Calculate mean, variance and standard deviation for the following distribution.
Classes | $30-40$ | $40-50$ | $50-60$ | $60-70$ | $70-80$ | $80-90$ | $90-100$ |
${f_i}$ | $3$ | $7$ | $12$ | $15$ | $8$ | $3$ | $2$ |
Let the assumed mean $A =65 .$ Here $h=10$
We obtain the following Table from the given data :
Class |
Frequency ${f_i}$ |
Mid-point ${x_i}$ |
${y_i} = \frac{{{x_i} - 65}}{{10}}$ | ${y_i}^2$ | ${f_i}{y_i}$ | ${f_i}{y_i}^2$ |
$30-40$ | $3$ | $35$ | $-3$ | $9$ | $-9$ | $27$ |
$40-50$ | $7$ | $45$ | $-2$ | $4$ | $-14$ | $28$ |
$50-60$ | $12$ | $55$ | $-1$ | $1$ | $-12$ | $12$ |
$60-70$ | $15$ | $65$ | $0$ | $0$ | $0$ | $0$ |
$70-80$ | $8$ | $75$ | $1$ | $1$ | $8$ | $8$ |
$80-90$ | $3$ | $85$ | $2$ | $4$ | $6$ | $12$ |
$90-100$ | $2$ | $95$ | $3$ | $9$ | $6$ | $18$ |
$N=50$ | $-15$ | $105$ |
Therefore $\bar x = A + \frac{{\sum {{f_i}{y_i}} }}{{50}} \times h = 65 - \frac{{15}}{{50}} \times 10 = 62$
Variance ${\sigma ^2} = \frac{{{h^2}}}{{{N^2}}}\left[ {N{{\sum {{f_i}{y_i}} }^2} - {{\left( {\sum {{f_i}{y_i}} } \right)}^2}} \right]$
$=\frac{(10)^{2}}{(50)^{2}}\left[50 \times 105-(-15)^{2}\right]$
$=\frac{1}{25}[5250-225]=201$
and standard deviation $(\sigma)=\sqrt{201}=14.18$
The variance of first $50$ even natural numbers is
For two data sets, each of size $5$, the variances are given to be $4$ and $5$ and the corresponding means are given to be $2$ and $4$, respectively. The variance of the combined data set is
Find the mean and variance for the data $6,7,10,12,13,4,8,12$
The mean of the numbers $a, b, 8,5,10$ is $6$ and their variance is $6.8$. If $M$ is the mean deviation of the numbers about the mean, then $25\; M$ is equal to
Let the mean of the data
$X$ | $1$ | $3$ | $5$ | $7$ | $9$ |
$(f)$ | $4$ | $24$ | $28$ | $\alpha$ | $8$ |
be $5.$ If $m$ and $\sigma^2$ are respectively the mean deviation about the mean and the variance of the data, then $\frac{3 \alpha}{m+\sigma^2}$ is equal to $..........$.