Calculate the mean, variance and standard deviation for the following distribution:
Class | $30-40$ | $40-50$ | $50-60$ | $60-70$ | $70-80$ | $80-90$ | $90-100$ |
$f_i$ | $3$ | $7$ | $12$ | $15$ | $8$ | $3$ | $2$ |
From the given data, we construct the following Table
Class |
Freq $\left( {{f_i}} \right)$ |
Mid-point $\left( {{x_i}} \right)$ |
${f_i}{x_i}$ | ${\left( {{x_i} - \bar x} \right)^2}$ | ${f_i}{\left( {{x_i} - \bar x} \right)^2}$ |
$30-40$ | $3$ | $35$ | $105$ | $729$ | $2187$ |
$40-50$ | $7$ | $45$ | $315$ | $289$ | $2023$ |
$50-60$ | $12$ | $55$ | $660$ | $49$ | $588$ |
$60-70$ | $15$ | $6$ | $975$ | $9$ | $135$ |
$70-80$ | $8$ | $75$ | $600$ | $169$ | $1352$ |
$80-90$ | $3$ | $85$ | $255$ | $529$ | $1587$ |
$90-100$ | $2$ | $95$ | $190$ | $1089$ | $2178$ |
$50$ | $3100$ | $10050$ |
Thus Mean $\bar x = \frac{1}{N}\sum\limits_{i = 1}^7 {{f_i}{x_i}} = \frac{{3100}}{{50}} = 62$
Variance $\left( {{\sigma ^2}} \right) = \frac{1}{N}\sum\limits_{i = 1}^7 {{f_i}{{\left( {{x_i} - \bar x} \right)}^2}} $
$ = \frac{1}{{50}} \times 10050 = 201$
and Standerd deviation $\left( \sigma \right) = \sqrt {201} = 14.18$
Let ${x_1}\;,\;{x_2}\;,\;.\;.\;.\;,{x_n}$ be $n$ observations, and let $\bar x$ be their arithmaetic mean and ${\sigma ^2}$ be the variance
Statement $-1$ :Variance of $2{x_1}\;,2\;{x_2}\;,\;.\;.\;.\;,2{x_n}$ is $4{\sigma ^2}$ .
Statement $-2$: Arithmetic mean $2{x_1}\;,2\;{x_2}\;,\;.\;.\;.\;,2{x_n}$ is $4\bar x$.
Find the mean and variance for the data
${x_i}$ | $6$ | $10$ | $14$ | $18$ | $24$ | $28$ | $30$ |
${f_i}$ | $2$ | $4$ | $7$ | $12$ | $8$ | $4$ | $3$ |
Find the standard deviation of the first n natural numbers.
The standard deviation of $25$ numbers is $40$. If each of the numbers is increased by $5$, then the new standard deviation will be