Measures of Variability

The Variance

The variance is approximately equal to the average squared distance of each observation about the mean and is generally denoted by s2. This is most easily seen in its formula, which is given by

where

xi represents the individual observation from the ith subject; the mean of the data is given by x;

σ is the summation sign, which indicates that you sum over everything that follows it. The limits below and above the sign indicate where you start, and stop, the summation, respectively. As it is written above, it says you should begin summing with the squared deviation of x1 from the mean and stop with the squared deviation of xn from the mean; and

n represents the number of observations in your data set.

For illustrative purposes, consider the data in Table 1 above. To compute the variance of the height measurements, do the following:

• 1.
  Calculate the mean height x.
• 2.
  For each observation, calculate the deviation from the mean xi − x.
• 3.
  Square the deviation of each observation from the mean (xi − xþ2.
• 4.
  Sum over all the squared deviations.
• 5.
  Divide this sum by n − 1, where n is the sample size.

Table 1 gives the quantities described in Steps 2 and 3 above for each observation. For this set of data the procedure above gives

The Standard Deviation

The standard deviation is the most commonly used measure of variability for data that follow a bell-shaped (or normal) distribution. It is generally denoted by s, and is simply the square root of the variance. Formally, it is given by the formula

where the quantities in the formula are defined in the same way as described above for the variance. If we consider the data in Table 1, we can calculate the standard deviation easily using the value that we calculated for the

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