Variability

Psy 207: Introduction to Statistics Erwin Segal Numerical measures of sets
II Variability Homework: Click here

Variability is about how measures of individuals within a set differ from one another. Measures of sets representing these differences are as important for statistics and understanding sets as are the measures of central tendency. The use of a measure of variability in conjunction with one of central tendency gives the reasearcher and the reader important information for understanding the properties of a population and for comparing different populations. We will identify four different measures of variability.

Range--The simplest measure; it is the difference between the highest and the lowest score value. Range = X(high) – X(low)

deviation score-- this is a very important concept, that of a. This score gives the distance and direction of a score from the mean in the units of measure. Deviation scores are usually represented with lower class letters, e.g., x. Computationally, .

If x_i is positive we know that the score is larger than the mean, if negative, x_i is smaller than the mean. Deviation scores are the primary measures used in identifying measures of variability.

What seems to be a sensible computation is the Average deviation. That is, the average of the deviation scores . This however has a serious problem. It always equals zero.

A seldom used alternative is the Mean absolute deviation . This could be a reasonable measure of variability, but for technical reasons it is almost never used.

Computing AD and MAD for Men's heights, we have:

Variance, Standard Deviation, and Sum of Squares

Two other measures of variability, Variance and Standard Deviation, which are also based on deviation scores are the primary measures.

Variance is conceptually the average of sum of the squared deviations from the mean. The symbol for the variance of a population is s² , the symbol for the variance of a sample, when used to estimate the variance of a population) is s² . The formula for a variance is or ; or

(SS is explained below). If you divide by n, rather than n-1 to compute the variance of a random sample the result tends to be too small. Dividing by n-1 exactly corrects for that error. If the group that the variance is computed for is the whole population, than the average of the squared deviations from the mean is the population variance.

The variance is a very important statistic and is used very widely in statistics, but it has one serious drawback as far as communicating variability is concerned. It has different units than those that measure the individuals of the sample or population. Since the deviation scores are squared, the variance is in square units. Square inches in the example we are using.

Standard deviation = the positive square root of the variance; either, , or depending upon whether it is computed on a population or a random sample. This is the most popular descriptive measure of variability. It has mathematical utility, and numerically has the appropriate units. The standard deviation is a reliable measure of variability.
SS = sum of squares = numerator of the variance, i.e., sum of squared deviations from the mean. This term, and variants thereof, is widely used in statistical computation, so it is important to be familiar with it. SS is usually not computed by its definitional formula , but rather by one of several computational formulae. I prefer this one:

. The X's are the direct measures on the individuals.

Computing Variance and Standard Deviation for the heights of men using the definitional formula

= 197.2381; ; .

If our only concern is the 21 men's heights s² is the appropriate variance, if, however, the 21 men were a random sample from a larger population s² is the appropriate variance. The standard deviations are s = 3.065, and
s = 3.140.

Computing Variance and Standard Deviation for the heights of men using the computational formula

The actual measures directly enter into the computation.

The standard deviations are s = 3.06 and s = 3.14 respectively. Variances and standard deviations from the women’s data can be computed in the same ways. Compute the population and the sample variances and also both standard deviations.
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