses
Many different types of graphs exist, and each has something different about it that
makes it useful in a unique way. Here we will be looking at just two types of graph: bar
graphs and line graphs.
Here is a bar graph and a line graph plotting the patient's hourly temperature data that
we looked at in Table 8.
Figure 8 Bar graph and line graph of the same data (a patient's temperature measured
over a period of time)
From either of these graphs, you can quickly see that the patient's temperature is
gradually declining through the day, and by the seventh hour (19:00) it is at a normal
level of 36.9 °?C. Representing the data as a graph also allows you to estimate what
the patient's temperature probably was at the fifth hour of the measurement period
(17:00), when someone forgot to take a reading. If you imagine a straight line
connection between the temperature values at the fourth and sixth hours (16:00 and
18:00 respectively) this line would intersect the fifth time point at about 37.3 °?C.
Which type of graph is best to use? To help answer this question you can consider the
list shown in Table 9.
Table 9 Choosing the best graph for your data - advantages and
disadvantages of bar and line graphs
Graph
type
Advantages
Bar and
line
trends in data can be seen clearly (how one variable affects the other)
Bar and
line
easy to use the value of one of the variables to determine the value of the other
variable
Bar and
line
enables predictions to be made about results of data you don't have yet
Bar
best for 'discrete' variables (those that change in jumps, with no 'in between'
values)
Line
best for 'continuous' variables (those that change smoothly)
You will see from Table 9 that we have identified two different types of variable, and
these are defined by the way in which their numerical values change. Discrete
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variables can only have specific values within any given range (e.g. 1, 2, 3).
Continuous variables are not limited in this way, and can have any value within a
range.
SAQ 4
Find some examples of continuous data and discrete data in your workplace.
Answer
Examples of continuous variables could include: temperature, blood pressure and
pH. Examples of discrete variables might be: blood type, numbers of patients and
needle size.
3.4 Bar graphs
The following graph (Figure 9) records how the outside diameter of a hypodermic
needle is related to the needle gauge number.
Figure 9 Relationship between needle diameter and gauge size
This is an example of data that do not vary continuously, but instead change in discrete
jumps, and discrete data are often best represented using a bar chart. (An alternative
name for a bar chart is a histogram.) To express discrete data as a line chart would be
misleading, since it would give the impression that gauge number and needle diameter
are changing smoothly all of the time, which is not the case (there are no half- or
quarter-gauge sizes, only whole numbers).
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SAQ 5
If you wanted a needle with a diameter of 2 mm, what would be the required needle
gauge number?
Answer
Figure 9 shows that no such needle exists. A 2-mm-diameter needle falls between
gauge 14 (2.1 mm) and gauge 15 (1.8 mm). However, because 2.1 mm is closer to
2 mm than 1.8 mm is, you'd probably choose the gauge 14 needle.
3.5 Line graphs
To illustrate how to create and use line graphs, we will use the example of a
calibration curve.
A calibration curve is a type of line graph in which the response of a measuring device
to a series of known concentrations of a substance is plotted. You can then make a
measurement of an unknown sample - in the case we're about to examine, blood
serum samples from new-born infants - and use the calibration curve to work out what
concentration of substance is present.
SAQ 6
Think of a type of machine that can perform measurements to give readings that
can then be used to make a standard curve.
Answer
A spectrophotometer provides these types of measurement data, which can be
used to produce standard curves.
Box 5 Spectrophotometers explained
A spectrophotometer is an instrument that determines how much light of a particular
colour is absorbed by a liquid sample. The more there is of a coloured substance in the
solution, the more light will be absorbed (i.e. the less light passes through the solution).
After measuring how much light is absorbed by a series of solutions containing known
concentrations of the coloured substance, you can draw a graph of this data and use it
to calculate the concentration of that substance in an unknown sample from a
measurement of how much light it absorbs.
Here's how a spectrophotometer works:
1
White light from a bulb (source) is focused into a narrow beam by passing it
through a thin slit.
2
A prism is used to split the beam of white light into its component colours, in
the same way that water droplets can split sunlight into its component colours
to make a rainbow. Different colours of light have a different wavelength: the
distance between the peaks of the light waves, measured in nanometres
(where 1 nanometre is 10-9 metres). For an idea of scale, individual virus
particles range in size from about 20-300 nm).
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3
A second thin slit, just after the prism, can be moved from side to side to select
just one colour of light to pass through to the sample.
4
The light passes through a container with the liquid sample inside (usually the
light passes through 1 cm thickness of the liquid).
5
A light detector measures how much light is transmitted through the sample,
and compares this with how much light was emitted by the source. The
difference between these values gives a measure of how much light was
absorbed by the sample: i.e. the absorbance (A), often also called the optical
density (OD). The absorbance varies with wavelength, so measurements of
this type always specify the wavelength of light that was shone through the
sample. In the following example with C-reactive protein, the wavelength was
450 nm, so this would be quoted as A450 or OD450.
This process is summarised below in Figure 10, which also gives an indication of the
wavelengths of different parts of the visible light spectrum.
Figure 10 A spectrophotometer measures how much light of a certain wavelength is
absorbed by a liquid
In this particular example, we are looking at how the concentration of C-reactive
protein (CRP: a blood component produced in response to infection) changes the
intensity of a blue-coloured test solution: the more CRP present, the more intense the
blue colour becomes. The intensity of the blue colour is determined using a
spectrophotometer to measure the Optical Density at 450 nm (O.D. 450 nm). More
specifically, this is a measure of the amount of a blue light with a wavelength of 450
nanometres (450 nm) that is absorbed by a known thickness of the solution before the
light reaches a detector.
When making a calibration curve, the following points need to be considered:
1
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The range of blood serum CRP concentrations that might be encountered in
infants, and the level of infection that this correlates with, are as follows:
normal: <2 μg/ml
sepsis: >22 μg/ml
dying: >120 μg/ml
Where < means 'less than' and > means 'more than'.
(Romagnoli et al., 2001)
From these values, a range of known standards containing between 0-100 μg/ml
CRP should cover the most likely range of infant serum CRP concentrations.
2
Am I within the working range of the instrument?
Many instruments are only accurate over a specific range of values. As we
continue to increase the CRP concentration, the blue dye will become more and
more intense. However, this can't continue forever and after a point the solution
will be saturated with blue dye. Beyond that the solution can't get any more
intense, no matter how much CRP we add. As we begin to reach this saturation
point, the measurements will plateau out to the maximum value as CRP is
increased. The most reliable part of the calibration curve covers the middle range
of concentrations, where it is closest to being a straight line. For this reason, it is
called the linear part of the graph.
3
How random or 'noisy' is the assay and the measuring device?
Depending on the precision of the measuring device you may not get the same
reading every time from the same sample. To compensate for this it is a good idea
to repeat the sample measurement two or three times and then calculate the
average or mean value. For example, if you repeated the measurement of the
same sample once, then you'd add both results together and then divide the
resulting figure by 2 to find the average value.
With these points in mind, Figure 11 shows a table and graph of OD450 values
measured using a series of known concentrations of CRP.
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Figure 11 Calibration curve for CRP
In general, the x-axis is used to represent a variable that changes in a consistent way,
or in a way that you can control (here, it's the known concentrations of CRP that you
have measured out). The y-axis is normally used to represent a variable that you
measure but may not be able to affect directly (in this case the optical density of the
CRP test solutions that you read from the measuring device).
The small circular symbols on the graph mark the measured data points and they are
joined by a curved line.
Notice that the graph shows a positive correlation between the two variables: the
more CRP that is present, the larger the optical density reading. Other types of data
may show a negative correlation between the variables, whereby one of the
measured entities decreases as the other increases, e.g. the further you drive your car,
the less petrol you have in the petrol tank.
Questions relating to Figure 11
The following OD450 values were measured from serum samples taken from three
babies: 0.12, 0.40, and 1.74.
SAQ 7
Use the calibration curve shown in Figure 11 to estimate the serum concentration of
CRP for each sample.
Answer
From the curve you can deduce the values are about 2 g/ml, 7 g/ml, and 90 g/ml,
respectively.
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SAQ 8
Does the existing scale allow you to confidently estimate these values?
Answer
You probably found it difficult to make accurate CRP readings for the OD450 values
of 0.12 and 1.74. This is because the scale of the y-axis (the OD450 values) is too
compressed. If the graph were redrawn, with the y-axis maybe twice its current
height (but still running from 0 to 2) then this would provide a better resolution.
SAQ 9
One of your estimated concentrations might be particularly inaccurate; which one
do you think this is likely to be and why?
Answer
Apart from the problem already mentioned concerning the size of the y-axis, the
OD450 measurement of 1.74 is likely to be inaccurate. This is because the
measuring assay is becoming saturated, as you can see by the calibration curve
flattening off towards a constant high value in this region. The colour of the test
solution is almost at its maximum intensity and changes in CRP concentration
make almost no difference to the OD450 reading.
SAQ 10
What do you infer about the health of each baby from these results?
Answer
Earlier on in this section you were told what level of infection corresponded with a
particular range of serum CRP concentrations:
normal: <2 μg/ml
sepsis: >22 μg/ml
dying: >120 μg/ml
You have calculated the CRP values of three babies to be about 2 μg/ml, 7 μg/ml,
and 90 μg/ml. Using these figures for reference, you can see that the baby with a
serum CRP concentration of 2 μg/ml is healthy, although at the high end of the
range for a healthy baby. The baby with a CRP concentration of 7 μg/ml is beyond
the normal level and would be expected to have a mild fever, but is not yet
dangerously unwell. The baby with a CRP concentration of 90 μg/ml is very
seriously ill and will require intensive care.
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3.6 Graphs with unusual scales - graphing
exponentials
3.6.1 Radioactivity and bugs!
Many natural processes involve repeated doublings or halving at regular intervals. You
may have come across this already in your work, in the context of bacterial growth or
radioactivity. In this section, we are going to look in more detail at bacterial growth and
radioactivity and we will be using graphs to examine how the numbers of bacteria or
numbers of radioactive atoms change over time.
3.6.2 Exponential increase: bacteria
Bacteria are single-celled organisms. Many different types of bacteria exist and they
populate almost every environment on earth, from deep oceans to soil to human
intestines. Several bacteria are beneficial to us: for instance, our gut bacteria can help
to break down foodstuffs that we would otherwise find difficult to digest. However,
some bacteria produce harmful toxins and if they grow in an uncontrolled way in our
bodies this can have serious health consequences.
If a bacterium is growing under perfect conditions, then it will divide in two at a constant
rate, which is called the doubling time (generally every 12 minutes to 24 hours,
depending on the type of bacterium). Each of the newly produced bacteria will then
themselves divide in two, and so on. Thus, from one original bacterium, the following
number of offspring will be generated during each round of division: 2, 4, 8, 16, 32, 64,
128, etc. In other words, a doubling at each division: 1 × 2 = 2, 2 × 2 = 4, 4 × 2 = 8, 8 ×
2 = 16, 16 × 2 = 32, 32 × 2 = 64, 64 × 2 = 128.
From your earlier knowledge of exponentsyou might realise that this series of numbers
can also be represented as a power series of exponentials: 21, 22, 23, 24, 25, 26, 27.
Hence, this type of growth is often referred to as 'exponential growth'. In reality, it's rare
to find organisms undergoing exponential growth, except at the beginning of an
infection where growth conditions are the closest to perfect. Later on, factors such as
cell death, nutrient availability, waste product production and overcrowding can all
restrict the growth rate.
3.6.3 Exponential decrease: radioactive decay
The most familiar example of exponential decrease is provided by radioactive decay.
Radioactivity is a natural phenomenon that is used routinely in many medical
applications, from imaging (radioactive tracers in PET scanning) to therapy (radiotherapy to destroy tumours). During radioactive decay, the number of radioactive
atoms halves at a constant rate, called the half-life. For instance, the radioactive
isotope 11C, pronounced 'carbon 11', has a half-life of 1224 seconds (a little over 20
minutes). After 1224 seconds, there would only be half of the starting amount of 11C
remaining. After another 1224 seconds there would be only half of this amount
remaining, i.e. 1/4 of the starting amount, and so on. Thus, the fraction of the starting
material that remains after each half-life follows this series: 1/2, 1/4, 1/8, 1/16, 1/32, 1/
64, etc. Just as we saw with exponential increase, this sequence can be re-written as a
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power series of exponentials, except this time they are fractions: 1/21, 1/22, 1/23, 1/24,
1/25, 1/26.
Exponential decay occurs in many more situations than just radioactivity. For instance,
most drugs become metabolised in the body according to an exponential decay
pattern. The clearance of most substances from the blood by the kidneys (or their
clearance from the blood using a dialysis machine) also follows an exponential decay
pattern.
Box 6 Radioactivity - a brief explanation of atomic instability
At the core of every atom is a nucleus, made up of a fixed number of protons and
neutrons, and orbiting the nucleus are electrons. The number of protons present defines
what the element is (hydrogen, oxygen, gold, etc.) whilst the number of protons plus the
number of neutrons defines the isotope number of that element.
Although each isotope of the same element has a different configuration of protons and
neutrons in its nucleus, most interactions between atoms involve the electrons that orbit
at a relatively large distance around the nucleus. Because the behaviour of the
electrons isn't greatly altered by changing the number of neutrons, isotopes of the same
element generally look and behave the same as each other, it's just that their weights
are very slightly different because of the different numbers of neutrons present. Almost
every element has many different isotopes and these often exist together naturally as a
mixture. The air you're breathing right now contains a mixture of three different oxygen
isotopes: 16O, 17O, and 18O.
Let's look at one element, carbon, in a little more detail. The nucleus of a carbon atom
normally has 6 protons and 6 neutrons. These two numbers added together give the
isotope number, so this would be 12C (pronounced, 'carbon 12'). However, other
isotopes of carbon exist, with different numbers of neutrons.
There is a form of carbon that has 6 protons and 5 neutrons, 11C. However, this
arrangement of protons and neutrons is unstable. The nucleus of a 11C atom is poised,
like an over-wound spring, to suddenly re-arrange. When this happens, a burst of
energy is released that is detected as radioactivity, and in the process, one of the
protons is converted into a neutron. The resulting atom has a much more stable
nucleus, containing 5 protons and 6 neutrons, but is now a different element (remember,
the number of protons defines the element) with its own unique properties; it has
become an atom of 11B (boron 11). Some nuclear arrangements are inherently more
unstable than others are, and this explains why different radioactive isotopes undergo
these rearrangements (often called 'radioactive decay') at different rates.
3.6.4 Representing exponential relationships using graphs
What do exponential increase and decrease look like when plotted as a graph?
Although exponentials describe anything that continually doubles or halves, the
specific assumption of 'exponential increase' and 'exponential decay' are that these
happen during a constant time interval. If the time taken for doubling or halving
remains constant, then an exponential increase looks like the thick blue line in
Figure 12, which shows the number of bacteria present in a millilitre of growth medium
(marked on the left side of the graph), counted each hour.
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Figure 12 Graph showing an exponential increase of bacteria numbers over time
You can see that at 3 hours there were about 10 bacteria/ml, at 4 hours there were
about 20 bacteria/ml and at 5 hours there were about 40 bacteria/ml, so, in this case
the bacteria are dividing with a doubling time of around 1 hour. In practice, it's unlikely
that any cell type would be doubling, rather conveniently, at each time point you had
chosen to make a measurement. Instead, it's best to estimate the doubling time by
reading off the time at which there were a known number of cells e.g. 50 bacteria/ml,
and then reading off the time when that number of bacteria had doubled to 100
bacteria/ml. The time difference between these values gives the doubling time (often
given the symbol, Td) or if we were dealing with exponential decay, it would give the
half-life (often given the symbol, t1/2). In this case, there were 50 bacteria/ml at about
5.5 hours and 100 bacteria/ml at about 6.5 hours, so the bacteria doubled in about 6.5
- 5.5 = 1 hour. Try this for yourself to estimate how long it took to go from 100 bacteria/
ml to 200 bacteria/ml. If the bacteria are still growing exponentially then you should get
the same value for the doubling time.
Because an exponential increase can also be represented as a power series, if we
drew a graph showing the time at which there were 21, 22, 23, 24, 25, 26, 27, etc.
bacteria present then this would produce a straight line if the bacteria were doubling
exponentially. In practice, it's extremely unlikely that we would be lucky enough to
make a measurement when there were exactly 4 (i.e. 22) or 16 (24) or 128 (27) bacteria
present. More likely, we might find that we had 130 bacteria/ml instead of 128 bacteria/
ml. Nevertheless, it is possible to convert our existing data of bacteria number
measured at fixed time points into a straight-line graph. To do this we need to convert
our bacteria number values into logarithms.
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A logarithm isn't a lumberjack pop group, but is a mathematical method to calculate the
values of exponents. In essence, a logarithm asks the question 'what power do I need
to raise something to in order to get the answer I want?' For instance, 2 raised to what
power, 'p', gives the value 128? (i.e. 128 = 2p). We've already seen that 128 = 27, so
the logarithm of 128 with base 2 is 7.
A word about bases: you're already familiar with the decimal system of numbering, in
which exponents are of the type 'ten to the power something' (e.g. 103, 106 etc.). In the
decimal system, 10 is the base number. However, we could actually choose any base
we like. Things that change by halving or doubling are best expressed using base 2,
i.e. you'd express exponents in a base 2 system of numbering as 'two to the power
something' (23, 26, etc.).
For many complex historical and mathematical reasons, most logarithms use a base
value of 2.718, a special mathematical constant. For instance, if you use the logarithm
button on a calculator (marked 'ln', or 'natural logarithm') to find the logarithm of 128
you are asking it to calculate the value of 'p' in the equation 128 = 2.718p. This
calculation gives the answer p = 4.852.
In Figure 12 we have converted the measurements of the numbers of bacteria/ml into
their logarithm values (marked on the right side of the graph) and have plotted these
against time, using a red colour. Because one axis is logarithmic and the other is
normal, this is called a semi-logarithmic graph. Notice that plotting the logarithmic data
gives a straight line, as we predicted it would, and this confirms that the growth of the
bacteria is exponential. If their growth had started to slow down then the semilogarithmic plot would deviate from a straight line at that point and begin to flatten off. It
is for this reason - that it's easier to tell by eye if a line is straight than to tell if it's an
exponential curve - that semi-logarithmic graphs are used to spot this kind of pattern.
The steepness, or gradient, of the line in a semi-logarithmic graph can be used to
determine how fast the bacteria are dividing, i.e. to find the doubling time, or in the
case of exponential decay to find the half-life. You may come across specific names for
this gradient: for exponential increases, the gradient is also known as the growth
constant, whilst for exponential decreases the gradient is known as the decay
constant.
Just like the gradient of a road, the gradient of a line graph tells you how far you would
go up (or down) on the y-axis (change in y: abbreviated to ΔY) if you moved one unit
along on the x-axis (change in x: abbreviated to ΔX). (The symbol 'Δ' is the Greek letter
'delta', and is used to denote a difference or change.) On this graph, it's difficult to read
accurately how much the y-axis logarithm value changes per hour, but it looks to be
somewhere between 0.6 and 0.7. A more accurate estimate can be made if we look
over a wider range. Over the entire 8 hours of measurement, the logarithm of the cell
density changes by about 5.5, so that gives a gradient of 5.5 ÷8 = 0.6875, which is
the growth constant.
The doubling time or half-life is given by dividing the logarithm of 2 by the gradient of
the graph, i.e.
.
Using a calculator we can confirm that ln(2) = 0.6931. Therefore Td = 0.6931 ÷ 0.6875
= 1.008 hours. This agrees well with our estimate from the exponential curve that the
bacteria are doubling every (one) hour.
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3.6.5 Using the gradient of a semi-logarithmic graph to calculate
doubling time or half-life
Knowing the equation
allows you to perform several useful calculations without
needing to make a graph, and we'll look at one such example in a moment.
First, let's return to the gradient of the exponential increase graph in Figure 12. This
gradient is positive: as you move left to right along the x-axis the graph climbs, and the
values on the y-axis increase. What would be the gradient if this graph were reversed,
to show an exponential decrease? In that situation, the graph would fall from left to
right and the y-axis values would become smaller as you moved along the x-axis, so
the gradient would have a negative value.
There are some basic rules for dealing with negative numbers that you need to be
aware of before moving onto the next example, which deals with exponential decay.
Box 7 Being positive about negative numbers
A negative number is any value less than zero. Look at the thermometer scale below.
−4 °?C −3 °?C −2 °?C −1 °?C 0 °?C +1 °?C +2 °?C +3 °?C +4 °?C
The freezing point of water is marked as 0 °?C. Temperatures above freezing go up
incrementally in positive numbers. As temperatures get colder and colder below 0 °?C
their values also increase, but in negative numbers. The temperature difference
between 0 °?C and 1 °?C is 1 °?C. Similarly, the temperature difference between 0 °?C
and −1 °?C is also 1 °?C. The temperature difference between −1 °?C and +1 °?C is
2 °?C. Using a scale like this we can come up with some general rules for addition and
subtraction involving negative numbers.
Addition and subtraction involving negative numbers
Using the thermometer scale above, you should be able to verify for yourself that the
following sums are correct:
3 − 4 = −1
−2 + 1 = −1
−1 − 1 = −2
Now here's the tricky one: subtracting a negative number is the same as adding a
positive number, so:
−5 − (−3) = −5 + 3 = −2
Another helpful way to view the same calculation is as follows:
−5 − (−3) = −(5 − 3) = −(2) = −2
Multiplication and division involving negative numbers
If you multiply 2 × 2, you get 4, so, a positive number multiplied by a positive number,
gives a positive number. However, what do you get if you multiply 2 × −2?
The rule is that a positive number multiplied by a negative number (or the other way
round), gives a negative number, so 2 × −2 = −4 (and −2 × 2 = −4).
The general idea is that if you multiply (or divide) a positive number by a negative
number then that changes the sign of your eventual answer; or, put another way:
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multiplying or dividing the same signs together results in a positive number, whilst doing
the same with opposite signs give a negative number.
positive × positive = positive
positive × negative = negative
negative × positive = negative
negative × negative = positive
positive ÷ positive = positive
positive ÷ negative = negative
negative ÷ positive = negative
negative ÷ negative = positive
These rules will come in handy as you work through the following example question.
Example 1
The radioactive isotope iodine 131 (131I) is used in medicine to diagnose conditions of
the thyroid gland. 131I has a half-life of 8 days. If the hospital has just received a
delivery of 10 g 131I: how much will remain after 25 days?
As an initial check, we can work out a rough answer. Each half-life takes 8 days,
therefore, after 8 + 8 + 8 = 24 days the 131I will have gone through three half-lives.
Since we lose half of the 131I after each half-life, out of the 10 g starting material there
will be 1.25 g left after 3 half-lives.
We want to know how much 131I will remain after 25 days, so we would estimate that
there should be a little less than 1.25 g remaining.
Now let's work out the answer accurately using some of the principles we would apply
if we were plotting data of this radioactive decay on a graph like that shown in
Figure 12.
On a semi-logarithmic graph we know that,
where Td can represent doubling time
or half-life; this distinction becomes important in a moment. We are concerned here
with the half-life, so let's re-name Td as 'half-life'.
If we multiply both sides of the equation by gradient, and divide both sides of the
equation by half-life, then this re-arranges to give
Using a calculator to find the logarithm of 2 gives, ln(2) = 0.6931
The half-life is 8 days, so
We know that the gradient of the graph can also be calculated in a different way,
directly from the graph itself, since the gradient of any straight-line graph is defined
as
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Now the distinction between doubling time and half-life becomes important. The initial
equation we used doesn't distinguish between exponential increase or decrease.
However, because we are dealing with an exponential decay (the amount of 131I is
decreasing with time) the gradient will be negative. The actual numerical value, the
steepness, of the gradient is unchanged; it's just that we are moving 'downhill'
(exponential decrease) rather than uphill (exponential increase), so this value must
have a negative sign.
Therefore,
= −0.0866
Rearranging gives, ΔY = −0.0866 × ΔX
The time interval we are interested in is ΔX = 25 days, so
Using the general rules we established in Box 7, we know that because we are
multiplying a positive number by a negative number the answer will be a negative
number, thus, ΔY = −0.0866 × 25 = −2.165
The y-axis is on a logarithmic scale, so this value that we have deduced for ΔY
represents the logarithm of the change in the amount of 131I that has occurred during
25 days.
The starting amount of 131I is 10 g, so in order to be able to compare this with our value
for ΔY (which is in logarithms) it too needs to be converted into a logarithm, and the
logarithm of 10 is ln(10) = 2.3026.
During 25 days, this starting amount of 131I will be reduced by ΔY, therefore: the
remaining 131I after 25 days = 2.3026 - 2.165 = 0.1376
Once more, this value for the remaining amount of 131I is a logarithm because the
values that we have used to calculate it have been in logarithmic form. In order to
convert it to a conventional number you need to use the 'anti-logarithm' button on your
calculator (shown as ex on most calculators). You should find that ex(0.1376) = 1.148,
which is how many grams of 131I are remaining after 25 days.
Finally, always check that your calculated value fits with your initial estimate from
counting half-lives. Our initial rough answer was for there to be a little less than 1.25 g
131
I remaining after 25 days. Reassuringly, our accurate answer of 1.148 g is exactly in
keeping with the rough calculation.
SAQ 11
The elimination of a drug from the blood, due to metabolism and excretion, follows an
exponential decay with a half-life of 4 hours. Below a blood plasma concentration of
0.2 mcg/ml the drug is not effective. (Blood consists of cells floating in a liquid. This
liquid component of the blood is called plasma and can be produced by centrifuging a
blood sample to sediment out all of the cells.) If a patient is dosed intravenously to a
blood plasma concentration of 1.5 mcg/ml at 13:00 one day, at what time will the blood
plasma concentration of drug have become reduced to a level where the drug is no
longer effective and needs to be re-administered?
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Answer
To answer this, first work out a rough answer by counting how many half-lives are
needed to get from the initial concentration of drug to the ineffective concentration of
drug:
Somewhere between the second half-life (4 + 4 = 8 hours) and the third half-life (4 + 4
+ 4 = 12 hours), the plasma concentration will reach 0.2 mcg/ml and the drug will
become ineffective.
Using the equation
we can find the gradient, as we have previously seen, by rearranging to give:
We are dealing with a decrease, so the slope of the gradient on the graph will be
negative, therefore
We want to know at what time to re-administer the drug, and time is on the x-axis, so
we need to make ΔX the subject of the equation: rearranging gives, ΔX = ΔY ÷
−0.1733.
We can calculate ΔY because we already know the start and end concentrations of the
drug. They are 1.5 mcg/ml at the start and 0.2 mcg/ml at the end. As such, ΔY = ln(1.5)
− ln(0.2) = (0.4055) − (−1.6094) = 0.4055 + 1.6094 = 2.0149 (remembering that
subtracting a negative number is like adding a positive number).
Because ΔY describes a decrease, this number should be represented as a negative
number, i.e. −2.0149, then ΔX = −2.0149 ÷ −0.1733 = 11.627 hours.
So, after 11.627 hours the drug concentration will have decreased from 1.5 mcg/ml to
0.2 mcg/ml, and the drug will have to be re-administered. This fits with our original
estimate of somewhere between 8 and 12 hours.
3.7 Descriptive statistics
3.7.1 Averages: finding the middle of a group of numbers
The average of a group of numbers (which is sometimes called the 'mean') represents the
balancing point or middle of the data. It is found by adding together all of the individual
data values and dividing by the sample size.
For example, I ask five friends how many children they have, and I get the answers: 0, 1,
2, 4, and 1. The total number of children is 0 + 1 + 2 + 4 + 1 = 8 and there are 5 friends, so
the average number of children per friend is 8 ÷ 5 = 1.6. This has been represented
graphically in Figure 13.
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Figure 13 The average is the balancing point of the data
Notice that the average doesn't tell you anything about the range of values that were
measured (the maximum and minimum values within the given group of numbers). For
instance, someone with their head in the oven and their feet in the freezer might have an
average temperature of 37 °?C, even though their hair is on fire and their feet are frozen!
3.8 Descriptive statistics
3.8.1 Standard deviation: finding how reproducible a series of
measurements are
Even if we know the maximum and minimum and middle values in a group of numbers, we
still don't have a clear idea about the distribution of values within that range: are most of
the values all bunched up at one end or spread evenly across the results?
For instance, if I count my pulse rate on the hour every hour, nine times over the course of
a day, I might get the following values for the number of beats per minute (bpm): 61, 59,
60, 62, 60, 100, 59, 63, 61. The average result is 65 bpm and range of values is 59-100
bpm. From looking solely at the range you might get the impression that my heart rate
fluctuates wildly throughout the day. In fact, my heart rate is remarkably constant, and the
value of 100 bpm was a reading, taken after running up the stairs just before 14:00.
The way to find out whether a series of measurements are all tightly grouped together or
are spread out more evenly is to make a graph that shows how often a particular value
was recorded. This type of graph is called a frequency distribution, because it shows how
frequently particular values were recorded.
For instance, in the list of my pulse rate measurements from above:
62, 63, and 100 bpm were recorded once
59, 60 and 61 bpm were recorded twice.
These data have been plotted on the bar graph shown in Figure 14.
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Figure 14 A frequency distribution histogram showing how often my pulse rate was
measured to be a particular value
With enough measurements, this type of graph eventually resembles a bell-shape, often
called a Gaussian curve or a normal distribution, where the most common value is at the
top of the curve and there's a spread of less and less common results (some larger and
some smaller) on either side. For example if I'd continued to make pulse rate
measurements, I would soon have found that my measurement of 100 bpm was a one-off
and in fact, most of the measurements were centred around 65 bpm.
Where results are very regularly reproduced and don't deviate much from the mean value
(high precision), the bell-shaped curve is steep and narrow (like the top graph in
Figure 15) and this indicates a small standard deviation from the mean value (as
suggested by the condensed spread of values on the x-axis). In contrast, when the results
are more variable (low precision), the bell-shaped curve is relatively spread and flat like
the bottom graph in Figure 15, and this indicates a large standard deviation from the mean
value.
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3 Handling data
Figure 15 A normal frequency distribution looks like a bell-shaped curve. Top: data
showing a small standard deviation from the mean value. Bottom: data showing a large
standard deviation from the mean value
The exact value of the standard deviation for a group of numbers is calculated using a
complex equation that you are not required to know. Suffice to say that in my pulse rate
data above, the mean value is 65 bpm and the standard deviation is 13.2 bpm. Because
the standard deviation indicates the spread of data both greater and less than the mean
value, it is shown with a 'plus or minus' symbol. Thus, the mean value with the standard
deviation is 65 ± 13.2 bpm.
About 68% of all the results occur within one standard deviation of the mean value on the
horizontal, x-axis, and this figure is represented by the red areas on both of the graphs in
Figure 15). About 95% of the results lie within two standard deviations from the mean (the
red plus the green areas on these two graphs), and about 99% of the results lie within
three standard deviations of the mean value (the red, green and blue areas).
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Conclusion
This information can be used to find out if measurements are unusual or not. For instance,
we know that 95% of the measurements should be within 2 standard deviations of the
mean value, meaning that only 5% of the results will fall outside of two standard
deviations. Because the graph is symmetrical, this 5% includes results that are both larger
and smaller than the mean value. If we are only interested in results larger than the mean
value, then we can see that only 5 ÷ 2 = 2.5% of results occur outside of the green area to
the right of the graph. i.e. only 2.5% of the results would be expected to be more than 2
standard deviations greater than the mean value.
In my pulse rate data, one standard deviation was 13.2 bpm and the mean value was 65
bpm. Therefore a pulse rate two standard deviations larger than the mean would be (13.2
× 2) + 65 = 91.4 bpm. As such, I would expect any pulse rate of 91.4 bpm or above to
occur less than 2.5% of the time. If today I measured my pulse rate on 5 occasions and it
was above 91 bpm on one occasion then that could happen by chance, but if it was this
high on subsequent measurements then I should become increasingly worried, since 2
out of the 5 measurements made (i.e. 40%) were above 91 bpm.
Conclusion
This free course provided an introduction to studying Health & Wellbeing. It took you
through a series of exercises designed to develop your approach to study and learning at
a distance, and helped to improve your confidence as an independent learner.
Keep on learning
Study another free course
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Acknowledgements
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Acknowledgements
Except for third party materials and otherwise stated (see terms and conditions), this
content is made available under a
Creative Commons Attribution-NonCommercial-ShareAlike 4.0 Licence
Course image: David J Morgan in Flickr made available under
Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence.
Grateful acknowledgement is made to the following sources for permission:
Sections 1.10 and 3.8: www.CartoonStock.com
This extract is taken from S110 © 2007 The Open University.
All other materials included in this course are derived from content originated at the Open
University.
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