Uncertainties and Error Propagation (2024)

Contents

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)

1. Systematic and random errors.

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.

2. Determining random errors.

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

The least count is the smallest division that is marked on the instrument. Thus a meter stick will have a least count of 1.0 mm, a digital stop watch might have a least count of 0.01 sec.

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

(b) Estimated Uncertainty

Often other uncertainties are larger than the ILE. We may try to balance a simple beam balance with masses that have an ILE of 0.01 grams, but find that we can vary the mass on one pan by as much as 3 grams without seeing a change in the indicator. We would use half of this as the estimated uncertainty, thus getting uncertainty of ±1.5 grams.

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.

See Also
How to Calculate Relative Uncertainty - isobudgetsThe Significance of Measurement Uncertainties5 Ways to Make Measurements and Uncertainty Approachable for Students1.6: Uncertainties in Scientific Measurements

(c) Average Deviation: Estimated Uncertainty by Repeated Measurements

The statistical method for finding a value with its uncertainty is to repeat the measurement several times, find the average, and find either the average deviation or the standard deviation.

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.

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
7.4	-0.2	0.2	0.04
8.1	0.5	0.5	0.25
7.9	0.3	0.3	0.09
7.0	-0.6	0.6	0.36
<t> = 7.6	<t-<t>>= 0.0	<|t-<t>|>= 0.4	= 0.247 Std. dev = 0.50

A simple average of the times is the sum of all values (7.4+8.1+7.9+7.0) divided by the number of readings (4), which is 7.6 sec. We will use angular brackets around a symbol to indicate average; an alternate notation uses a bar is placed over the symbol.

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 non-zero 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 (N-1) 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 built-in 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.

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
15.4	0.06667	0.004445
15.2	0.26667	0.071112
15.6	0.13333	0.017777
15.7	0.23333	0.054443
15.5	0.03333	0.001111
15.4	0.06667	0.004445
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

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

See Also

Uncertainties and Error Propagation

(d) Conflicts in the above

In some cases we will get an ILE, an estimated uncertainty, and an average deviation and we will find different values for each of these. We will be pessimistic and take the largest of the three values as our uncertainty. [When you take a statistics course you should learn a more correct approach involving adding the variances.] For example we might measure a mass required to produce standing waves in a string with an ILE of 0.01 grams and an estimated uncertainty of 2 grams. We use 2 grams as our uncertainty.

The proper way to write the answer is

1. Choose the largest of (i) ILE, (ii) estimated uncertainty, and (iii) average or standard deviation.
2. Round off the uncertainty to 1 or 2 significant figures.
3. Round off the answer so it has the same number of digits before or after the decimal point as the answer.
4. Put the answer and its uncertainty in parentheses, then put the power of 10 and unit outside the parentheses.

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

(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 (=7.6-0.4) 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.

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

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.

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

Send any comments or corrections to Vern Lindberg.