50
Reliability of Measurements
Simple expressions for x and average deviation can be made by using the
symbol 2, which means "the sum of" whatever follows it:
and
SU - x
Average deviation = —
Lj -
L
(5-1)
In each case, the sum of n values is understood. One important point to remember: x may be calculated for any number of measurements, but only for an
infinite number of measurements will x = p, the "true value."
PROBLEM:
Five persons measure the length of a room, getting values of 10.325 m, 10.320 m,
10.315 m, 10.313 m, and 10.327 m. Find the average value and the average deviation.
SOLUTION:
Add the separate values and divide by 5 to get the arithmetical mean. Set opposite
each value its deviation from the average, without regard to sign. Take the average
of these deviations.
Measurement
Deviation
x t
x t — x
10.325 m
0.005 m
10.320
0.000
10.315
0.005
10.313
0.007
10.327
0.007
SJT, = 51.600 m
0.024 m = 2|jr, - x\
^ = 10.320 m
^'~ *' = 0.0048 m = Average deviation
The average is 10.320 m, with an average deviation of 0.0048 m. It is proper
to write the average as 10.320 m, because the deviation affects digits in only the
third decimal place. It is not correct to give the length as 10.32 m; this implies
that the measurement is uncertain in the second decimal place. Average deviation is one of the simplest measures of reliability of measurements (the spread
of experimental values), but a better estimate of reliability can be made with
standard deviation.