3.8 Measures of Dispersion 47- interquartile range;
- mean absolute deviation; and
- standard deviation.
We have already encountered and defi ned the interquartile range
because of its importance in illustrating the spread of the data in a box -
and - whisker plot. So we will now move on to the defi nitions of (2) and
(3) above.
The mean absolute deviation of a sample is defi ned as follows:
The mean absolute deviation (MAD) for a sample is the average
absolute difference between the sample mean and the observed
value. Its formula is:MAD i Ei E n
n
=−∑= 1 , ,where Ei=the i th observation, E is the sample mean, and n is the
sample size. The MAD like the median is less sensitive to outliers
than the next measure we discuss, the standard deviation.The standard deviation of a sample is the square root of the estimate
of the variance. The variance measures the average of the squared
deviations from the mean. The sample estimate of variance would then
be given by the formula VEEn=∑in= 1 ()i−
2
, where E is the sample
mean, and Ei is the i th observation. Usually, we would divide by n as
in the formula above, but for normally distributed observations from a
random sample, this estimate is proportional to a chi - square random
variable with n − 1 degrees of freedom. * The expected value † of a chi -
square random variable is equal to its degrees of freedom. So if instead
of n in the denominator we used n − 1 and label the estimate as S 2 , then
( n − 1) S 2 / σ 2 is known to have exactly a chi - square distribution with
n − 1 degrees of freedom, where σ 2 is the true population variance.
Therefore, as previously discussed, its expected value E [( n − 1 )
S 2 / σ 2 ] = n − 1, or E [ S 2 / σ 2 ] = 1. But this means E [ S 2 ] = σ 2. So S 2 is an† Expected value is a statistical term for the mean of a distribution.* Think of the degrees of freedom as a parameter that determines the shape of the chi - square
distribution.