# Measures of Centre

When some measurements are taken and a set of numbers is obtained, (for example income, voting preference, number of children, or number of product defects), often it is useful to know something about what the ‘typical’ or ‘middle’ value looks like. These are known as measures of central tendency, or simply the ‘average’. There are, however, three common distinctly different ways of calculating the ‘average’ of a sample, each with its own particular use and interpretation.

· Mean: the arithmetic average, calculated by summing all values and dividing by the number of values.

· Median: the middle value, with half the values being higher and half lower.

· Mode: the value that occurs most often, or with greatest probability.

Mean, median, and mode can all be equal, but they need not be, and thus it is very important when interpreting statistical data to be sure which form of ‘average’ is being reported. Of particular importance is the distinction between mean and median. The two values can differ quite drastically when the sample contains some very high or very low values.

Consider, for example, a sample of the incomes of five different households. Suppose that four of the households are middle income, earning \$55, \$60, \$60, and \$65 thousand per year, while the the fifth household is very rich, earning two million dollars per year. To find the mean, we add up all these values and divide by five, yielding a result of \$448 thousand. While useful for some purposes, this figure clearly Clearly thus does not represent the ‘typical’ house at all, since most houses in the sample have incomes very much lower than this. The reason this occurs is because the mean is being ‘pulled up’ by the single very wealthy household. If we now calculate the median, or the middle value, we find that it is \$60, since two households earn less than this amount while and two earn more. This is clearly far more more representative of the typical household income than was the mean, even though both could potentially be reported as the ‘average’. Finally, in this example the mode is also \$60, since this value occurs twice while all the others occur only once. This simple example highlights the importance of understanding the differences between these three measures of center, and knowing which is being used in a given situation.