2.2 ESTIMATING RANDOM ERRORS
2.2 Estimating Random Errors
Look at Figure 2. How far apart are lines A and B? You're
measuring this by laying a transparent rule, with a metric scale,
across the two lines and observing where each strikes the
scale. Line B seems to fall exactly on the 11.9 cm mark. Line
A falls somewhere between 4.2 and 4.3 cm, a little closer to
the former. Should you read the position of line A as 4.2 cm?
To do so isn't exactly wrong, but it wastes some of the information that's available to you. Look closely and you can see that
A intercepts the scale at a point about 1/3 or 1/4 mm past the
4.2 cm mark; you should probably read it as 4.23 cm. You
always read a scale to some fraction — 1/2 or 1/5 or 1/10 — of
its smallest marked subdivision.
How closely does the measurement process determine this
number? Looking at the scale, I'd say that one might easily
read it as high as 4.25 or as low as 4.22 cm, and quote the
reading as ±0.02 cm. If the line had been thicker, or less
straight, or if the ruler were thick, there'd be more room for
variation in the outcome, and your error estimate would be
bigger. The point is that some estimate of the range of uncertainty is an essential part of the measurement. In this case,
you can get your estimate just by looking at what you're doing.
Likewise, call the scale position of line B 11.90 ± 0.02 cm;
and the distance from A to B is
(11.90 ± 0.02) - (4.23 ± 0.02) = 7.67 ± 0.03 cm
Figure 2
How did I estimate the uncertainty in the difference? We'll take
up that question in detail a little later on, but in the example
here, notice that the result is not what I'd get if I just added the errors in the two measured quantities.
This procedure (figure the error in the result caused by each measured variable, acting alone, then add
them) is sometimes an acceptable one, although it's actually not correct. (It assumes the different
contributions are "working together", whereas in fact they are independent; thus adding overestimates
the error in the overall result. Better practice is to use the methods of "propagation of errors", of which
more later.)
In the example of Figure 2, the two lines were straight and parallel, and the only source of random
error was the reading of the ruler scale. But now suppose you want to measure the height of the little
man in Figure 3. Again, you'd do it by aligning the ruler with his height, and reading off where each end
of him hits the scale. In the original drawing, the readings are 1.89 and 13.57 cm, so his height is
2-3
2-4
2.2 ESTIMATING RANDOM ERRORS
as the difference, or 11.68 cm.
But the process isn't quite so crisp and clear this time,
is it? You have some significant leeway in where you say
his "ends" are, and just what direction is "along his height".
It's no longer obvious how to estimate the uncertainty in
the measurement, although it's clearly bigger than just that
due to reading the ruler scale. One way to gauge the
uncertainty is to make several independent trials of the
experiment, and see how much the results vary. The
crucial word here is "independent": you have to be sure
that one result doesn't in any way bias the way you look at
the next one.
Figure 3
I had several students measure Charlie's height with the same ruler. The results were
11.66 11.62 11.69 11.69 11.72
11.61 11.77 11.64 11.66 11.73
11.66 11.61 11.57 11.78 11.72
11.60 cm.
When we repeat the same measurement several times, the best overall estimate of the "true" result
we can make is the mean (that is, the simple average) of all the values. If we measured x, N times, and
got x1, x2, x3, . . , xN,
the mean of the N values is
x=
1 
x1 + x 2 + . . . . + x N
=
xi
N
N i
(1)
For the 16 measurements of Charlie's height, above,
x=
11.66 + 11.62 + 11.69 + . . .+ 11.60cm
= 11.677cm
16
But the point is that I've learned more than this. From how much these numbers fluctuate, I get a
sense of how big the random errors in this measurement are. Individual trials vary from the mean by
anything up to around 0.1 cm, either way. (As I expected, this measurement is substantially less precise
than that of the first example.) There are various ways one can express quantitatively the amount of
variation observed. Far the most important is the standard deviation. This is defined as
(2)
. . . where _
x is the mean of the N measured values. For the data on Charlie's height, above,
sx =
(11.66 - 11.677 )2 + (11.62 - 11.677 )2 + . . . .
= 0.060 cm
(16 - 1)
is the standard deviation1 of x.
Strictly speaking this is a "sample standard deviation" — an estimate, based on a finite sample
of data, of the actual standard deviation of a probability distribution.
1
2.2 ESTIMATING RANDOM ERRORS
2-5
What does the standard deviation tell you about your data? Looking at (2), we see that it is
(approximately) the square root of the average value of the squared deviation-from-the-mean-value of
the x's. (It would be exactly this, the "root mean square" deviation, if only there were an N rather than N
- 1 in the denominator. There are very good boring reasons why it's N - 1 rather than N, which we can
talk about some other year.) Thus s is a typical or representative value of how far a single such trial of x
is likely to deviate from the mean. The mean is the best estimate available of the "true" value of x.
If you tried the experiment N times and got the same value of x every time, Equation (2) would give
you zero. This does not mean sx = 0; there's no such thing as an experiment without random error!
What it means is that your measurements weren't made precisely enough to show you the fluctuations;
you needed to read the meter to one more decimal place, or whatever.
Notice again that sx is an indication of how much a single trial of your experiment is likely to deviate
from the "true" (as estimated by the mean) value. However, what we're probably most interested in is
how well the average of all N measurements has determined the true value. This is not at all the same
thing: sixteen measurements taken together must nail down the result better than any single
measurement can. What we need to know is, in effect, how to calculate the standard error (also called
the standard deviation of the mean) of N repeated trials of an experiment. This turns out to be
sx =
sx
N
(3)
(Let me postpone proving this until we've considered "propagation of errors" in Section 3.4, because
that makes it easy.) If we go back to the set of measurements of the height of the little man in Figure 2,
the standard error is
sx =
0.060 cm
= 0.015 cm
16
and the correct way to quote the result of these measurements is
x = 11.677 ± 0.015 cm
or perhaps
11.68 ± 0.02 cm
—  NOT 11.677 ± 0.060 cm. When I give the mean value as the best overall result, the
representative uncertainty I associate with it must be the standard deviation of the mean value2.
Finally: sx too is only an estimate, drawn from a finite sample of data, of the "true" standard deviation
σ; and this estimate too is subject to uncertainty. From propagation of errors I can work out the
standard deviation of sx — that is, an estimate of how well N measurements of x determine the standard
deviation of x. What I'll get is
s( s x ) _
sx
N
The only reason for bringing this up is to point out that, unless the number of trials N is quite large,
repeated measurements determine the standard deviation only rather roughly. Thus it's almost never
worth quoting a standard deviation or a standard error to more than one significant digit, and I probably
should have given Charlie's height, above, as 11.68 ± 0.02 cm.
2
Does this help at all to make the distinction clear? If you were asked to predict
the result of the next single trial, you'd say 11.68 ± .06 cm.
(4)