Physics 4BL Laboratory 1
Uncertainties and statistics
I Background Information
I.1 Uncertainty in Measurements
Any experimentally measured quantity x cannot be known exactly due to error in our measurement.
Therefore, when a measured quantity is reported, for example in a scientiﬁc journal or an academic
lab report, one presents the measured quantity as
x
=
x
best
±
δx,
where
x
best
is our best guess at the quantity x, and
δx
quantiﬁes the uncertainty in our measure-
ment. Uncertainties in measurement are often classiﬁed into two distinct categories, systematic
and random. The measurements performed in this laboratory introduce us to analysis methods
for treating the latter type, which is statistical in nature. As a result, our conﬁdence level on the
outcome is improved by making more measurements, and the exercises here demonstrate how to
treat this aspect quantitatively. It is worth emphasizing that appreciating the impact of random
uncertainties is fundamental to the process of measuring any physical quantity. Whether it seems
relevant for a given situation is quantitative: is the conﬁdence level in the measured outcome small
or large when compared to what is needed?
Lets assume N measurements of x are made, yielding values
x
1
,x
2
,...,x
N
. A large number of such
measurements, called an ensemble, has certain useful properties such as the mean value and the
spread of the values around it, called the standard deviation. In nearly all situations our best
estimate of x is the mean value of these measurements:
x
=
1
N
N
X
i
=1
x
i
(1)
As more measurements are made, we expect that our measured mean value
x
will approach the
“true” value X, which is deﬁned as the mean value if an inﬁnite number of measurements were
made. That is,
X
= lim
N
→∞
x
= lim
N
→∞
1
N
N
X
i
=1
x
i
.
What we seek is a quantitative measure of how far the measured mean
x
may deviate from the
“true” value X. Standard practice is to use a quantity called the standard deviation of the mean,
denoted by
σ
x
as such a measure. Then, when one publishes (as in your lab reports for this class)
a measured quantity, it is written as
x
=
x
±
σ
x
signifying that we are conﬁdent X, the quantity we ultimately want to know, lies within some range
of values around our measured
x
, due to random error. In the ﬁrst part of the present laboratory,
you will examine how to calculate our measure of random error, called the standard deviation of
the mean, and how it is aﬀected by changing the ensemble size, N.
1