Statistics Examples

Example 1 : If mean of the seies $x_1$, $x^2$, ….. , $x_n$ is $\bar{x}$, then the mean of the series $x_i$ + 2i, i = 1, 2, ……, n will be-

Solution : As given $\bar{x}$ = $x_1 + x_2 + …. + x_n\over n$

If the mean of the series $x_i$ + 2i, i = 1, 2, ….., n be $\bar{X}$, then

$\bar{X}$ = $(x_1+2) + (x_2+2.2) + (x_3+2.3) + …. + (x_n + 2.n)\over n$

    = $x_1 + x_2 + …. + x_n\over n$ + $2(1+2+3+….+n)\over n$

    = $\bar{x}$ + $2n(n+1)\over 2n$

    = $\bar{x}$ + n + 1.

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Example 2 : A student obtained 75%, 80%, 85% marks in three subjects. If the marks of another subject are added then his average marks can not be less than-

Solution : Total marks obtained from three subjects out of 300 = 75 + 80 + 85 = 240

if the marks of another subject is added then the total marks obtained out of 400 is greater than 240

if marks obtained in fourth subject is 0 then

    minimum average marks = $240\over 400$$\times$100 = 60%

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Example 3 : The mean and variance of 5 observations of an experiment are 4 and 5.2 respectively. If from these observations three are 1, 2 and 6, then remaining will be-

Solution : As given $\bar{x}$ = 4, n = 5 and ${\sigma}^2$ = 5.2. If the remaining observations are $x_1$, $x_2$ then

${\sigma}^2$ = $\sum{(x_i – \bar{x})}^2\over n$ = 5.2

$\implies$ ${(x_1-4)}^2 + {(x_2-4)}^2 + {(1-4)}^2 + {2-4)}^2 + {(6-4)}^2\over 5$ = 5.2

$\implies$ ${(x_1-4)}^2 + {(x_2-4)}^2$ = 9     …..(1)

Also $\bar{x}$ = 4 $\implies$ $x_1 + x_2 + 1 + 2 + 6\over 5$ = 4 $\implies$ $x_1 + x_2$ = 11     …..(2)

from eq.(1), (2)     $x_1$, $x_2$ = 4, 7

Practice these given statistics examples to test your knowledge on concepts of statistics.