The mean and standard deviation of some data for the time taken to complete . a test are calculated with the following results:

Number of observations $=25,$ mean $=18.2$ seconds, standard deviation $=3.25 s$

Further, another set of 15 observations $x_{1}, x_{2}, \ldots, x_{15},$ also in seconds, is now available and we have $\sum_{i=1}^{15} x_{i}=279$ and $\sum_{i=1}^{15} x_{i}^{2}=5524 .$ Calculate the standard deviation based on all 40 observations.

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Given, $n_{1}=25, \bar{x}_{i}=18.2, \sigma_{1}=3.25$

$n_{2}=15, \sum_{i=1}^{15} x_{i}=279$ and $\sum_{i=1}^{15} x_{i}^{2}=5524$

For first set $\Sigma x_{i}=25 \times 18.2=455$

$\therefore$

$\sigma_{1}^{2}=\frac{\Sigma x_{i}^{2}}{25}-(18.2)^{2}$

$\Rightarrow \quad(3.25)^{2}=\frac{\Sigma x_{i}^{2}}{25}-(18.2)^{2} \Rightarrow 10.5625+331.24=\frac{\Sigma x_{i}^{2}}{25}$

$\Rightarrow \quad \Sigma x_{i}^{2}=25 \times(10.5625+331.24)=25 \times 341.8025=8545.0625$

For combined SD of the 40 observations, $n=40$.

Now $\quad \sum_{i=1}^{40} x_{i}^{2}=5524+8545.0625=14069.0625$

and $\quad \sum_{i=1}^{40} x_{i}=455+279=734$

$\therefore \quad SD =\sqrt{\frac{14069.0625}{40}-\left(\frac{734}{40}\right)^{2}}=\sqrt{351.1726-(18.35)^{2}}$

$=\sqrt{351.726-336.7225}=\sqrt{15.0035}=3.87$

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  • [JEE MAIN 2022]

Find the mean and variance for the following frequency distribution.

Classes $0-10$ $10-20$ $20-30$ $30-40$ $40-50$
Frequencies $5$ $8$ $15$ $16$ $6$

The data is obtained in tabular form as follows.

${x_i}$ $60$ $61$ $62$ $63$ $64$ $65$ $66$ $67$ $68$
${f_i}$ $2$ $1$ $12$ $29$ $25$ $12$ $10$ $4$ $5$

One set containing five numbers has mean $8$ and variance $18$ and the second set containing $3$ numbers has mean $8$ and variance $24$. Then the variance of the combined set of numbers is