Mean from deviations about an arbitrary value: The algebraic sum of deviations of 20 observations measured from 23 is 70. What is the mean of these observations?

Introduction / Context:
When deviations are measured from an arbitrary number a, the sum of observations is sum(xi) = n * a + sum(xi - a). Knowing the deviation sum lets us recover the true mean quickly.

Given Data / Assumptions:

• n = 20 observations
• Reference a = 23
• Sum of deviations = Σ(xi - 23) = 70

Concept / Approach:
Compute Σxi = n * a + Σ(xi - a) = 20 * 23 + 70, then divide by n to obtain the mean.

Step-by-Step Solution:
Σxi = 20 * 23 + 70 = 460 + 70 = 530Mean = 530 / 20 = 26

Verification / Alternative check:
Equivalently, the mean equals 23 + (70 / 20) = 23 + 3.5 = 26.5? Careful: 70/20 = 3.5, but we must use Σxi = 460 + 70 = 530, and 530/20 = 26; direct “add 70/20 to 23” only works if 23 is the previous mean, not merely a reference. Hence the formula via totals is the robust path and yields 26.

Why Other Options Are Wrong:
24, 25, and 27 do not match Σxi / n; 23 is just the reference about which deviations were computed, not the mean.

Common Pitfalls:
Adding deviation average to the reference as if the reference were the prior mean. Always compute Σxi correctly first.

Final Answer:
26