If both the standard deviation and mean of data `x_(1),x_(2),x_(3),……x_(50)` are 16, then the mean of the data set `(x_(1)-4)^(2),(x_(2)-4)^(2),(x_(3)-4)^(2),….(x_(50)-4)^(2)` is

Options

A. 200
B. 100
C. 400
D. 1600

Answer

To find the mean of the data set \((x_1 - 4)^2, (x_2 - 4)^2, (x_3 - 4)^2, \ldots, (x_{50} - 4)^2\), we can follow these steps:

### Step 1: Understand the given data
We are given that the mean \(\mu\) and the standard deviation \(\sigma\) of the data set \(x_1, x_2, \ldots, x_{50}\) are both 16.

### Step 2: Calculate the mean of the new data set
We need to find the mean of the transformed data set \((x_i - 4)^2\). The mean of a data set is calculated as:

\[ \text{Mean} = \frac{1}{n} \sum_{i=1}^{n} x_i \]

For our transformed data set, we can express the mean as:

\[ \text{Mean of } (x_i - 4)^2 = \frac{1}{50} \sum_{i=1}^{50} (x_i - 4)^2 \]

### Step 3: Expand the expression
We can expand \((x_i - 4)^2\):

\[ (x_i - 4)^2 = x_i^2 - 8x_i + 16 \]

### Step 4: Substitute into the mean formula
Now, substituting this back into the mean calculation, we have:

\[ \text{Mean} = \frac{1}{50} \sum_{i=1}^{50} (x_i^2 - 8x_i + 16) \]

This can be separated into three parts:

\[ \text{Mean} = \frac{1}{50} \left( \sum_{i=1}^{50} x_i^2 - 8 \sum_{i=1}^{50} x_i + \sum_{i=1}^{50} 16 \right) \]

### Step 5: Calculate each component
1. **Sum of \(x_i\)**: Since the mean is 16, we have:

\[ \sum_{i=1}^{50} x_i = 50 \times 16 = 800 \]

2. **Sum of \(16\)**: This is simply:

\[ \sum_{i=1}^{50} 16 = 50 \times 16 = 800 \]

3. **Sum of \(x_i^2\)**: We can use the relationship between variance, mean, and standard deviation. The variance \(\sigma^2\) is given by:

\[ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2 \]

This can be expanded to:

\[ \sigma^2 = \frac{1}{n} \left( \sum_{i=1}^{n} x_i^2 - n\mu^2 \right) \]

Rearranging gives us:

\[ \sum_{i=1}^{n} x_i^2 = n\sigma^2 + n\mu^2 \]

Substituting \(n = 50\), \(\sigma = 16\), and \(\mu = 16\):

\[ \sum_{i=1}^{50} x_i^2 = 50 \times 16^2 + 50 \times 16^2 = 50 \times 256 + 50 \times 256 = 25600 \]

### Step 6: Substitute back into the mean formula
Now we substitute back into our mean formula:

\[ \text{Mean} = \frac{1}{50} \left( 25600 - 8 \times 800 + 800 \right) \]

Calculating this gives:

\[ \text{Mean} = \frac{1}{50} \left( 25600 - 6400 + 800 \right) = \frac{1}{50} \left( 25600 - 6400 + 800 \right) = \frac{1}{50} \left( 19800 \right) = 396 \]

### Final Answer
The mean of the data set \((x_1 - 4)^2, (x_2 - 4)^2, \ldots, (x_{50} - 4)^2\) is **396**.

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