The mean and variance of 10 observations are 9 and 34.2, respectively. If 8 of these observations are 2, 3, 5, 10, 11, 13, 15, 21, then the mean deviation about the median of all the 10 observations is

The mean and variance of 10 observations are 9 and 34.2 respectively. Find the mean deviation about the median

Q. The mean and variance of 10 observations are 9 and 34.2, respectively. If 8 of these observations are 2, 3, 5, 10, 11, 13, 15, 21, then the mean deviation about the median of all the 10 observations is

(A) 7

(B) 4

(C) 5

(D) 6

Correct Answer: 5

Explanation

Mean of 10 observations is 9, so the total sum is:

Total sum = 10 × 9 = 90

Sum of the given 8 observations:

2 + 3 + 5 + 10 + 11 + 13 + 15 + 21 = 80

Hence, the sum of the remaining two observations is:

90 − 80 = 10

Variance is given by:

σ² = (Σx² / n) − (mean)²

So,

34.2 = (Σx² / 10) − 81

Σx² = 1152

Sum of squares of the given 8 observations:

2² + 3² + 5² + 10² + 11² + 13² + 15² + 21² = 1094

Let the remaining observations be a and b. Then:

a + b = 10 a² + b² = 1152 − 1094 = 58

Solving, we get:

a = 3, b = 7

So the full data set is:

2, 3, 3, 5, 7, 10, 11, 13, 15, 21

Median = (5th + 6th terms)/2 = (7 + 10)/2 = 8.5

Mean deviation about the median is:

(1/10) Σ |x − 8.5| = 5

Hence, the mean deviation about the median is 5.

Explanation

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