Copyright July 1, 1999 (Rev Aug. 03) 
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)
Graphing Manual 
Vernier Caliper Manual 
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 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.
How can we estimate the uncertainty of a measured quantity? Several approaches can be used, depending on the application.
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

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. 
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.
(t<t>)^{2}  
0.04  
0.25  
0.09  
0.36  
<t> = 7.6  <t<t>>= 0.0  <t<t>>= 0.4  <(t<t>)^{2}>
= 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). 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 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, 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 (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. 
=AVEDEV(A2:A5)  Find the average deviation of the numbers in the range of cells. 
=STDEV(A2:A5)  Find the sample standard deviation of the numbers in the range of cells. 
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.
(x <x>)^{2}  
0.004445  

0.071112  
0.017777  
0.054443  
0.001111  
0.004445  
St. dev. ±0.17512 
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.
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 
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
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.
(a) What is the correct way to write the mass of the object including
its uncertainty? (b) What is the mistake in each incorrect one? Answer

Problem: I measure a length with a meter stick with aleast 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 
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.
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.
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> ± 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/cm^{3}. The textbook value is 2.71 g/cm^{3}. 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/cm^{3} 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 10^{8} m/s and 3.00 x 10^{8} m/s
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.
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. 
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.
Derivation: We will assume that the uncertainties are arranged
so as to make z as far from its true value as possible. Average deviations Δz = Δx + Δy in both cases With more than two numbers added or subtracted we continue to add the uncertainties. 
Using simpler average errors  Using standard deviations  
Eq. 1a  Eq. 1b 
Example: w = (4.52 ± 0.02) cm, x = ( 2.0 ± 0.2) cm, y = (3.0 ± 0.6) cm z = x + y  w = 2.0 + 3.0  4.5 = 0.5 cm
Notice that we round the uncertainty to one significant figure and round the answer to match. 
For multiplication by an exact number, multiply the uncertainty by the same exact number. 
Example: x = (3.0 ± 0.2) cm. Find C = 2
p x = 18.850 cm C = (18.8 ± 1.3) cm We round the uncertainty to two figures since it starts with a 1, and round the answer to match. 
Example: x = (2.0 ± 0.2) cm, y = (3.0 ± 0.6) cm. Find z = x  2y. z = x  2y = 2.0  2(3.0) = 4.0 cm
The 0 after the decimal point in 4.0 is significant and must be written in the answer. The uncertainty on the left starts with a 1 and is kept to two significant figures. (More on rounding in Section 7.) 
Derivation: We can derive the relation for multiplication easily. Take the largest values for x and y, that is z + Δz = (x + Δx)(y + Δy) = xy + x Δy + y Δx + Δx Δy Usually Δx << x and Δy << y so that the last term is much smaller than the other terms and can be neglected. Since z = xy, Δz = y Δx + x Δy which we write more compactly by forming the relative error, that is the ratio of Δz/z, namely 
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 
Example: w = (4.52 ± 0.02) cm, x =
(2.0 ±
0.2) cm. Find z = w x.

Example: x = ( 2.0 ± 0.2) cm, y = (3.0 ± 0.6) sec Find z = x/y.
Note that in this case we round off our answer to have no more decimal places than our uncertainty. 
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)cm^{2}, y = (3.0 ± 0.6) cm. Find.
Because the uncertainty begins with a 1, we keep two significant figures and round the answer to match. 
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

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)
Example: Consider S = x cos (θ) for x = (2.0 ± 0.2) cm, θ = 53 ± 2 °. S = (2.0 cm) cos 53° = 1.204 cm To get the largest possible value of S we would make x larger, (x + Δx) = 2.2 cm, and θ smaller, (θ  Δθ) = 51°. The largest value of S, namely (S + ΔS), is (S + ΔS) = (2.2 cm) cos 51° = 1.385 cm. The difference between these numbers is ΔS = 1.385  1.204 = 0.181 cm which we round to 0.18 cm. Then S = (1.20 ± 0.18) cm. 
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.  
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.) 
Example Round off z = 12.0349 cm and Δz = 0.153 cm. Since Δz begins with a 1, we round off Δz to two significant figures:

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 10^{6} s and Δz = 2 x 10^{4} s,
we should write our answer as
z = (1.43± 0.02) x 10^{6} 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 10^{6} ± 2 x 10^{4}) s. Don't write like this!
Problem: Express the following results in proper rounded form, x ± Δx. (i) m = 14.34506 grams, Δm = 0.04251 grams. 
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 10^{3} cm (which has one significant figure), or 8.00 x 10^{3} 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."
Problem: How many significant figures are there in each of the following? Answer

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^{3}sec.
(iii) M = 7.35 x 10^{22} kg ΔM = 2.6 x 10^{20} 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 10^{8} m/s and 3.00 x 10^{8} 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/s^{2}, x = (3.61 ± 0.18) m, y = (650 ± 20) m/s.
(iii) z = x^{3} 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) m^{2}.
(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 10^{6} (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?
(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 10^{22} 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) m^{3} iv) 18.8 ± 0.6) m^{3} 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.)
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. Δc 
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 or uncertainty. Δx 
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, Δx. At the 67% confidence level the range of possible true values is from <x>  Δx to <x> + Δx. See Confidence Level . 
Relative Error  The ratio of absolute error to the average, Δx/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.Symbol is σ 
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. Δx 