Related Formula
Mean x = (Σ xᵢ)/(n)$\bar{x} = \frac{\sum x_i}{n}$
Variance σ² = (Σ xᵢ²)/(n) - ( x)²$\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$
Mean Deviation about Median = (Σ |xᵢ - M|)/(n)$\frac{\sum |x_i - M|}{n}$ where M$M$ is the median.
Core Logic
Let the two missing observations be a$a$ and b$b$.
From the given mean:
(2 + 3 + 5 + 10 + 11 + 13 + 15 + 21 + a + b)/(10) = 9$$\frac{2 + 3 + 5 + 10 + 11 + 13 + 15 + 21 + a + b}{10} = 9$$
(80 + a + b)/(10) = 9 a + b = 10$$\frac{80 + a + b}{10} = 9 \implies a + b = 10$$
From the given variance:
(Σ xᵢ²)/(10) - 9² = 34.2$$\frac{\sum x_i^2}{10} - 9^2 = 34.2$$
(2² + 3² + 5² + 10² + 11² + 13² + 15² + 21² + a² + b²)/(10) = 34.2 + 81 = 115.2$$\frac{2^2 + 3^2 + 5^2 + 10^2 + 11^2 + 13^2 + 15^2 + 21^2 + a^2 + b^2}{10} = 34.2 + 81 = 115.2$$
Step 1: Solve for a and b
Sum of squares of knowns:
4 + 9 + 25 + 100 + 121 + 169 + 225 + 441 = 1094$$4 + 9 + 25 + 100 + 121 + 169 + 225 + 441 = 1094$$
1094 + a² + b² = 1152$$1094 + a^2 + b^2 = 1152$$
a² + b² = 58$a^2 + b^2 = 58$
We know a+b = 10 b = 10-a$a+b = 10 \implies b = 10-a$.
a² + (10-a)² = 58$$a^2 + (10-a)^2 = 58$$
2a² - 20a + 100 = 58 2a² - 20a + 42 = 0$$2a^2 - 20a + 100 = 58 \implies 2a^2 - 20a + 42 = 0$$
a² - 10a + 21 = 0 (a-3)(a-7) = 0$$a^2 - 10a + 21 = 0 \implies (a-3)(a-7) = 0$$
Thus, the missing observations are 3 and 7.
Step 2: Find the Median
Arrange all 10 observations in ascending order:
2, 3, 3, 5, 7, 10, 11, 13, 15, 21$2, 3, 3, 5, 7, 10, 11, 13, 15, 21$
Since there are 10 observations, median M$M$ is the average of the 5th and 6th terms:
M = (7 + 10)/(2) = 8.5$$M = \frac{7 + 10}{2} = 8.5$$
Find absolute deviations |xᵢ - M|$|x_i - M|$:
|2-8.5|=6.5$|2-8.5|=6.5$
|3-8.5|=5.5$|3-8.5|=5.5$
|3-8.5|=5.5$|3-8.5|=5.5$
|5-8.5|=3.5$|5-8.5|=3.5$
|7-8.5|=1.5$|7-8.5|=1.5$
|10-8.5|=1.5$|10-8.5|=1.5$
|11-8.5|=2.5$|11-8.5|=2.5$
|13-8.5|=4.5$|13-8.5|=4.5$
|15-8.5|=6.5$|15-8.5|=6.5$
|21-8.5|=12.5$|21-8.5|=12.5$
Sum of absolute deviations:
6.5 + 5.5 + 5.5 + 3.5 + 1.5 + 1.5 + 2.5 + 4.5 + 6.5 + 12.5 = 50$$6.5 + 5.5 + 5.5 + 3.5 + 1.5 + 1.5 + 2.5 + 4.5 + 6.5 + 12.5 = 50$$
Mean Deviation = (50)/(10) = 5$$\text{Mean Deviation} = \frac{50}{10} = 5$$
Chapter Mix
Class 11 Mathematics: Statistics
