Let one observation equal to the mean is added to $n$ observations. If the variance changes from 78 to 72, then $n$ is equal to
Step-by-Step Solution
Key Concept: Adding a constant to all observations shifts the mean by the same amount, leaving the variance and sum of squared deviations unchanged.
When adding a constant $c$ to all observations, the new mean becomes $\bar{x}' = \bar{x} + c$. The sum of squared deviations becomes $\sum(x_i + c - (\bar{x} + c))^2 = \sum(x_i - \bar{x})^2$, which remains unchanged. The variance also remains constant since variance depends only on deviations from the mean, not on the absolute values of the data.
Correct Answer: 12