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 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.

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

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.

(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.

(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.

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). 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, Δt.

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.

A second example is shown in Table 2 for a measurement of length. The average, average deviation, and standard deviation are shown at the bottom of the table.

We round the uncertainty off 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.

(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.

(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.

Standard Error = Standard Deviation/√N = σ/√N

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.

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> ± Δx, but that 95% lie within <x> ± 2 Δx. 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. In 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) g/cm³. The textbook value is 2.71 g/cm³. 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) g/cm³ 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 σ1.2. The central shaded region is the area under the curve between (<x> - σ) and (x + σ), and roughly 67% of the time a measurement will be in this range. The wider shaded region represents (<x> - 2σ) and (x + 2σ), 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 108 m/s and 3.00 x 108 m/s

4. Relative and Absolute Errors

The quantity Δz is called the absolute error while Δz/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.
 

5. Propagation of Errors, Basic Rules

Suppose two measured quantities x and y have uncertainties, Δx and Δy, as determined above: we would report x ± Δx, and y ± Δy. From the measured quantities a new quantity, z, is calculated from x and y. What is the uncertainty, Δz, in z? For the purposes of this course we will use a simplified version of the proper statistical treatment. The formulas for a full statistical treatment (using standard deviations) will also be given. The guiding principle in all cases is to consider the most pessimistic situation.

The examples included in this section also involve the proper rounding of answers, which is covered in more detail in Section 6. The examples use the propagation of errors using average deviations on the left and standard deviations on the right.

(a) Addition and Subtraction: z = x + y     or    z = x - y

Using simpler average errors		Using standard deviations
	Eq. 1a		Eq. 1b

(b) Multiplication and Division: z = x y    or    z = x/y

The same rule holds for multiplication, division, or combinations, namely add all the relative errors to get the relative error in the result.

Using simpler average errors		Using standard deviations
	Eq. 2a		Eq.2b

(c) Products of powers: z = x^m yⁿ .

The results in this case are

Using simpler average errors		Using standard deviations
	Eq. 3a		Eq.3b

Example:  w = (4.52 ± 0.02) cm, A = (2.0 ± 0.2)cm2, y = (3.0 ± 0.6) cm. Find. The second relative error, (Δy/y), is multiplied by 2 because the power of y is 2.  The third relative error, (ΔA/A), is multiplied by 0.5 since a square root is a power of one half. So Δz = 0.49 (28.638 cm2) = 14.03 cm2 which we round to 14 cm2 z = (29 ± 14) cm2 Using Eq. 3b, z=(29 ± 12) cm2 Because the uncertainty begins with a 1, we keep two significant figures and round the answer to match.

(d)  Mixtures of multiplication, division, addition, subtraction, and powers.

If z is a function which involves several terms added or subtracted we must apply the above rules carefully.  This is best explained by means of an example.

Example:  w = (4.52 ± 0.02) cm, x = (2.0 ± 0.2) cm, y = (3.0 ± 0.6) cm z = wx +y2 = 18.0 cm2 First we compute v = wx as in the example in (b) to get v = (9.0 ± 0.9) cm2. Next we compute Finally, we compute Δz = Δv + Δ(y2) = 0.9 + 3.6 = 4.5 cm2 rounding to 4 cm2 Hence z = (18 ± 4) cm2. We have v = wx = (9.0 ± 0.9) cm.  The calculation of the uncertainty in y2 is the same as that shown to the left. Then from Eq. 1b  Δz =  3.7 cm2 z = (18 ± 4) cm2.

(e) Other Functions: e.g.. z = sin x. The simple approach.

For other functions of our variables such as sin(x) we will not give formulae. However you can estimate the error in z = sin(x) as being the difference between the largest possible value and the average value. and use similar techniques for other functions.
Thus Δ(sin x) = sin(x + Δx)- sin(x)

(f) Other Functions: Getting formulas using partial derivatives

The general method of getting formulas for propagating errors involves the total differential of a function. Suppose that z = f(w, x, y, ...) where the variables w, x, y, etc. must be independent variables!

The total differential is then

We treat the dw = Δw as the error in w, and likewise for the other differentials, dz, dx, dy, etc. The numerical values of the partial derivatives are evaluated by using the average values of w, x, y, etc. The general results are

Using simpler average errors
	Eq. 4a.
Using standard deviations
	Eq. 4b

Example: Consider S = x cos (θ) for x = (2.0 ± 0.2) cm, θ = (53 ± 2) °= (0.9250 ± 0.0035) rad. S = 2.0 cm cos 53° = 1.204 cm Hence S = (1.20 ± 0.13) cm (using average deviation approach) or S = (1.20 ± 0.12) cm (using standard deviation approach.)

6. Rounding off answers in regular and scientific notation.

When the answer is given in scientific notation, the uncertainty should be given in scientific notation with the same power of ten. Thus, if

z = 1.43 x 10⁶ s and Δz = 2 x 10⁴ s,

we should write our answer as

z = (1.43± 0.02) x 10⁶ s.

This notation makes the range of values most easily understood. The following is technically correct, but is hard to understand at a glance.

z = (1.43 x 10⁶ ± 2 x 10⁴) s. Don't write like this!

7. Significant Figures

A significant figure is any digit 1 to 9 and any zero which is not a place holder. Thus, in 1.350 there are 4 significant figures since the zero is not needed to make sense of the number. In a number like 0.00320 there are 3 significant figures --the first three zeros are just place holders. However the number 1350 is ambiguous. You cannot tell if there are 3 significant figures --the 0 is only used to hold the units place --or if there are 4 significant figures and the zero in the units place was actually measured to be zero.

How do we resolve ambiguities that arise with zeros when we need to use zero as a place holder as well as a significant figure? Suppose we measure a length to three significant figures as 8000 cm. Written this way we cannot tell if there are 1, 2, 3, or 4 significant figures. To make the number of significant figures apparent we use scientific notation, 8 x 10³ cm (which has one significant figure), or 8.00 x 10³ cm (which has three significant figures), or whatever is correct under the circumstances.

We start then with numbers each with their own number of significant figures and compute a new quantity. How many significant figures should be in the final answer? In doing running computations we maintain numbers to many figures, but we must report the answer only to the proper number of significant figures.

In the case of addition and subtraction we can best explain with an example. Suppose one object is measured to have a mass of 9.9 gm and a second object is measured on a different balance to have a mass of 0.3163 gm. What is the total mass? We write the numbers with question marks at places where we lack information. Thus 9.9???? gm and 0.3163? gm. Adding them with the decimal points lined up we see

9.9 ????
0.3163?
10.2 ???? = 10.2 gm.

In the case of multiplication or division we can use the same idea of unknown digits. Thus the product of 3.413? and 2.3? can be written in long hand as

3.413?
2.3?
   ?????
 10239?
 6826?
7.8????? = 7.8

The short rule for multiplication and division is that the answer will contain a number of significant figures equal to the number of significant figures in the entering number having the least number of significant figures. In the above example 2.3 had 2 significant figures while 3.413 had 4, so the answer is given to 2 significant figures.

It is important to keep these concepts in mind as you use calculators with 8 or 10 digit displays if you are to avoid mistakes in your answers and to avoid the wrath of physics instructors everywhere. A good procedure to use is to use use all digits (significant or not) throughout calculations, and only round off the answers to appropriate "sig fig."

8. Problems on Uncertainties and Error Propagation.

Try the following problems to see if you understand the details of this part . The answers are at the end.

(a) 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.

(b) Express the following results in proper rounded form, x ± Δx.

(i) m = 14.34506 grams, Δm = 0.04251 grams.

(ii) t = 0.02346 sec, Δt = 1.623 x 10^-3sec.

(iii) M = 7.35 x 10²² kg ΔM = 2.6 x 10²⁰ kg.

(iv) m = 9.11 x 10^-33 kg Δm = 2.2345 x 10^-33 kg

(c) Are the following numbers equal within the expected range of values?

(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 10⁸ m/s and 3.00 x 10⁸ m/s

(d) Calculate z and Δz for each of the following cases.

(i) z = (x - 2.5 y + w) for x = (4.72 ± 0.12) m, y = (4.4 ± 0.2) m, w = (15.63 ± 0.16) m.

(ii) z = (w x/y) for w = (14.42 ± 0.03) m/s², x = (3.61 ± 0.18) m, y = (650 ± 20) m/s.

(iii) z = x³ for x = (3.55 ± 0.15) m.

(iv) z = v (xy + w) with v = (0.644 ± 0.004) m, x = (3.42 ± 0.06) m, y = (5.00 ± 0.12) m, w =    (12.13 ± 0.08) m².

(v) z = A sin y for A = (1.602 ± 0.007) m/s, y = (0.774 ± 0.003) rad.

(e) How many significant figures are there in each of the following?

(i) 0.00042   (ii) 0.14700   (ii) 4.2 x 10⁶   (iv) -154.090 x 10^-27

(f) I measure a length with a meter stick which has a least count of 1 mm I measure the length 5 times with  results in mm of 123, 123, 124, 123, 123 mm. What is the average length and the uncertainty in length?

Answers for Section 8:

(a) (4.342 ± 0.018) grams

(b) i) (14.34 ± 0.04) grams      ii) (0.0235 ± 0.0016) sec or (2.35 ± 0.16) x 10^-2 sec
   iii) (7.35 ± 0.03) x 10²² kg   iv) (9.11 ± 0.02) x 10^-31 kg

(c) Yes for (i) and (iii), no for (ii)

(d) i) (9.4 ± 0.8) m   ii) (0.080 ± 0.007) m/s   iii) (45 ± 6) m³   iv) 18.8 ± 0.6) m³   v) (1.120 ± 0.008 m/s

(e) i) 2   ii) 5   iii) 2   iv) 6

(f) (123 ± 1) mm (I used the ILE = least count since it is larger than the average deviation.)

9. Glossary of Important Terms

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