There are two
commonly used definitions of the standard
The population standard deviation:
σx=(1/N × Σdi2)½
and the sample standard deviation:
σx=(1/(N-1) × Σdi2)½
where: σx is the standard deviation, N is the number
of measurements, and di is the difference between measurement
i and the mean.
Both methods are frequently used. The sample standard deviation
corrects the tendency to understate the uncertainty in measurements,
especially when the sample size is small. This can be understood for the
extreme case where there is only one measurement available (N=1). In
this case the population standard deviation gives the absurd result
σx=0, whereas the sample standard deviation is
undefined (0/0); a proper representation of the ignorance of the
standard deviation after just one measurement.
For practical purposes, the difference between the two definitions is
insignificant. However, when you calculate the standard deviation, you
need to be aware of the two definitions and report the one you are
using. Also keep this in mind when you are using the standard
deviation function in your favorite spreadsheet; e.g.
Excel uses the STDEV function for the sample standard deviation
and STDEVP function for the population standard deviation.
Measurement Measured value Deviation di2
i xi di=xi-xmean
1 17 0.8 0.64
2 15 -1.2 1.44
3 16 -0.2 0.04
4 17 0.8 0.64
5 16 -0.2 0.04
xmean = 16.2 Σdi2= 2.80
In this example, the population standard deviation
=(1/5 × 2.80)½
=0.75, and the
sample standard deviation
=(1/4 × 2.80)½
Use the sample standard deviation for calculating confidence intervals.