STATISTICS
It should be noted that, ̄x, ̄xhandx ̄rmswould remain well defined even if some
sample values were negative, but the value of ̄xgcould then become complex.
The geometric mean should not be used in such cases.
Calculate ̄xg, ̄xhandx ̄rmsfor the sample given in table 31.1.The geometric mean is given by (31.3) to be
̄xg= (188. 7 × 204. 7 ×···× 200 .0)^1 /^8 = 184. 4.The harmonic mean is given by (31.4) to be
̄xh=8
(1/ 188 .7) + (1/ 204 .7) +···+(1/ 200 .0)
= 183. 9.
Finally, the root mean square is given by (31.5) to be
x ̄rms=[ 1
8 (188.^7
(^2) + 204. 72 +···+ 200. 02 )]^1 /^2 = 185. 5 .
Two other measures of the ‘average’ of a sample are itsmodeandmedian.The
mode is simply the most commonly occurring value in the sample. A sample may
possess several modes, however, and thus it can 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 valuesxi(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. IfNis odd then the median is simply equal to
x(N+1)/ 2 , whereas ifNis even the median of the sample is usually taken to be
1
2 (xN/^2 +x(N/2)+1).
Find the mode and median of the sample given in table 31.1.
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 valuesN= 8, which is even, the median of the sample is
1
2 (x^4 +x^5 )=
1
2 (188.7 + 189.8) = 189.^25 .
31.2.2 Variance and standard deviation
The variance and standard deviation both give a measure of the spread of values
in a sample about the sample meanx ̄.Thesample varianceis defined by
s^2 =
1
N
∑N
i=1
(xi− ̄x)^2 , (31.6)