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| VI. | |
Measures of Variability |
The investigator frequently is concerned with the variability of the distribution, that is, whether the measurements are clustered tightly around the mean or spread over the range. One measure of this variability is the difference between two percentiles, usually the 25th and the 75th percentiles. The pth percentile is a number such that p percent of the measurements are less than or equal to it; in particular, the 25th and the 75th percentiles are called the lower and upper quartiles, respectively. The pth percentile is readily found from the cumulative-frequency graph, (Fig. 1) by running a horizontal line through the p percent mark on the vertical axis on the graph, then a vertical line from this point on the graph to the horizontal axis; the abscissa of the intersection is the value of the pth percentile.
The standard deviation is a measure of variability that is more convenient than percentile differences for further investigation and analysis of statistical data. The standard deviation of a set of measurements x1, x2, …, xn, with the mean is defined as the square root of the mean of the squares of the deviations; it is usually designated by the Greek letter sigma (σ). In symbols
The square, σ2, of the standard deviation is called the variance. If the standard deviation is small, the measurements are tightly clustered around the mean; if it is large, they are widely scattered.
