Question

For a statistical data \( x_1, x_2, \dots, x_{10} \) of 10 values, a student obtained the mean as 5.5 and \[ \sum_{i=1}^{10} x_i^2 = 371. \] He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. 
The variance of the corrected data is:

• A. 7
• B. 4
• C. 9
• D. 5

Hint:

To find the variance after correcting some values, adjust the sum of squares and the sum of the values accordingly, then apply the formula for variance.

Answer 1

The Correct Option is A

Given: 

\( \sum x_i = (5.5) \times 10 = 55 \), and \( \sum x_i^2 = 371 \).

Corrected Mean Calculation:

\[ \text{Corrected Mean} = \frac{\sum x_i'}{10} = \frac{55 + 6 + 8 - (4 + 5)}{10} = 6. \]

Sum of Squared Corrected Values:

\[ \sum (x_i')^2 = 371 + 6^2 + 8^2 - (4^2 + 5^2) = 471 - 41 = 430. \]

Variance Calculation:

\[ \text{Variance} = \frac{\sum (x_i')^2}{10} - \left( \frac{\sum x_i'}{10} \right)^2. \] Substituting the values: \[ \text{Variance} = \frac{430}{10} - 36 = 43 - 36 = 7. \]

Final Answer:

The correct option is \( \boxed{7} \).