A sample of 25 observations has a standard deviation of 4. The sum of the squared deviations from the sample mean is:
A) 21
B) 25
C) 100
D) 384
E) 400

Question

Quizplus · Accepted Answer

We know that the formula for calculating the sum of squared deviations from the sample mean is:
SS = Σ(x - x̄)²
where x is each individual observation, x̄ is the sample mean, and Σ represents the sum of the deviations for all observations.

Since we are not given the actual values of the observations, we cannot calculate SS directly. However, we do know that the standard deviation of the sample is 4. We also know that the formula for calculating the standard deviation is:
s = sqrt[Σ(x - x̄)² / (n - 1)]
where s is the sample standard deviation, n is the sample size, and again Σ represents the sum of the deviations for all observations.

Rearranging this formula to solve for Σ(x - x̄)², we get:
Σ(x - x̄)² = (n - 1) * s²
Plugging in the given values, we get:
Σ(x - x̄)² = 24 * 4²
Σ(x - x̄)² = 384

Therefore, the answer is D (384).

A Sample of 25 Observations Has a Standard Deviation of 4