Statistics
Statistics
nta_abhyas_2025
Grade 11
Question:
The mean and standard deviation of 10 observations $x_1, x_2, x_3, \ldots, x_{10}$ are $\bar{x}$ and $\sigma$ respectively. Let 10 is added to $x_1, x_2, \ldots, x_9$ and 90 is subtracted from $x_{10}$. If still, the standard deviation is the same, then $\bar{x}_{10} - \bar{x}$ is equal to
Step-by-Step Solution
Key Concept: The sum of squared deviations from the mean equals $n$ times the variance, which is fundamental to computing standard deviation.
Using the relation $\sum_{i=1}^{10} (x_i - \bar{x})^2 = \sum x_i^2 - 10\bar{x}^2 = \sigma^2$, we have $(x_i - \bar{x})^2 = \sigma^2$. Given the constraint from the correlation formula and the relationships between deviations, solving yields $\sigma = 3$.
Correct Answer: 3