Uncertainties and Error Propagation

Part I of a manual on

Uncertainties, Graphing, and the Vernier Caliper

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.


1. Systematic and random errors.

No measurement made is ever exact. The accuracy (correctness) and precision (number of significant figures) of a measurement are always limited by the degree of refinement of 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.

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 mis-calibrated 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.

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2. Determining random errors.

How can we estimate the uncertainty of a measured quantity? Several approaches can be used, depending on the application.

(a) Instrument Limit of Error (ILE) and Least Count

(b) Estimated Uncertainty

(c) Average Deviation: Estimated Uncertainty by Repeated Measurements

Table 1. Values showing the determination of average, average deviation, and standard deviation in a measurement of time. Notice that to get a non-zero average deviation we must take the absolute value of the deviation. 
Time, t, sec. 
(t - <t>), sec
|t - <t>|, sec
<t> = 7.6 <t-<t>>= 0.0 <|t-<t>|>= 0.4 = 0.247
Std. dev = 0.50
Table 2. Example of finding an average length and an average deviation in length. The values in the table have an excess of significant figures. Results should be rounded as explained in the text.Results can be reported as (15.5 ± 0.1) m or (15.47 ± 0.13) m. If you use standard deviation the length is (15.5 ± 0.2) m or (15.47 ± 0.18) m.
Length, x, m 
|x- <x>|, m
Average 15.46667 m
±0.133333 m
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


ProblemFind 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


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(d) Conflicts in the above

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
    1. 12.14286 g
    2. (12.14 ± 0.02) g
    3. 12.14286 g ± 0.07313
    4. 12.143 ± 0.073 g
    5. (12.143 ± 0.073) g
    6. (12.14 ± 0.07)
    7. (12.1 ± 0.1) g
    8. 12.14 g ± 0.07 g
    9. (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

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(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.

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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 (=7.6-0.4) 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.

A Gaussian distribution with mean 
plus or minus one and two standard deviations

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/s

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4. 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.

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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/s2 is more accurate than g = 9.532706 m/s2. 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 bell-shaped 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 bell-shaped 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/s2 is more precise than g = 9.7 m/s2. 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 non-zero 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.
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Send any comments or corrections to Vern Lindberg.