Find the mean and variance of the frequency distribution given below:
$\begin{array}{|l|l|l|l|l|} \hline x & 1 \leq x<3 & 3 \leq x<5 & 5 \leq x<7 & 7 \leq x<10 \\ \hline f & 6 & 4 & 5 & 1 \\ \hline \end{array}$
Find the mean and variance of the frequency distribution given below:
$\begin{array}{|l|l|l|l|l|} \hline x & 1 \leq x<3 & 3 \leq x<5 & 5 \leq x<7 & 7 \leq x<10 \\ \hline f & 6 & 4 & 5 & 1 \\ \hline \end{array}$
$\begin{array}{|c|c|c|c|c|} \hline x & f _{ i } & x _{ i } & f x _{ i } & f x _{ i }^{ 2 } \\ \hline 1-3 & 6 & 2 & 12 & 24 \\ \hline 3-5 & 4 & 4 & 16 & 64 \\ \hline 5-7 & 5 & 6 & 30 & 180 \\ \hline 7-10 & 1 & 8.5 & 8.5 & 72.25 \\ \hline \text { Total } & n=16 & & \Sigma f_{i} x_{i}=66.5 & \Sigma f_{i}{ }_{i}^{2}=340.25 \\ \hline \end{array}$
$\therefore \quad$ Mean $=\frac{\Sigma f_{i} x_{i}}{\Sigma f_{i}}=\frac{66.5}{16}=4.13$
And variance $=\sigma^{2}=\frac{\Sigma f_{i} x_{i}^{2}}{\Sigma f_{i}}-\left(\frac{\Sigma f_{i} x_{i}}{\Sigma f_{i}}\right)^{2}=\frac{340.25}{16}-(4.13)^{2}$
$\quad=21.2656-17.0569=4.21$
Similar Questions
If the mean and variance of five observations are $\frac{24}{5}$ and $\frac{194}{25}$ respectively and the mean of first four observations is $\frac{7}{2}$, then the variance of the first four observations in equal to
If the mean and variance of five observations are $\frac{24}{5}$ and $\frac{194}{25}$ respectively and the mean of first four observations is $\frac{7}{2}$, then the variance of the first four observations in equal to
- [JEE MAIN 2024]
Find the variance and standard deviation for the following data:
${x_i}$
$4$
$8$
$11$
$17$
$20$
$24$
$32$
${f_i}$
$3$
$5$
$9$
$5$
$4$
$3$
$1$
Find the variance and standard deviation for the following data:
| ${x_i}$ | $4$ | $8$ | $11$ | $17$ | $20$ | $24$ | $32$ |
| ${f_i}$ | $3$ | $5$ | $9$ | $5$ | $4$ | $3$ | $1$ |
The variance of the first $n$ natural numbers is
The variance of the first $n$ natural numbers is
If the mean of the frequency distribution
Class:
$0-10$
$10-20$
$20-30$
$30-40$
$40-50$
Frequency
$2$
$3$
$x$
$5$
$4$
is $28$ , then its variance is $........$.
If the mean of the frequency distribution
| Class: | $0-10$ | $10-20$ | $20-30$ | $30-40$ | $40-50$ |
| Frequency | $2$ | $3$ | $x$ | $5$ | $4$ |
is $28$ , then its variance is $........$.
- [JEE MAIN 2023]
Let $x_1, x_2, x_3, x_4, .......... , x_n$ be $n$ observations and let $\bar x$ be their arithmetic mean and $\sigma ^2$ be their variance.
Statement $-1$ : Variance of observations $2x_1, 2x_2, 2x_3, ......, 2x_n$ is $4\sigma ^2$ .
Statement $-2$ : Arithmetic mean of $2x _1, 2x_2, 2x_3, ......, 2x_n$ is $4\bar x$ .
Let $x_1, x_2, x_3, x_4, .......... , x_n$ be $n$ observations and let $\bar x$ be their arithmetic mean and $\sigma ^2$ be their variance.
Statement $-1$ : Variance of observations $2x_1, 2x_2, 2x_3, ......, 2x_n$ is $4\sigma ^2$ .
Statement $-2$ : Arithmetic mean of $2x _1, 2x_2, 2x_3, ......, 2x_n$ is $4\bar x$ .