Statistical or Random Errors
Every measurement an experimenter makes is uncertain to some degree. The uncertainties are of two kinds: (1)
random errors, or (2) systematic errors. For example, in measuring the time required for a weight to fall to the
floor, a random error will occur when an experimenter attempts to push a button that starts a timer
simultaneously with the release of the weight. If this random error dominates the fall time measurement, then if
we repeat the measurement many times (
N
times) and plot equal intervals (bins) of the fall time
t
i
on the
horizontal axis against the number of times a given fall time
t
i
occurs on the vertical axis, our results (see
histogram below) should approach an ideal bellshaped curve (called a Gaussian distribution) as the number of
measurements
N
becomes very large.
The best estimate of the
true
fall time
t
is the
mean
value (or average value) of the distribution
⟨
t
⟩
:
If the experimenter squares each deviation from the mean, averages the squares, and takes the square root of
that average, the result is a quantity called the "rootmeansquare" or the "standard deviation"
s
of the
distribution. It measures the random error or the statistical uncertainty of the individual measurement
t
i
:
About twothirds of all the measurements have a deviation less than one
s
from the mean and 95% of all
measurements are within two
s
of the mean. In accord with our intuition that the
uncertainty of the mean
should
be smaller than the uncertainty of any single measurement, measurement theory shows that in the case of
random errors the standard deviation of the mean,
s
m
, is given by:
PHY133  Classical Physics I
Laboratory
Error
Analysis
⟨
t
⟩
= (
Σ
N
i=1
t
i
)/
N
.
(1)
s
=
√
[
Σ
N
i=1
(
t
i

⟨
t
⟩
)
2
/ (
N
1) ].
(2)
Page 1 of 5
http://sbhep1.physics.sunysb.edu/~rijssenbeek/PHY133_ErrorAnalysis.html
9/19/2003
PHY117/8  Error Analysis
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m
=
s
/
√
N
,
where
N
again is the number of measurements used to determine the mean. Then the result of the
N
measurements of the fall time would be quoted as
t
=
⟨
t
⟩
±
s
m
.
Whenever you make a measurement that is repeated
N
times, you are supposed to calculate the mean value and
its standard deviation as just described. For a large number of measurements this procedure is somewhat
tedious. If you have a calculator with statistical functions it may do the job for you. There is also a simplified
prescription for estimating the random error which you can use. Assume you have measured the fall time about
ten times. In this case it is reasonable to assume that the largest measurement
t
max
is approximately +2
s
from
the mean, and the smallest
t
min
is 2
s
from the mean. Hence:
s
≈
¼ (
t
max

t
min
)
is an reasonable estimate of the uncertainty in a single measurement. The above method of determining
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 Fall '03
 Rijssenbeek
 Physics, random error, Fall time

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