In an experiment with 15 observations on x , then following results were available: \sum x^2 = 2830 , \sum x = 170 One observation that was 20 was found to be wrong and was replaced by the correct value 30. Then the corrected variance is :

Statistics question

2003 · Shift 0 · Q74

2003 · Shift 0 · Q74

View written solutionFree

Correct answer: D

Given data for 15 observations
$\quad \sum x = 170, \quad \sum x^2 = 2830$$ One observation was wrongly taken as $20$ and is corrected to $30$.$
Correct the sum

Since $20$ is replaced by $30$ ,

$\sum x_{\text{correct}} = 170 - 20 + 30 = 180$
Correct the sum of squares

$\sum x^2_{\text{correct}} = 2830 - 20^2 + 30^2$

$= 2830 - 400 + 900 = 3330$
Use the variance formula

For variance of $n$ observations,

$\sigma^2 = \frac{\sum x^2}{n} - \left(\frac{\sum x}{n}\right)^2$

Substitute corrected values:

$\sigma^2 = \frac{3330}{15} - \left(\frac{180}{15}\right)^2$

$= 222 - 12^2$

$= 222 - 144 = 78$
Match with options

The corrected variance is

$\boxed{78.00}$

So the correct option is D.

2003 · Shift 0 · Q74

2003 · Shift 0 · Q74

View written solutionFree

Correct answer: D

Given data for 15 observations
$\quad \sum x = 170, \quad \sum x^2 = 2830$$ One observation was wrongly taken as $20$ and is corrected to $30$.$
Correct the sum

Since $20$ is replaced by $30$ ,

$\sum x_{\text{correct}} = 170 - 20 + 30 = 180$
Correct the sum of squares

$\sum x^2_{\text{correct}} = 2830 - 20^2 + 30^2$

$= 2830 - 400 + 900 = 3330$
Use the variance formula

For variance of $n$ observations,

$\sigma^2 = \frac{\sum x^2}{n} - \left(\frac{\sum x}{n}\right)^2$

Substitute corrected values:

$\sigma^2 = \frac{3330}{15} - \left(\frac{180}{15}\right)^2$

$= 222 - 12^2$

$= 222 - 144 = 78$
Match with options

The corrected variance is

$\boxed{78.00}$

So the correct option is D.