Shortcuts for the mean

When the class marks are big, f × x makes huge numbers. Two shortcuts give exactly the same mean with smaller numbers.

Assumed mean method. Guess a middle class mark and call it a. Find how far each class mark is from it: dᵢ = xᵢ – a. Then:

x̄ = a + Σfᵢdᵢ / Σfᵢ

Step deviation method. If all the dᵢ share a common factor h (usually the class size), divide it out: uᵢ = dᵢ / h. The u's become small numbers like –2, –1, 0, 1, 2. Then:

x̄ = a + h × Σfᵢuᵢ / Σfᵢ

Example: wages of 50 workers. The class marks are 510, 530, 550, 570, 590. Take a = 550 and h = 20, so u = –2, –1, 0, 1, 2. Σfu = –12, so the mean = 550 + 20 × (–12/50) = ₹545.20.

Any a gives the same answer. Choose a different a and every dᵢ changes, but they change by exactly as much as a did, so the final mean is the same. (Your working will look different from a friend's who chose another a. That's fine.)

Which method? All three give the same answer, so pick whichever keeps the arithmetic easy. Use step deviation when the numbers are big and the classes are equal.

A missing frequency? Put the unknown f into the table, write the mean formula with it, set it equal to the given mean, and solve.