If the mean of the data: 7,8,9,7,8,7,$\lambda$,8 is 8, then the variance of this data is :-
Finding Lambda using Mean
The dataset is given as: 7, 8, 9, 7, 8, 7, $\lambda$, 8. There are 8 observations (n=8).
The mean ($\mu$) is given as 8.
The formula for the mean is: $\mu = \frac{\sum x_i}{n}$
Substituting the values:
$8 = \frac{7 + 8 + 9 + 7 + 8 + 7 + \lambda + 8}{8}$
$8 = \frac{54 + \lambda}{8}$
Multiply both sides by 8:
$64 = 54 + \lambda$
Solve for $\lambda$:
$\lambda = 64 - 54 = 10$
Calculating Variance
Now the complete dataset is: 7, 8, 9, 7, 8, 7, 10, 8.
The mean ($\mu$) is 8.
The formula for variance ($\sigma^2$) is: $\sigma^2 = \frac{\sum (x_i - \mu)^2}{n}$
Calculate the squared differences from the mean:
- $(7 - 8)^2 = (-1)^2 = 1$
- $(8 - 8)^2 = (0)^2 = 0$
- $(9 - 8)^2 = (1)^2 = 1$
- $(7 - 8)^2 = (-1)^2 = 1$
- $(8 - 8)^2 = (0)^2 = 0$
- $(7 - 8)^2 = (-1)^2 = 1$
- $(10 - 8)^2 = (2)^2 = 4$
- $(8 - 8)^2 = (0)^2 = 0$
Sum of the squared differences:
$\sum (x_i - \mu)^2 = 1 + 0 + 1 + 1 + 0 + 1 + 4 + 0 = 8$
Calculate the variance:
$\sigma^2 = \frac{8}{8} = 1$
The variance of the data is 1.
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