In an experiment with 10 observations on x the following results are available `Sigmax^(2)=354 and Sigma x=58`. If one observation 8 that was found to be wrong and was replaced by the corrected value 10, then the corrected variance is

Options

A. 5
B. 3
C. 4
D. 6

Answer

To find the corrected variance after replacing the wrong observation in the dataset, we can follow these steps:

### Step 1: Identify the given values
We have:
- Number of observations, \( n = 10 \)
- \( \Sigma x^2 = 354 \)
- \( \Sigma x = 58 \)
- The wrong observation is \( 8 \) and the corrected observation is \( 10 \).

### Step 2: Calculate the corrected sum of observations
To find the corrected sum of observations, we need to remove the wrong observation and add the corrected one:
\[ \Sigma x_c = \Sigma x - \text{(wrong observation)} + \text{(corrected observation)} \]
Substituting the values:
\[ \Sigma x_c = 58 - 8 + 10 = 60 \]

### Step 3: Calculate the corrected sum of squares
Next, we calculate the corrected sum of squares by removing the square of the wrong observation and adding the square of the corrected observation:
\[ \Sigma x_c^2 = \Sigma x^2 - (\text{wrong observation})^2 + (\text{corrected observation})^2 \]
Substituting the values:
\[ \Sigma x_c^2 = 354 - 8^2 + 10^2 = 354 - 64 + 100 = 390 \]

### Step 4: Calculate the corrected variance
The formula for variance is given by:
\[ \text{Variance} = \frac{\Sigma x_c^2}{n} - \left(\frac{\Sigma x_c}{n}\right)^2 \]
Substituting the corrected values:
\[ \text{Variance} = \frac{390}{10} - \left(\frac{60}{10}\right)^2 \]
Calculating the terms:
\[ \text{Variance} = 39 - 6^2 = 39 - 36 = 3 \]

### Final Answer
The corrected variance is \( 3 \).

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