NIMCET 2023 — Mathematics PYQ

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Question

NIMCET 2023 — Mathematics PYQ

NIMCET | Mathematics | 2023

The mean of 5 observation is 5 and their variance is 124. If three of the observation are 1, 2 and 6; then the mean deviation from the mean of the data is

Choose the correct answer:

Explanation

1. Finding the remaining two observations:

Let the observations be $1, 2, 6, a, b$ .

Mean ( $\bar{x}$ ) = 5.

\frac{1 + 2 + 6 + a + b}{5} = 5

9 + a + b = 25 \implies a + b = 16 \dots (i)

2. Using the Variance formula:

Variance ( $\sigma^2$ ) = 12.4.

\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2

12.4 = \frac{1^2 + 2^2 + 6^2 + a^2 + b^2}{5} - (5)^2

12.4 + 25 = \frac{41 + a^2 + b^2}{5}

37.4 \times 5 = 41 + a^2 + b^2

187 - 41 = a^2 + b^2 \implies a^2 + b^2 = 146 \dots (ii)

3. Solving for $a$ and $b$ :

Using $(a+b)^2 = a^2 + b^2 + 2ab$ :

(16)^2 = 146 + 2ab \implies 256 - 146 = 2ab \implies 2ab = 110 \implies ab = 55

The numbers whose sum is 16 and product is 55 are 5 and 11.

So, the observations are: $1, 2, 5, 6, 11$ .

4. Calculating Mean Deviation from Mean ( $\bar{x}=5$ ):

\text{M.D.}(\bar{x}) = \frac{\sum |x_i - \bar{x}|}{5}

$|1 - 5| = 4$
$|2 - 5| = 3$
$|5 - 5| = 0$
$|6 - 5| = 1$
$|11 - 5| = 6$

\text{Sum of deviations} = 4 + 3 + 0 + 1 + 6 = 14

\text{Mean Deviation} = \frac{14}{5} = 2.8

Correct Option: 3

Choose the correct answer:

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