Mean of grouped data

In grouped data, we do not know each value, so each class is represented by its midpoint (x): for the class 40-60, the midpoint is 50.

Direct method: X̄ = Σfx ÷ N, where f is the frequency of each class and N = Σf.

Example: daily wages of 20 workers: 4 earn 40-60 (midpoint 50), 10 earn 60-80 (70), and 6 earn 80-100 (90). Σfx = 200 + 700 + 540 = 1,440, so X̄ = 1,440 ÷ 20 = ₹72.

Indirect method: choose an assumed mean A, the midpoint of a class near the middle; work out d = x − A for each class. Then X̄ = A + Σfd ÷ N. With A = 70: d = −20, 0, 20; Σfd = −80 + 0 + 120 = 40; X̄ = 70 + 40 ÷ 20 = 72.

With step deviations, divide each d by the class interval i (here 20) to get u = −1, 0, 1. Then X̄ = A + (Σfu ÷ N) × i = 70 + (2 ÷ 20) × 20 = 72.

Choosing the assumed mean near the middle keeps the numbers small.