1
Errors are always found in measurements.
There are four main factors which contribute to measurement errors:
1. Reading the scale errors:
This error depends on the accuracy of the scale being used; the finer the scale, the less the error.
In general, a measurement is only accurate to no more than the smallest scale, or in some cases,
1/2 that value.
2. System errors:
These errors are due to flaws in either your equipment, your experiment plan, or in the way you
use the equipment.
Examples of equipment flaws include:
- using a meter stick with one end worn short.
- using a warped meter stick.
- measuring up from the surface of a lab table that is not flat.
- school centigram balances are really only accurate to ±0.1 g, even if they are
called centigram balances; therefore round off such measurements to the nearest
tenth of a gram.
---> e.g. you took the mass of an object with one of these centigram balances to be 125.63 g,
so realistically, this value should read
.
Examples of flaws in equipment use include:
- the use of a stopwatch; the readout gives a lot of digits but the hand held
method used is inaccurate.
---> when using your stopwatch, your reaction time is about 0.2 sec. Your
error is at least half your reaction time because you both turn on the watch
and turn it off.
---> measurements must be rounded down to reflect the level of accuracy in the
measurement.
---> e.g. a hand-held time reading of 12.360 s has a real error of ±
s and
therefore should read
- the use of a meter stick; most classroom pieces can be read to four figures, but a
hand held meter stick can’t give this level of accuracy so the rule-of-thumb is to
measure your meter stick to three figures if over 10 cm and two figures if under
10 cm.
Examples of flaws in experimental plans include:
- the type of measurements made. Bigger measurements are better than smaller
ones, and it is a system error to make small measurements when you could
design to make larger ones.
---> e.g. students were asked to determine the mass of a type of coin, using a
balance of accuracy of ± 0.2 g.
- one student measured a single coin at 15.5 g; determine the % error
- another student measured 20 of the same type of coin, obtaining a mass
of 312.7 g and an average of 15.6 g per coin; determine the % error:
3. Random errors:
2
Every time a person does a measurement that person does it just a little bit differently; this is also
true if different people perform the same measurement. This results in slightly different
measurements. Therefore, take several measurements and take an average!
As an example, you are measuring how far a spring can shoot a cart across the lab floor. It takes a
few trials to get consistent results and then you get similar results in each trial. The
measurements are as follows:
1.54 m, 3.67 m, 2.57 m, 1.23 m, 3.74 m, 3.69 m, 3.72 m, 3.75 m, 3.68 m.
What should you do with these measurements?
(a) Which ones could we discard and why?
(b)
Take an average of the measurements kept in (a).
Another example is the measurement of a coin’s mass previously discussed.
4. Errors Introduced by Using Math:
Because measurements have errors, when measurements are multiplied or divided, or added or
subtracted, the resultant answer must be no more accurate than the least accurate input.
We have rules of manipulation to ensure this, which are called the rules of significant figures.
Multiplying or dividing
The answer must have the same quantity of significant digits as the least accurate input.
Ex.
17.74 m x 0.576 m x 5.6 m = 57.222144 ---> 57 m3
Adding or Subtracting Significant Figures
The accuracy of a sum or difference is determined by the least accurate place value of the digits
in the input data. Examine the example below.
468.32
accurate to
the nearest
hundredth
+
236.4
=
only accurate
to the nearest
tenth
704.72
round off to the
tenth’s place
value = 704.7
Subtracting
Ex. 468.92
---------accurate to nearest hundredth
-24.
---------accurate to nearest unit
444.92 ---------round off to nearest unit; rounded answer = 445
SIGNIFICANT FIGURES: All readings/calculations in science should be taken to the
figure which is doubtful and no further. Hence each figure in a reading/calculation should be
a significant figure. In most cases zeros preceding digits are not significant, but merely
indicate the proper position of the decimal point.
e.g. 6.48 --> has 3 significant figures. The 8 is the first uncertain figure.
How many sig. figs. are in the following?
00.77 ---> __ sig. figs.
0.0031 ---> __ sig. figs.
3
3.200 ---> __ sig. figs.
100.01 --> __ sig. figs.
0.00316 --> __ sig. figs.
A better method of determining the number of sig. figs. that a particular number has is to
place that number into scientific notation.
e.g.
0.0036
-->
0.0010072 -->
10000
---->
3.16 x 10-3
1.0072 x 10-3
1 x 104
{ 3 sig. figs.}
{5 sig. figs.}
{1 sig. fig. or maybe 5.}
Note that in the absence of a decimal point, any trailing zeroes may or may not be significant.
READINGS:
In taking measurements {temp, distance, etc...} significant figures are those digits our
measuring devices can determine or estimate. Significant figures are those digits you are sure
of plus the first doubtful figure only.
20
15
10
5
A Graduate cylinder appears as shown above. What digits of the volume measurement do you
know for sure? What digit can you estimate? So we have a volume reading of __?__ ml, in
which one of the digits is __?__, but all of the digits are __?__.
The Plan to Minimize Errors in Experiments
(1)
Any measurement should be done more than once to see if it can be
repeated. Then, several measurements should be taken and averaged to get
rid of random errors.
(2)
Make your measurement large! When measuring small objects, a large
quantity of those objects should be measured together and then averaged.
This reduces the % error.
(3)
Round off measurements to reflect the level of accuracy. A three figure
measurement is usual.
(4)
Use the rules of significant figures when adding, subtracting, multiplying
or dividing. Remember the number of digits shown in a measurement
indicates it’s level of accuracy.