For a sample size of 10 observations `x_(1), x_(2),...... x_(10)`, if `Sigma_(i=1)^(10)(x_(i)-5)^(2)=350 and Sigma_(i=1)^(10)(x_(i)-2)=60`, then the variance of `x_(i)` is

Answer

To find the variance of the sample observations \( x_1, x_2, \ldots, x_{10} \), we will use the given equations:

1. \( \sum_{i=1}^{10} (x_i - 5)^2 = 350 \)
2. \( \sum_{i=1}^{10} (x_i - 2) = 60 \)

### Step 1: Calculate \( \sum_{i=1}^{10} x_i \)

From the second equation, we can express \( \sum_{i=1}^{10} x_i \):

\[ \sum_{i=1}^{10} (x_i - 2) = 60 \]
This expands to:
\[ \sum_{i=1}^{10} x_i - \sum_{i=1}^{10} 2 = 60 \]
Since \( \sum_{i=1}^{10} 2 = 20 \) (because there are 10 terms),
\[ \sum_{i=1}^{10} x_i - 20 = 60 \]
Thus,
\[ \sum_{i=1}^{10} x_i = 60 + 20 = 80 \]

### Step 2: Calculate \( \sum_{i=1}^{10} x_i^2 \)

Using the first equation, we can expand it:
\[ \sum_{i=1}^{10} (x_i - 5)^2 = 350 \]
Expanding this gives:
\[ \sum_{i=1}^{10} (x_i^2 - 10x_i + 25) = 350 \]
This can be rearranged to:
\[ \sum_{i=1}^{10} x_i^2 - 10\sum_{i=1}^{10} x_i + 250 = 350 \]
Substituting \( \sum_{i=1}^{10} x_i = 80 \):
\[ \sum_{i=1}^{10} x_i^2 - 10 \cdot 80 + 250 = 350 \]
This simplifies to:
\[ \sum_{i=1}^{10} x_i^2 - 800 + 250 = 350 \]
Thus,
\[ \sum_{i=1}^{10} x_i^2 - 550 = 350 \]
So,
\[ \sum_{i=1}^{10} x_i^2 = 350 + 550 = 900 \]

### Step 3: Calculate the Mean \( \bar{x} \)

The mean \( \bar{x} \) is given by:
\[ \bar{x} = \frac{\sum_{i=1}^{10} x_i}{10} = \frac{80}{10} = 8 \]

### Step 4: Calculate the Variance

The variance \( \sigma^2 \) is calculated using the formula:
\[ \sigma^2 = \frac{\sum_{i=1}^{10} x_i^2}{n} - \bar{x}^2 \]
Substituting the values we have:
\[ \sigma^2 = \frac{900}{10} - 8^2 \]
Calculating this gives:
\[ \sigma^2 = 90 - 64 = 26 \]

### Final Answer

The variance of \( x_i \) is \( \sigma^2 = 26 \).

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