Elementary Uncertainty Analysis
Introduction
Say you wish to measure the length of a table. You pull out your trusty meter stick and
measure it to be 1.669m long. Most people walk away and think they’ve got it.
However, is it exactly 1.669m long? In other words, is it 1.66900000000000…m long?
Probably not. So obviously there is a range of values for its length that we are
comfortable saying that the actual or true value lies between.
Estimating Uncertainties
So how do we determine this range of values? The best way to do this is to measure it
many times (the more, the better). It is extremely likely that you will not get the same
measurement each time; there will be a range of values for the table’s length.
So why do you get a range of values every time you measure it? Believe it or not, even if
you measure it perfectly each time, you will still not get the same value each time! There
are actual (complicated) physical processes that prevent this measurement from coming
out exactly the same each time, no matter what that measurement is. Human error in
using a measuring device can expand this range of values, but even if you do your best to
eliminate this possibility (and all good experimentalists do) you will still get a range of
values.
So make many measurements of something. The average of these measurements would
be your value, and the number you would use for the uncertainty is the standard deviation
in these measurements. The formula for standard deviation is
𝑁
1
Standard deviation in x = √
∑(𝑥̅ − 𝑥𝑖 )2
𝑁−1
𝑖=1
Where 𝑁 is the number of measurements, 𝑥̅ is the average of all your measurements, and
𝑥𝑖 is the result of the ith measurement. (Some of you have calculators that can give you
the standard deviation automatically.) So, for example, let’s say you measure your
table’s length 5 times and get 1.669m, 1.668m, 1.670m, 1.667m, and 1.667m. The
average of these values is 1.6688m and the standard deviation is
1
√
[(1.6688 − 1.669)2 + (1.6688 − 1.668)2 + (1.6688 − 1.670)2 + (1.6688 − 1.667)2 + (1.6688 −
5−1
= 0.0013 m
Thus, you would write the length of the table as
1.6688 ± 0.0013 m.
Richard A. Thomas – UST Physics
Quite often, though, we don’t have time to measure things many times. If you measure
something only once, you have to estimate the uncertainty in your measurement. In other
words, you make a guess based on experience: how big should your uncertainty limits be
so that if you had time to measure it many times most of your measurements would fall
within these uncertainty limits of your single measurement? Most of you, however, do
not have a lot of experience estimating uncertainties. So instead, go by this rule of
thumb: use half the smallest demarcation on your measuring instrument as the
uncertainty. So, for example, if your meter stick’s finest markings are 1mm (0.001m)
apart, then you can go with an uncertainty of ± 0.5mm (±0.0005m). You would then
write the length of the table as:
1.669 ± 0.0005m
Uncertainty Propagation
But now, what if we want to know the top area of the table? If we measure the width to
be 1.424 ± 0.0005m, the area is 1.669m × 1.424m = 2.3767m2. But what is the
uncertainty in this area measurement?
One way to do this is to find out, based on your length and width uncertainties, what the
maximum and minimum areas would be. So, in our above example:
max area = 1.6695m × 1.4245m = 2.3782m2
min area = 1.6685m × 1.4235m = 2.3751m2
This defines a range of values; the area could be as big as 2.3782m2 or as small as
2.3751m2. The value in the middle of this range is 2.37665m2, and the two extrema are
about 0.00155m2 on either side of this value. Rounding off, one would write the area as
2.3767 ± 0.0016 m2.
Mixing uncertainties like this is called “uncertainty propagation.”
One does NOT need to calculate BOTH the maximum and minimum values of the table
area. A shorthand way of calculating the uncertainty in the area of the table is to just
calculate the maximum value and subtract off the measured table area:
uncertainty in area = ΔA = max area – area
= 2.3782m2 – 2.3767m2
= 0.0015m2
Note that this doesn’t give you exactly the same uncertainty, but that’s ok. This quickand-dirty method actually overestimates the uncertainty anyway, so you are fine using it.
NOTES:
Richard A. Thomas – UST Physics
1) The method of determining the uncertainty in the area of the table based on the
uncertainties of the length and width described above actually overestimates the
real uncertainty. The correct way of doing uncertainty propagation is more work
and takes longer. For Physics 111/112, though, the quick-and-dirty way shown
above will be just fine.
2) Sometimes it may be difficult to judge exactly where on the ruler the edge of the
table corresponds to. If this is the case, you should go with a larger uncertainty.
If, however, the finest markings are far apart and you think you can interpolate
between them, you may use a smaller uncertainty that half the smallest
demarcation. Just use your best judgment.
3) A good experimentalist ALWAYS makes their measurements as precise as
possible, with the smallest uncertainty possible.
4) The uncertainty should never have more than 2 significant figures. Also, the last
decimal place in your value should not go beyond the last decimal place in your
uncertainty. So, for example, 2.377 ± 0.002 m2 is fine, but 2.3767 ± 0.002 m2 is
bad form.
5) You are often asked to measure something and compare it to something else; are
they the same? The answer is “yes” if the ranges defined by their uncertainties
overlap. The answer is “no” if they do not. So, for example, 3.64 ± 0.04 m2 and
3.66 ± 0.05 m2 do match (are the “same”); 3.64 ± 0.04 m2 and 3.75 ± 0.05 m2 do
not match (are not the “same”).
6) If your values did not match the value you expected them to match, then go back
and see if you did anything wrong in your measurements or calculations. A good
experimentalist will check and double-check their results. When they are certain
they have not made any mistakes, they will stand by their measurements even if
the rest of the world disagrees with them. So don’t always accept “given” values
as gospel. Maybe some approximation was used in obtaining a “given” value,
and your measurements were precise enough to show that the “real” value is
actually different from the “given” one derived from an approximation. Often it
makes things even more interesting when the experimental measurement doesn’t
match a theoretical prediction. You should try to figure out why it is different if
possible. However, there is always the possibility that one of your measuring
tools is inaccurate (e.g., your meter stick is actually not 1 m long), so many
experimentalists will spend quite a bit of time making sure that this is not the
case.
Richard A. Thomas – UST Physics