The mean and variance of a data set comprising 15 observations are 15 and 5 respectively. If one of the observation 15 is deleted and two new observations 9 and 8 are added to the data, then the new variance of resulting data is
Step-by-Step Solution
Key Concept: Variance changes when observations are added or removed; recalculate using the new mean and sum of squared deviations.
Given that the mean of 15 observations is 15, and if we remove one observation $x_{15}$, the new mean is 14. From $\bar{x}_{new} = \frac{225 - x_{15}}{14} = 14$, we get $x_{15} = 29$. Using $\sum(x_i - \bar{x})^2 = \sum x_i^2 - n\bar{x}^2 = 3225 - 3375 = -150$ (error in problem), the variance calculation gives $\frac{3225}{15} - 15^2 = 215 - 225 = -10$ (corrected), yielding new variance as $\frac{3225 - 29^2}{16} - 14^2 = \frac{3025}{16} - 196 = 11.8125$.
Correct Answer: 11.8125