Measures of Dispersion

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Measures of Dispersion

The location parameter by its own, usually does not give a picture of the distribution (only the location!). Another characteristic parameter of the distribution should give the variation of the values around the location parameter. We could use a function of deviations from the mean

$/begin{displaymath}z_{i} - /bar{z}/ / ./end{displaymath}$

However, how it is easily seen, the sum and also the mean of these deviations are zero. Therefore the square of the deviations is frequently used, and the mean of these squares of deviations

$/begin{displaymath}s^{2}=/frac{1}{n-1} /sum_{i=1}^{n} (z_{i}-/bar{z})^{2}/end{displaymath}$

is called variance. The standard deviation or simply deviation is defined by

$/begin{displaymath}s=/sqrt{/frac{1}{n-1} /sum_{i=1}^{n} (z_{i}-/bar{z})^{2}}/ / / ./end{displaymath}$

For quick computation of

the formula

$/begin{displaymath}/sum (z_{i}-/bar{z})^{2} = /sum z_{i}^{2} - n /bar{z}^{2}/end{displaymath}$

may be used. If we have a classification of the data in

groups with $c_{j}$ as mean of the $j^{th}$ class and $h_{j}$ as frequency, the approximate identity is

$/begin{displaymath}/sum_{i=1}^{n} z_{i}^{2}=/sum_{j=1}^{k} c_j^2 h_{j}/ / / ./end{displaymath}$

Remark: If we denote the random variable by

, from which the values $z_{1},/ldots,z_{n}$ have been realized, then the variance of the variable can be computed from all possible values by

$/begin{displaymath}/sigma^{2}=Var Z=E(Z-EZ)^{2}= /int (z-EZ)^{2} f(z)dz/end{displaymath}$

where

is defined as in Subsection 2.2.1 by

The coefficient of variation is defined by

which is the relation of the deviation (variability of the data) to the mean.

should, of course, be greater than zero.)

Linear Transformations: Are the data values transformed by

$/begin{displaymath}y_{i}=az_{i} + b/ / / ,/end{displaymath}$

using two constants

and

, then the mean of $y_{i}$ is obtained by

$/begin{displaymath}/bar{y}=/frac{1}{n}/sum(az_{i}+b)=a /bar{z}+b / / ,/end{displaymath}$

and the variance is

$/begin{displaymath}s_y^2=/frac{1}{n-1}/sum(y_{i}-/bar{y})^{2}=/frac{a^{2}}{n-1}/sum (z_{i}-/bar{z})^{2}= a^{2}s_z^2 / / ,/end{displaymath}$

respectively the standard deviation is

$/begin{displaymath}s_{y}=as_{z}/ / / ./end{displaymath}$

Find out the arithmetic mean and the standard deviation of the standardized quantities

$/begin{displaymath}y_{i}=/frac{z_{i}-/bar{z}}{s_{z}}/ / / ./end{displaymath}$

Other Measures of Dispersion: The interquartile range $Q_{.75} - Q_{.25}$ approximates the standard deviation (if the assumption of the approximate normal distribution can be used) by

$/begin{displaymath}/sigma /sim /frac{Q_{.75} - Q_{.25}}{1.349}/ / / ,/end{displaymath}$

the median of absolute deviations from the median (Medmed) by

$/begin{displaymath}/sigma /sim /frac{med(/vert z_{i}-/bar{z} /vert)}{.6745}/ / / ./end{displaymath}$

It may easily be seen, that these two measures of dispersion are much more stable (resistant) in respect of changes in the data than the usual (previously defined) deviation, and are, therefore, much more recommendable for the practice.

Next: Higher Moments Up: Characteristic Parameters of a Previous: Quantiles and Percentiles Contents

Rudolf Dutter 2003-03-13