Central tendency and the mean of ungrouped data

Statistical techniques help us summarise data. There are three main kinds: measures of central tendency, of dispersion (spread), and of relationship.

A measure of central tendency is a single value that represents the whole set, lying near its centre. The three are the mean, the median and the mode.

The mean (average) of ungrouped data, by the direct method:

X̄ = Σx ÷ N: add all the values, and divide by how many there are.

Example: the rainfall at five stations is 820, 940, 760, 1,010 and 870 mm. Σx = 4,400, so the mean = 4,400 ÷ 5 = 880 mm.

By the indirect method, you first subtract a convenient constant A (an assumed mean) from each value; these are the deviations, d. Then X̄ = A + Σd ÷ N. With A = 800, the deviations are 20, 140, −40, 210 and 70; Σd = 400; so X̄ = 800 + 400 ÷ 5 = 880 mm.

Both methods give the same mean; the indirect one makes the arithmetic easier with big numbers.