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Chapter 3: Measures of Variability
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Notes:
When computing the value of the variance or the standard deviation,
the data values can be population values or sample values.
Hence we can compute the sample variance or the standard deviation.
Both population and sample data values are assumed to be finite.
Following is the definition of the sample variance.
Definition: Sample Variance
The sample variance is an approximate average of the squared deviations of
the data values from the sample mean.
That is, the deviations of the data values from the sample mean are first
computed, then the values for these deviations are squared, and the
approximate average of these square values are found. The average is
approximate because we divide not by the sample size
but by
– 1.
Remarks:
1.
We divide by the quantity
–1 in order to make the sample variance
an unbiased estimator of the population variance.
2.
An estimator is said to be unbiased if the average value of the
estimator is equal to the parameter it is estimating. For example, the
sample mean
̅
is an unbiased estimator of the population mean
.
That is, if we take all possible samples of a fixed size and find the
means of these samples, then the average of all these sample means
will be equal to the population mean.
3.
The sample variance uses the squares of the deviations from the mean,
as this will eliminate the effect of the signs (as was also the case when
we used the absolute value of the deviations in computing the
).
Generally, if there are
data values in the sample, with
x
representing the
data values, and
̅
representing the sample mean, then the variance of the set
of sample values, denoted by
is given by