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Step-by-Step Solution
Step 1: Understand the Given Data
We have 50 observations, denoted by
$x_1, x_2, \dots, x_{50}$.
Their mean is 16, and their standard deviation is also 16.
Step 2: Express the Mean and Sum of Observations
The mean of the 50 observations is given by
$ \displaystyle \frac{\sum_{i=1}^{50} x_i}{50} = 16.$
Thus,
$$
\sum_{i=1}^{50} x_i = 16 \times 50 = 800.
$$
Step 3: Express the Standard Deviation
The standard deviation is given by
$ \sigma = \sqrt{ \frac{\sum_{i=1}^{50} x_i^2}{50} \;-\; \mu^2 }, $
where $ \mu = 16 $ is the mean. We know
$ \sigma = 16. $
Hence,
$$
16 \;=\; \sqrt{\frac{\sum_{i=1}^{50} x_i^2}{50} \;-\; 16^2} \quad \Rightarrow \quad
16^2 = \frac{\sum_{i=1}^{50} x_i^2}{50} \;-\; 256.
$$
Therefore,
$$
256 = \frac{\sum_{i=1}^{50} x_i^2}{50} - 256
\quad \Rightarrow \quad
\frac{\sum_{i=1}^{50} x_i^2}{50} = 256 + 256 = 512.
$$
So,
$$
\sum_{i=1}^{50} x_i^2 = 512 \times 50 = 25600.
$$
Step 4: Set Up the Required Mean Expression
We want the mean of
$(x_i - 4)^2$ for $i = 1, 2, \dots, 50$,
i.e.,
$$
\text{Required Mean} = \frac{\sum_{i=1}^{50} (x_i - 4)^2}{50}.
$$
Expand $(x_i - 4)^2$:
$$
(x_i - 4)^2 = x_i^2 - 8x_i + 16.
$$
Therefore,
$$
\sum_{i=1}^{50} (x_i - 4)^2
\;=\; \sum_{i=1}^{50} x_i^2 \;-\; 8 \sum_{i=1}^{50} x_i \;+\; \sum_{i=1}^{50} 16.
$$
Substituting the known sums:
$$
\sum_{i=1}^{50} x_i^2 = 25600, \quad
\sum_{i=1}^{50} x_i = 800, \quad
\sum_{i=1}^{50} 16 = 16 \times 50 = 800.
$$
We get
$$
\sum_{i=1}^{50} (x_i - 4)^2
= 25600 \;-\; 8 \times 800 \;+\; 800.
$$
Simplify step by step:
$$
-\,8 \times 800 = -6400,
$$
so
$$
25600 - 6400 = 19200, \quad 19200 + 800 = 20000.
$$
Hence,
$$
\sum_{i=1}^{50} (x_i - 4)^2 = 20000.
$$
Step 5: Calculate the Final Mean
Finally, the mean of $(x_i - 4)^2$ is:
$$
\frac{20000}{50} = 400.
$$
Answer: 400
