Copyright July 1, 2000 
Vern Lindberg

1. Systematic versus Random Errors
2. Determining Random Errors (a) Instrument Limit of
Error,
least count (b) Estimation
(c) Average
Deviation
(d) Conflicts (e) Standard
Error in the Mean
3. What does uncertainty tell me? Range of
possible values
4. Relative and Absolute error
5. Propagation of errors
(a) add/subtract
(b) multiply/divide (c) powers (d) mixtures of +*/
(e) other
functions
6. Rounding answers properly
7. Significant figures
8. Problems to try
9. Glossary of terms (all terms that are
bold
face and underlined)
Part II Graphing 
Part III The Vernier Caliper 
In this manual there will be problems for you to try. They are highlighted in yellow. 
There are also examples highlighted in green. 
Why should repeated measurements of a single quantity give different values? Mistakes on the part of the experimenter are possible, but we do not include these in our discussion. A careful researcher should not make mistakes! (Or at least she or he should recognize them and correct the mistakes.)
We use the synonymous terms uncertainty, error, or deviation to represent the variation in measured data. Two types of errors are possible. Systematic error is the result of a miscalibrated device, or a measuring technique which always makes the measured value larger (or smaller) than the "true" value. An example would be using a steel ruler at liquid nitrogen temperature to measure the length of a rod. The ruler will contract at low temperatures and therefore overestimate the true length. Careful design of an experiment will allow us to eliminate or to correct for systematic errors.
Even when systematic errors are eliminated there will remain a second type of variation in measured values of a single quantity. These remaining deviations will be classed as random errors, and can be dealt with in a statistical manner. This document does not teach statistics in any formal sense, but it should help you to develop a working methodology for treating errors.
The instrument limit of error, ILE for short, is the precision to which a measuring device can be read, and is always equal to or smaller than the least count. Very good measuring tools are calibrated against standards maintained by the National Institute of Standards and Technology.
The Instrument Limit of Error is generally taken to be the least count or some fraction (1/2, 1/5, 1/10) of the least count). You may wonder which to choose, the least count or half the least count, or something else. No hard and fast rules are possible, instead you must be guided by common sense. If the space between the scale divisions is large, you may be comfortable in estimating to 1/5 or 1/10 of the least count. If the scale divisions are closer together, you may only be able to estimate to the nearest 1/2 of the least count, and if the scale divisions are very close you may only be able to estimate to the least count.
For some devices the ILE is given as a tolerance or a percentage. Resistors may be specified as having a tolerance of 5%, meaning that the ILE is 5% of the resistor's value.
Problem: For each of the following
scales
(all in centimeters) determine the least count, the ILE, and read
the
length of the gray rod. Answer 
Another good example is determining the focal length of a lens by measuring the distance from the lens to the screen. The ILE may be 0.1 cm, however the depth of field may be such that the image remains in focus while we move the screen by 1.6 cm. In this case the estimated uncertainty would be half the range or ±0.8 cm.
Problem: I measure your height while you are standing by using a tape measure with ILE of 0.5 mm. Estimate the uncertainty. Include the effects of not knowing whether you are "standing straight" or slouching. Solution. 
Suppose we repeat a measurement several times and record the different values. We can then find the average value, here denoted by a symbol between angle brackets, <t>, and use it as our best estimate of the reading. How can we determine the uncertainty? Let us use the following data as an example. Column 1 shows a time in seconds.



^{} 



0.04 



0.25 



0.09 



0.36 
<t> = 7.6  <t<t>>= 0.0  <t<t>>= 0.4  = 0.247 Std. dev = 0.50 
Column 2 of Table 1 shows the deviation of each time from the average, (t<t>). A simple average of these is zero, and does not give any new information.
To get a nonzero estimate of deviation we take the average of the absolute values of the deviations, as shown in Column 3 of Table 1. We will call this the average deviation, Dt.
Column 4 has the squares of the deviations from Column 2, making the answers all positive. The sum of the squares is divided by 3, (one less than the number of readings), and the square root is taken to produce the sample standard deviation. An explanation of why we divide by (N1) rather than N is found in any statistics text. The sample standard deviation is slightly different than the average deviation, but either one gives a measure of the variation in the data.
If you use a spreadsheet such as Excel there are builtin functions that help you to find these quantities. These are the Excel functions.
=SUM(A2:A5)  Find the sum of values in the range of cells A2 to A5. 
=COUNT(A2:A5)  Count the number of numbers in the range of cells A2 to A5. 
=AVERAGE(A2:A5)  Find the average of the numbers in the range of cells A2 to A5. 
=AVEDEV(A2:A5)  Find the average deviation of the numbers in the range of cells A2 to A5. 
=STDEV(A2:A5)  Find the sample standard deviation of the numbers in the range of cells A2 to A5. 
For a second example, consider a measurement of length shown in Table 2. The average and average deviation are shown at the bottom of the table.





0.004445 


0.071112 


0.017777 


0.054443 


0.001111 


0.004445 


St. dev. ±0.17512 
We round the uncertainty to one or two significant figures (more on rounding in Section 7), and round the average to the same number of digits relative to the decimal point. Thus the average length with average deviation is either (15.47 ± 0.13) m or (15.5 ± 0.1) m. If we use standard deviation we report the average length as (15.47±0.18) m or (15.5±0.2) m.
Follow your instructor's instructions on whether to use average or standard deviation in your reports.
Problem Find the average, and average deviation for the following data on the length of a pen, L. We have 5 measurements,
(12.2, 12.5, 11.9,12.3, 12.2) cm. Solution
Problem: Find the average and the average deviation of the following measurements of a mass.
(4.32, 4.35, 4.31, 4.36, 4.37, 4.34) grams. Solution
The proper way to write the answer is
Problem: I make several measurements on the mass of an object. The balance has an ILE of 0.02 grams. The average mass is 12.14286 grams, the average deviation is 0.07313 grams. What is the correct way to write the mass of the object including its uncertainty? What is the mistake in each incorrect one? Answer
 12.14286 g
 (12.14 ± 0.02) g
 12.14286 g ± 0.07313
 12.143 ± 0.073 g
 (12.143 ± 0.073) g
 (12.14 ± 0.07)
 (12.1 ± 0.1) g
 12.14 g ± 0.07 g
 (12.14 ± 0.07) g
Problem: I measure a length with a meter stick with a least count of 1 mm. I measure the length 5 times with results (in mm) of 123, 123, 123, 123, 123. What is the average length and the uncertainty in length? Answer (e) Why make many measurements? Standard Error in the Mean.
We know that by making several measurements (4 or 5) we should be more likely to get a good average value for what we are measuring. Is there any point to measuring a quantity more often than this? When you take a statistics course you will learn that the standard error in the mean is affected by the number of measurements made.
The standard error in the mean in the simplest case is defined as the standard deviation divided by the square root of the number of measurements.
The following example illustrates this in its simplest form. I am measuring the length of an object. Notice that the average and standard deviation do not change much as the number of measurements change, but that the standard error does dramatically decrease as N increases.
Finding Standard Error in the Mean Number of Measurements, N Average Standard Deviation Standard Error 5 15.52 cm 1.33 cm 0.59 cm 25 15.46 cm 1.28 cm 0.26 cm 625 15.49 cm 1.31 cm 0.05 cm 10000 15.49 cm 1.31 cm 0.013 cm
For this introductory course we will not worry about the standard error, but only use the standard deviation, or estimates of the uncertainty.
3. What is the range of possible values?
When you see a number reported as (7.6 ± 0.4) sec your first thought might be that all the readings lie between 7.2 sec (=7.60.4) and 8.0 sec (=7.6+0.4). A quick look at the data in the Table 1 shows that this is not the case: only 2 of the 4 readings are in this range. Statistically we expect 68% of the values to lie in the range of <x> ± Dx, but that 95% lie within <x> ± 2 Dx. In the first example all the data lie between 6.8 (= 7.6  2*0.4) and 8.4 (= 7.6 + 2*0.4) sec. In the second example, 5 of the 6 values lie within two deviations of the average. As a rule of thumb for this course we usually expect the actual value of a measurement to lie within two deviations of the mean. If you take a statistics course you will talk about confidence levels.
How do we use the uncertainty? Suppose you measure the density of calcite as (2.65 ± 0.04) ^{}. The textbook value is 2.71 ^{}. Do the two values agree? Since the text value is within the range of two deviations from the average value you measure you claim that your value agrees with the text. If you had measured the density to be (2.65 ± 0.01)^{} you would be forced to admit your value disagrees with the text value.
The drawing below shows a Normal Distribution (also called a Gaussian). The vertical axis represents the fraction of measurements that have a given value z. The most likely value is the average, in this case <z> = 5.5 cm. The standard deviation is s = 1.2. The central shaded region is the area under the curve between (<x>  s) and (x + s), and roughly 67% of the time a measurement will be in this range. The wider shaded region represents (<x>  2s) and (x + 2s), and 95% of the measurements will be in this range. A statistics course will go into much more detail about this.
Problem: You measure a time to have a value of (9.22 ± 0.09) s. Your friend measures the time to be (9.385 ± 0.002) s. The accepted value of the time is 9.37 s. Does your time agree with the accepted? Does your friend's time agree with the accepted? Answer.
Problem: Are the following numbers equal within the expected range of values? Answer
(i) (3.42 ± 0.04) m/s and 3.48 m/s?
(ii) (13.106 ± 0.014) grams and 13.206 grams?
(iii) (2.95 ± 0.03) x ^{} m/s and 3.00 x ^{} m/s4. Relative and Absolute Errors
The quantity Dz is called the absolute error while Dz/z is called the relative error or fractional uncertainty. Percentage error is the fractional error multiplied by 100%. In practice, either the percentage error or the absolute error may be provided. Thus in machining an engine part the tolerance is usually given as an absolute error, while electronic components are usually given with a percentage tolerance.
Problem: You are given a resistor with a resistance of 1200 ohms and a tolerance of 5%. What is the absolute error in the resistance? Answer. 5. Propagation of Errors, Basic Rules
(a) Addition and Subtraction: z = x + y or z = x  y
(b) Multiplication and Division: z = x y or z = x/y
(c) Products of powers: ^{}.
(d) Mixtures of multiplication, division, addition, subtraction, and powers.
(e) Other Functions: e.g.. z = sin x. The simple approach.
(f) Other Functions: Getting formulas using partial derivatives
6. Rounding off answers in regular and scientific notation.
7. Significant Figures
8. Problems on Uncertainties and Error Propagation.
9. Glossary of Important Terms
Term  Brief Definition 
Absolute error  The actual error in a quantity, having the same units as the
quantity.
Thus if c = (2.95 ± 0.07) m/s, the absolute error is 0.07 m/s. See Relative Error. 
Accuracy  How close a measurement is to being correct. For gravitational acceleration near the earth, g = 9.7 m/s^{2} is more accurate than g = 9.532706 m/s^{2}. See Precision. 
Average  When several measurements of a quantity are made, the sum of the measurements divided by the number of measurements. 
Average Deviation  The average of the absolute value of the differences between each measurement and the average. See Standard Deviation. 
Confidence Level  The fraction of measurements that can be expected to lie within a given range. Thus if m = (15.34 ± 0.18) g, at 67% confidence level, 67% of the measurements lie within (15.34  0.18) g and (15.34 + 0.18) g. If we use 2 deviations (±0.36 here) we have a 95% confidence level. 
Deviation  A measure of range of measurements from the average. Also called error oruncertainty. 
Error  A measure of range of measurements from the average. Also called deviation or uncertainty. 
Estimated Uncertainty  An uncertainty estimated by the observer based on his or her knowledge of the experiment and the equipment. This is in contrast to ILE, standard deviation or average deviation. 
Gaussian Distribution  The familiar bellshaped distribution. Simple statistics assumes that random errors are distributed in this distribution. Also called Normal Distribution. 
Independent Variables  Changing the value of one variable has no effect on any of the other variables. Propagation of errors assumes that all variables are independent. 
Instrument Limit of Error (ILE) 
The smallest reading that an observer can make from an instrument. This is generally smaller than the Least Count. 
Least Count  The size of the smallest division on a scale. Typically the ILE equals the least count or 1/2 or 1/5 of the least count. 
Normal Distribution  The familiar bellshaped distribution. Simple statistics assumes that random errors are distributed in this distribution. Also called Gaussian Distribution. 
Precision  The number of significant figures in a measurement. For gravitational acceleration near the earth, g = 9.532706 m/s^{2} is more precise than g = 9.7 m/s^{2}. Greater precision does not mean greater accuracy! See Accuracy. 
Propagation of Errors  Given independent variables each with an uncertainty, the method of determining an uncertainty in a function of these variables. 
Random Error  Deviations from the "true value" can be equally likely to be higher or lower than the true value. See Systematic Error. 
Range of Possible True Values 
Measurements give an average value, <x> and an uncertainty, Dx. At the 67% confidence level the range of possible true values is from <x>  Dx to <x> + Dx. See Confidence Level . 
Relative Error  The ratio of absolute error to the average, Dx/x. This may also be called percentage error or fractional uncertainty. See Absolute Error. 
Significant Figures  All nonzero digits plus zeros that do not just hold a place before or after a decimal point. 
Standard Deviation  The statistical measure of uncertainty. See Average Deviation. 
Standard Error in the Mean 
An advanced statistical measure of the effect of large numbers of measurements on the range of values expected for the average (or mean). 
Systematic Error  A situation where all measurements fall above or below the "true value". Recognizing and correcting systematic errors is very difficult. 
Uncertainty  A measure of range of measurements from the average. Also called deviation or error. 
Send any comments or corrections to Vern Lindberg.