Standard Deviation

In the late 1860s, Sir Francis Galton formulated the law of deviation from an average, which has become one of the most useful statistical measures, known as the standard deviation, or SD as most often abbreviated. The standard deviation statistic is one way to describe the results of a set of measurements and, at a glance, it can provide a comprehensive understanding of the characteristics of the data set. Examples of some of the more familiar and easily calculated descriptors of a sample are the range, the median, and the mean of a set of data. The range provides the extent of the variation of the data, providing the highest and lowest scores but revealing nothing about the pattern of the data. And, either or both of the highest and lowest scores might be quite different from the scores in between. The median is the single middle score (or average of the two middle scores) that tends to be used to compensate for unusually high or low scores, but again, it still provides little information about the overall characteristics of the data. The mean is probably the most familiar, as well as easily calculated, descriptor of a given set of numbers. As individuals, we like to know where we fit in comparison to most other people. Countless surveys have been conducted, for example, to determine what “most” people earn or the taxes that most people pay. It is also useful to gauge a school's educational success by determining whether test scores are above or below average. On a more personal level, it can be interesting to know the average height of the players on one basketball team compared with another. However, although the mean is quick and easy to calculate, it is also subject to distortion by even a single extreme score, or outlier, in the data set. For example, the mean of a set of data can be the same; the mean height of two teams of basketball players might both be 7 feet tall. But, the variability of the players’ heights on each team could be quite different. The height of players on team 1 ranges from 6 feet 9 inches to 7 feet 3 inches. The height of team 2 might realistically cluster around 6 feet tall, but one of their players is 7 feet 6 inches tall. A comparison of the means alone hides the fact that most of team 2 is much shorter than team 1. To obtain insight as to which team might be more likely to win tonight's game, more information is needed about the true variability of the players’ heights.

To understand that variability, statisticians calculate the standard deviation, which is especially [Page 1422]important to research because, although the other measures described previously are useful, the standard deviation provides a more accurate picture of the distribution of measurements. This statistic is an indicator of the distance of individual measurements from the mean score. A low standard deviation indicates that the data points are clustered tightly around the mean value, whereas a high standard deviation indicates that the data are less precise and spread across a large range of values.

Figure 1 Differences in the Normal Curve as a Result of Differences in Distribution of the Data Are Illustrated for the Example of Groups 1 and 2

Figure 2 The Relationship of the Standard Deviation and the Normal Distribution of Data as Illustrated by the Group 1 Data Set

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