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Variance and standard deviation

It should be noted that, ¯x, ¯x j and ¯x r ms would remain well defined even if some sample values
were negative, but the value of ¯x g could then become complex. The geometric mean should not
be used in such cases.
Example 1.2. Compute ¯x g , ¯x h and ¯x rms for the sample given in table 2.1. ,

Solution. The geometric mean is given by (1.3) to be

¯ x g = (188.7 × 204.7 × · · · × 200.0) 1/8 = 184.4.

The harmonic mean is given by (1.4) to be
8
¯ x h = = 183.9.
(1/188.7) + (1/204.7) + · · · + (1/200.0)

Finally, the root mean square is given by (1.5) to be

[ ] 1/2
1
2
2
2
¯ x r ms = (188.7 + 204.7 + · · · + 200.0 ) = 185.5.
8
Two other measures of the ’average’ of a sample are itsmode andmedian. The mode is simply the
most commonly occurring value in the sample. A sample may possess several modes, however,
and it can thus be misleading in such cases to use the mode as a measure of the average of the
sample. The median of a sample is the halfway point when the sample values x i (i = 1, 2, …, N)
are arranged in ascending (or descending) order. Clearly, this depends on whether the size of the
sample, N, is odd or even. If N is odd then the median is simply equal to x (N+1)/2 , whereas if N is
1
even the median of the sample is usually taken to be (x N/2 + x (N/2)+1 ).
2
Example 1.3. Find the mode and median of the sample given in table 2.1. ,

Solution. From the table we see that each sample value occurs exactly once, and so any value
may be called the mode of the sample.
To find the sample median, we first arrange the sample values in ascending order and
obtain
166.3, 168.1, 169.0, 188.7, 189.8, 193.2, 200.0, 204.7.

Since the number of sample values N = 8, which is even, the median of the sample is
1 1
(x 4 + x 5 ) = (188.7 + 189.8) = 189.25.
2 2

Variance and standard deviation

The variance and standard deviation both give a measure of the spread of values in a sample about
the sample mean ¯x. The sample variance is defined by

N
1 ∑
2
2
s = (x i − ¯x) , (1.6)
N
i=1
and the sample standard deviation is the positive square root of the sample variance, i.e.
v
u N
u 1 ∑
2
s = t (x i − ¯x) . (1.7)
N
i=1

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