Introduction

Introduction

Before this chapter, you have used measures of central tendency — the mean, median, and mode — to summarise a data set with a single 'typical' value. But an average, on its own, does not tell the whole story. As the statisticians George Simpson and Fritz Kafka put it: 'An average does not tell the full story. It is hardly fully representative of a mass unless we know the manner in which the individual items scatter around it.'

Why the mean alone can mislead. Suppose three batsmen, X, Y and Z, each scored a total of 250 runs over the same five One Day International matches — so all three have exactly the same mean score of 50. Yet their individual innings look very different: X scored 90, 17, 104, 33, 6 and Z scored 112, 8, 96, 29, 5 — both swinging wildly between very high and very low scores — while Y scored a much steadier 40, 60, 55, 50, 45. Judged by the mean alone, all three batsmen look identical, but Y is clearly the more consistent, dependable one. What the mean is missing is exactly how much the data scatters around it, and that scatter is called dispersion.

A working definition. According to Spiegel, dispersion (or variation) is 'the degree to which numerical data tend to spread about an average value.'

What you already know. From earlier work you already have the concepts of a constant and a variable, the idea of an average, and how to compute the mean for both ungrouped and grouped data — all of that is reused here.

This chapter develops three of the commonly used measures of dispersion: range, variance, and standard deviation.