PHY133  Classical Physics I Laboratory
Error Analysis
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
⟩
:
⟨
t
⟩
= (
Σ
N
i=1
t
i
)/
N
.
(1)
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"
σ
of the distribution. It measures the random error or the statistical uncertainty of the
individual measurement
t
i
:
σ
=
√
[
Σ
N
i=1
(
t
i

⟨
t
⟩
)
2
/ (
N
1) ].
(2)
About twothirds of all the measurements have a deviation less than one
σ
from the mean and
95% of all measurements are within two
σ
of the mean. In accord with our intuition that the
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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,
σ
m
, is given by:
σ
m
=
σ
/
√
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
⟩
±
σ
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
σ
from the mean, and the smallest
t
min
is 2
σ
from the mean. Hence:
σ
≈
¼ (
t
max

t
min
)
is an reasonable estimate of the uncertainty in a single measurement. The above method of
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 Fall '03
 Rijssenbeek
 Physics, Laboratory Error Analysis

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