Question

In a series of 20 observations, 10 observations are each equal to k and each of the remaining half is equal to − k. If the standard deviation of the observations is 2, then write the value of k.

Answer 1

\[n = 20\]
\[ d_i = x_i - a\]
\[ = x_i - \frac{\sum x_i}{20}\]
\[ = x_i - 0\]
\[ = x_i \]
\[ \Rightarrow \sum d_i = \sum x_i = 0\]
\[ \Rightarrow \sum {d_i}^2 = 20 k^2 \]
\[ \Rightarrow \sigma^2 = \frac{\sum {d_i}^2}{n} - \left( \frac{\sum d_i}{n} \right)^2 \]
\[ = \frac{20 k^2}{20} - 0\]
\[ = k^2 \]
\[ \Rightarrow \sigma = 2 = \sqrt{k^2}\]
\[ \Rightarrow k = \pm 2\]