Central tendency and the arithmetic mean

A measure of central tendency sums up a whole set of data in one typical value, which makes comparison easy. When choosing a college, for example, students compare the average placement package of each one. The three common averages are the arithmetic mean, the median and the mode. (There are also the geometric and harmonic means.)

Each answers a different question. The book imagines Baiju, a farmer in the village of Balapur with 1 acre of land. Is his land above the average of all farmers in the village (the mean)? Does he own more than half of them (the median)? Does he own more than most of them (the mode)?

The arithmetic mean is the sum of all the values divided by their number:

X̄ = ΣX ÷ N

For example, four friends get pocket money of ₹200, ₹250, ₹300 and ₹450 a month. ΣX = 1200 and N = 4, so the mean is 1200 ÷ 4 = ₹300. Notice the mean need not be one of the values; it may even be a fraction, like 10.5.

When the numbers are large, two short-cut methods save work:

• Assumed mean method: choose any convenient value A (it need not be in the data), find each deviation d = X – A, and then X̄ = A + Σd ÷ N. With A = 250 for the pocket money: d = –50, 0, 50, 200; Σd = 200; X̄ = 250 + 200 ÷ 4 = 300, the same answer.
• Step deviation method: if all the deviations share a common factor c, divide them by it: d' = (X – A) ÷ c, and X̄ = A + (Σd' ÷ N) × c. With c = 50: d' = –1, 0, 1, 4; Σd' = 4; X̄ = 250 + (4 ÷ 4) × 50 = 300.

Whatever value of A you choose, the answer is always the same.