Question

If the median of the observations \(4,6,7,x,x+2,12,12,13\) arranged in an increasing order is 9,then the variance of these observations is 

• A. \(\dfrac{37}{4} \)
• B. \(\dfrac{38}{4} \)
• C. \(8\)
• D. \(9\)
• E. \(10\)

Answer 1

The Correct Option is A

Given: \(4,6,7,x,x+2,12,12,13\)

and mean(\(x̄\)) \(=9\)

Then as there are 8 observations , so 

\(\dfrac{x+x+2}{2}=9\)

\(⇒x=8\)

\(x_i\)	\((x_i-x̄)\)	\((x_i-x̄)^{2}\)
4	-5	25
6	-3	9
7	-2	4
8	-1	1
10	1	1
12	3	9
12	3	9
13	4	16
		74

Therefore, the two terms are \(8 , (8+2=)10\) respectively.

Now, Then Variance(\(σ^2\))\(=\dfrac{1}{8}×74= \dfrac{37}{4}\)

Answer 2

Let's place the observations in increasing order: \(4, 6, 7, x, x+2, 12, 12, 13\). \[ \frac{x + (x+2)}{2} = 9 \quad \Rightarrow \quad \frac{2x + 2}{2} = 9 \quad \Rightarrow \quad x + 1 = 9 \quad \Rightarrow \quad x = 8. \] Now, substituting \(x = 8\), the observations are: \[ 4, 6, 7, 8, 10, 12, 12, 13. \] We can now calculate the variance of these observations. 1. First, calculate the mean: \[ \text{Mean} = \frac{4 + 6 + 7 + 8 + 10 + 12 + 12 + 13}{8} = \frac{72}{8} = 9. \] 2. Next, calculate the squared differences from the mean: \[ (4 - 9)^2 = 25, \quad (6 - 9)^2 = 9, \quad (7 - 9)^2 = 4, \quad (8 - 9)^2 = 1, \] \[ (10 - 9)^2 = 1, \quad (12 - 9)^2 = 9, \quad (12 - 9)^2 = 9, \quad (13 - 9)^2 = 16. \] 3. Now, find the average of these squared differences (i.e., the variance): \[ \text{Variance} = \frac{25 + 9 + 4 + 1 + 1 + 9 + 9 + 16}{8} = \frac{74}{8} = \frac{37}{4} \]