PRELAB4: METRIC UNIT PREFIX AND ERROR ANALYSIS
INTRODUCTION
In this course it is often necessary to calculate frequencies by taking reciprocals of
periods. Sometimes the period will be is units of seconds. With sound signals, however,
the period will usually be in the millisecond range, and occasionally even the
microsecond range. Note that our Mean Reaction Time (RT) is approximately 180~200
msec to detect visual stimulus, and approximately 140-160 milliseconds to detect an
auditory stimulus.
Many students have difficulty taking reciprocal of period which is in milliseconds. For
example, they will write 1/(4.2 ms) = 0.24 Hz or even 0.24 mHz. This is not correct.
Remember that 4.2 ms is shorthand for 4.2 x 0.001 s = 0.0042 s:
1
1
4.2 x 0.001 s = 0.0042 s = 238 Hz
This homework will give you some practice take reciprocals of numbers which have
prefixes like milli- and micro- on their units. You will need a calculator to do this
homework. Since physical quantities can vary many factors of 10s, they are expressed in
standard “prefix” as follows:
T = terra = ×1012
G = giga = ×109
M = mega = ×106 = ×1,000,000
k = kilo = ×103 = ×1,000
m = milli = ×10−3 = ×0.001
μ (or u) = micro = ×10−6 = ×0.000001
n = nano = ×10−9
p = pico = ×10−12
These prefix will be needed in future labs. Please remember the prefix notation!
CALCULATION PROCEDURE
Method 1
The most direct way to calculate a reciprocal is simply to write the number without a
prefix before you take its reciprocal. This is what was done in the introduction to take the
reciprocal of 4.2 ms. Just write out 4.2 ms as 0.0042 s, put 0.0042 into your calculator
and push the 1/x button (x-1 on some calculators), or you can divide 1 by 0.0042.
The only problem with this approach is that you may make a mistake when translating
the prefix, for example, writing 4.2 ms = 0.042 s, which is not correct. But if you are
careful, you can use this method perfectly well.
Method 2
Since you will use your calculator to take the reciprocal, it is often more reliable to make
use of the scientific notation feature of your calculator to deal with the metric unit prefix.
A prefix like m (milli-) means “multiply by 10-3.” That is, 4.2 ms = 4.2 × 10−3 s.
On most calculators today, you can enter a number directly into the calculator using
scientific notation. This is done using a button which is marked either “EE” or “EXP,”
depending on what brand of calculator you have. To get
4.2 × 10−3 s into your calculator, you take the following steps:
4: Metric Unit Prefix Prelab–1
a)
4.2
b) EE
– Button for scientific notation (called EXP on some calculators).
c)
– Button to get a minus sign (called CHS on some calculators).
+/-
d) 3
The steps c) and d) are reversed in some calculators. At this point, your calculator display
should be something like: 4.2 –03 or 4.2E-3. Notice that this means 4.2 × 10-3, not 4.2-3.
Then, to take the reciprocal (to find the frequency) just hit the 1/x button. Your
calculator display may be 2.381 02, which means 2.381 × 102 = 238.1 Hz. (Note: some
calculators will display 238.1 directly on their screen after hitting the 1/x button.)
The advantage to this method is that you put the number part (4.2) directly into the
calculator, and then put the unit prefix (milli-) directly into the calculator as -03. You
never make a mistake counting zeros while moving the decimal point.
Calculations of period from frequencies are done in the same way. Say you have a
frequency of 2 kHz and you want to find the period:
1
1
= 5 x 10-4 s = 0.5 ms
T= f =
2  103 Hz
To do this on your calculator, you would push:
2
EE
3
to get a display of 2 03. Then your push 1/x, and the calculator displays 5 -04, which
means 5 × 10-4 = 0.0005. This is the same as 0.5 × 10−3, so the period is 0.5 ms.
Metric Unit Prefix Worksheet
If your answer is less than 0.01 or greater than 1000, then express your answer using a
prefix. For example, if f = 44 kHz, then write your answer as:
1
T = 44 x 1000 Hz = 0.000023 s =23 µs.
f = 16 Hz
T=
f = 16 kHz
T=
f = 750 Hz
T=
T = 0.010341 µs
f=
T = 0.167 ms
f=
4: Metric Unit Prefix Prelab–2
Uncertainties and Error Propagation
The accuracy (correctness) and precision (number of significant figures) of a
measurement are always limited by the apparatus used, by the skill of the observer, and
by the basic physics in the experiment. In doing experiments we are trying to establish
the best values for certain quantities, or trying to validate a theory. We must also give a
range of possible true values based on our limited number of measurements.
The variation in measured data is expressed in synonymous terms uncertainty, error, or
deviation. The Systematic error is the result of a mis-calibrated device, or a measuring
technique which always makes the measured value larger (or smaller) than the “true”
value. Careful design of an experiment will allow us to minimize or to correct for
systematic errors. The random errors can be dealt with in a statistical manner.
In your oscilloscope experiment, you can change the knob of voltage per division to
measure more precisely a small voltage value. For example, if you use 0.1 V/div then the
LEAST COUNT of the scope will be 0.02 V. Your measurement error is a fraction of
0.02 V, say ±0.01V, called the instrument limited error (ILE). If I measure a voltage 5
times, and each time measured is 0.36 V. My measurement is (0.36 ±0.01) V.
Besides ILE, the measured values may vary. For example, you measure the voltage of a
terminal of a mystery box, you may find 0.32 V, 0.34 V, 0.37 V, 0.37 V, 0.40V. The
average of these 5 measurements is 0.36 V, and the standard deviation is 0.03V. You
should choose the larger of the ILE and standard deviation. For example, since the ILE of
±0.005V is smaller than the standard deviation, your answer is (0.36 ±0.03) V.
measurements
1
2
3
4
5
average
Data (V)
0.32
0.34
0.37
0.37
0.40
0.36
|deviation| (V)
0.04
0.02
0.01
0.01
0.04
0.024
(ΔV)2 (Volt2)
0.0016
0.0004
0.0001
0.0001
0.0016
0.00095
The average of the last column is obtained by σ2=∑(∆V)2/(N-1), where the N is the
number of measurements, and σ is the standard deviation. Fill the table measured below
using a stop-watch and find the standard deviation. Note that the σ2 defined above is the
mean-square error for the ordinary least square, whereas the maximum-likelyhood
estimation (MLE) is =(N-1)/N σ2. The two estimates are quite similar in large samples;
σ2 is unbiased, while is biased but minimizes the mean squared error of the estimator.
In practice σ2 is used more often, since it is more convenient for the hypothesis testing.
The square root of σ2 is called the standard error of the regression (SER), or called
standard deviation.
When you see a number reported as (0.36 ± 0.03) V, your might thought that all the
readings lie between 0.33V (=0.36-0.03) and 0.39V (=0.36+0.03). A quick look at the
data shows that 3 of the 5 readings are in this range. Statistically we expect 68% of the
values to lie in the range of <V> ± σV, but that 95% lie within of <V> ± 2σV. In the first
example all the data lie between 0.30 (= 0.36 - 2*0.3) and 0.42 (= 0.36 + 2*0.3) V. As a
rule of thumb for this course we usually expect the actual value of a measurement to
4: Metric Unit Prefix Prelab–3
lie within two deviations of the mean. In the language of “statistics,” it is said to have
95% “confidence levels.” Now try to analyze the following data
measurements
1
2
3
4
5
average
Time t (sec.)
7.6
7.4
8.2
7.8
7.0
|deviation| (∆t) (s)
(Δt)2 (s2)
The SER or standard deviation is σ=
Significant figures
In Table 1, you find that the voltage between two terminals is 0.36 V. Your measurement
has two digits of significant figures. You will NOT write your answer as 0.360 V or 0.3
V. The basic rules are
1. Zeros at the beginning of a number are not significant. For example, 0.0254 m has
3 significant figures.
2. Zeros after the decimal point are significant. For example 0.360 and 0.3600 has
respectively 3 and 4 significant figures.
3. Zeros following a whole number may or may not be significant. For example, 500
kg may have 1 or 2 or 3 significant figures. To be precise, one writes 5.×102, or
5.0×102, or 5.00×102 for 1 or 2, or 3 significant figures.
4. Arithmetic operations such as multiplication, division, addition and subtraction
are round off to the smaller number of significant figures. For example
a) 5.3 m/(1.67 m/s) (=3.1736s) = 3.2 s
2 significant figures
b) 3.4 m × 3.65 m (= 8.76 m2) = 8.8 m2
2 significant figures
3
c) 725 m/0.125s (=5800 m/s) = 5.80×10 m/s
3 significant figures
d) In rare situation, the number of significant figures may increase by 1 in
addition. For example: 3.45+7.76=11.21. The answer has 4 significant figures.
Answer the following questions:
1. 3.40×102 m/s
significant figures:
2. 1200 miles
significant figures:
2
3. (3.40×10 m/s) * 0.2 s
significant figures:
4. 1021 miles/2.50hr
significant figures:
5. 15.2 + 3.3
significant figures:
6. 15.2 – 3.3
significant figures:
7. 15.2 - 8.
significant figures:
4: Metric Unit Prefix Prelab–4