2.1 Basic Average Formula

Average = (Sum of all observations)/(Number of observations)

Equivalently: Sum of observations = Average × Number of observations

This second form — Sum = Average × Count — is the single most useful rearrangement in this entire chapter. Nearly every shortcut in Section 3 is built by manipulating this equation instead of the original division form.

Solved Example 1: Find the average of 24, 36, 18, 42, and 30.

Sum = 24 + 36 + 18 + 42 + 30 = 150. Number of terms = 5. Average = 150 ÷ 5 = 30.

Solved Example 2: The average of 8 numbers is 25. If one number is excluded, the average of the remaining 7 numbers becomes 24. Find the excluded number.

Sum of 8 numbers = 25 × 8 = 200. Sum of remaining 7 numbers = 24 × 7 = 168. Excluded number = 200 − 168 = 32.

Solved Example 3: The average of 20 numbers is 0. How many of them can be greater than zero, at most?

If even one number were positive and the rest were zero or negative in such a way that the sum stays 0, this is possible. In fact, at most 19 numbers can be positive (the 20th must then be negative enough to bring the sum back to 0; all 20 cannot be positive because that would force a positive sum).

Important sub-rule — effect of adding a constant to every observation: If a fixed value k is added to every observation in a data set, the average also increases by exactly k. If every observation is multiplied by a constant k, the average is also multiplied by k. This sounds obvious but is the basis of several fast mental-math shortcuts used in Section 3, and it is frequently tested directly as a standalone question in MTS and Group D papers.

Solved Example 3(a): The average of 12 numbers is 45. If 5 is added to each number, find the new average.

New average = Old average + 5 = 45+5 = 50. (No need to touch the individual numbers or recompute any sum.)

Solved Example 3(b): The average of 8 numbers is 20. If each number is multiplied by 3, find the new average.

New average = Old average × 3 = 20×3 = 60.

Solved Example 3(c): The average of 7 numbers is 18. If two numbers are excluded, the average of the remaining 5 numbers becomes 16. Find the sum of the two excluded numbers.

Sum of 7 numbers = 18 × 7 = 126. Sum of remaining 5 numbers = 16 × 5 = 80. Sum of the two excluded numbers = 126 − 80 = 46.

This is a natural extension of Example 2 — the same “sum-subtraction” logic works regardless of how many observations are excluded at once, as long as you correctly track how many numbers remain.

Solved Example 3(d): The average of 10 numbers is 40. If each number is first increased by 5 and then the resulting number is multiplied by 2, find the new average.

Apply the two sub-rules in sequence, exactly as they would be applied to every individual number. First, adding 5 to each number raises the average to 40+5 = 45. Then multiplying every (already-increased) number by 2 doubles this average: 45×2 = 90. (Verify with a tiny check set: two numbers 38 and 42, average 40. After +5: 43, 47, average 45. After ×2: 86, 94, average 90. ✓)