Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# The Mean and Standard Deviation of 100 Observations Were Calculated as 40 and 5.1 Respectively by a Student Who Took by Mistake 50 Instead of 40 for One Observation. - Mathematics

The mean and standard deviation of 100 observations were calculated as 40 and 5.1 respectively by a student who took by mistake 50 instead of 40 for one observation. What are the correct mean and standard deviation?

#### Solution

$n = 100$

$\text{ Mean } = \bar{X} = 40$

$\sigma = SD = 5 . 1$

$\frac{1}{n}\sum x_i = \bar{X}$

$\Rightarrow \sum x_i = 100 \times 40 = 4000 \left( \text{ This is an incorrect reading due to misread values . } \right)$

$\text{ Corrected sum } , \sum x_i = 4000 - 50 + 40$

$= 3990$

$\Rightarrow \text{ Corrected mean }= \frac{\text{ Corrected sum } }{100}$

$= \frac{3990}{100}$

$= 39 . 9 . . . (1)$

To find the corrected SD:

$\sqrt{\text{ Variance } } = \sigma$

$\Rightarrow \sigma^2 = \left( 5 . 1 \right)^2 = \text{ Variance }$

$\text{ According to the formula } ,$

$\frac{1}{n} \sum_{} {x_i}^2 - \left( \bar{X} \right)^2 = \text{ Variance}$

$\Rightarrow \frac{1}{100} \sum_{} {x_i}^2 - \left( 40 \right)^2 = 26 . 01$

$\Rightarrow \frac{1}{100} \sum_{} {x_i}^2 - 1600 = 26 . 01$

$\Rightarrow \frac{1}{100} \sum_{} {x_i}^2 = 1626 . 01$

$\Rightarrow \sum_{} {x_i}^2 = 162601 \left( \text{ But, this is incorrect due t o misread values } \right)$

$\Rightarrow \text{ Corrected } \sum_{} {x_i}^2 = 162601 - {50}^2 + {40}^2$

$= 161701 . . . . (2)$

$\text{ Corrected variance } = \frac{1}{100}\text{ Corrected } \sum_{} {x_i}^2 - \left( \text{ Corrected mean } \right)^2$

$= \frac{161701}{100} - \left( 39 . 9 \right)^2 \left[\text{ using equations (1) and (2) } \right]$

$= 1617 . 01 - 1592 . 01$

$= 25$

$\text{ Corrected SD } = \sqrt{{\text{ Corrected variance} }}$

$= \sqrt{{25}}$

$= 5$

Corrected mean = 39.9
Corrected standard deviation = 5

Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 32 Statistics
Exercise 32.4 | Q 8 | Page 28