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 solve the problem, we will follow these steps:

### Step 1: Understand the given information
We have two equations based on the summations provided:
1. \(\sum_{i=1}^{10}(x_i - 5)^2 = 350\)
2. \(\sum_{i=1}^{10}(x_i - 2) = 60\)

### Step 2: Expand the first summation
We expand the first summation:
\[ \sum_{i=1}^{10}(x_i - 5)^2 = \sum_{i=1}^{10}(x_i^2 - 10x_i + 25) \]
This can be rewritten as:
\[ \sum_{i=1}^{10}x_i^2 - 10\sum_{i=1}^{10}x_i + 250 = 350 \]
Thus, we have:
\[ \sum_{i=1}^{10}x_i^2 - 10\sum_{i=1}^{10}x_i = 100 \quad \text{(Equation 1)} \]

### Step 3: Expand the second summation
Now we expand the second summation:
\[ \sum_{i=1}^{10}(x_i - 2) = \sum_{i=1}^{10}x_i - 20 = 60 \]
This implies:
\[ \sum_{i=1}^{10}x_i = 80 \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 2 into Equation 1
Substituting \(\sum_{i=1}^{10}x_i = 80\) into Equation 1:
\[ \sum_{i=1}^{10}x_i^2 - 10(80) = 100 \]
This simplifies to:
\[ \sum_{i=1}^{10}x_i^2 - 800 = 100 \]
Thus, we find:
\[ \sum_{i=1}^{10}x_i^2 = 900 \quad \text{(Equation 3)} \]

### Step 5: Calculate the mean
The mean \(\bar{x}\) is given by:
\[ \bar{x} = \frac{\sum_{i=1}^{10}x_i}{n} = \frac{80}{10} = 8 \]

### Step 6: Calculate the variance
The formula for variance \(Var(x)\) is:
\[ Var(x) = \frac{1}{n} \sum_{i=1}^{10}x_i^2 - \bar{x}^2 \]
Substituting the values we have:
\[ Var(x) = \frac{1}{10}(900) - 8^2 \]
Calculating this gives:
\[ Var(x) = 90 - 64 = 26 \]

### Final Answer
The variance of \(x_i\) is \(26\).

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