Find the standard deviation for the following data:
${x_i}$ | $3$ | $8$ | $13$ | $18$ | $25$ |
${f_i}$ | $7$ | $10$ | $15$ | $10$ | $6$ |
Let us form the following Table :
${x_i}$ | ${f_i}$ | ${f_i}{x_i}$ | ${x_i}^2$ | ${f_i}{x_i}^2$ |
$3$ | $7$ | $21$ | $9$ | $63$ |
$8$ | $10$ | $80$ | $64$ | $640$ |
$13$ | $15$ | $195$ | $169$ | $2535$ |
$18$ | $10$ | $180$ | $324$ | $3240$ |
$23$ | $6$ | $138$ | $529$ | $3174$ |
$48$ | $614$ | $9652$ |
Now, by formula $(3),$ we have
$\sigma = \frac{1}{N}\sqrt {N\sum {{f_i}x_i^2 - {{\left( {\sum {{f_i}{x_i}} } \right)}^2}} } $
$=\frac{1}{48} \sqrt{48 \times 9652-(614)^{2}}$
$=\frac{1}{48} \sqrt{463296-376996}$
$=\frac{1}{48} \times 293.77=6.12$
Therefore, Standard deviation $(c)=6.12$
Find the mean and variance for the data
${x_i}$ | $92$ | $93$ | $97$ | $98$ | $102$ | $104$ | $109$ |
${f_i}$ | $3$ | $2$ | $3$ | $2$ | $6$ | $3$ | $3$ |
If the variance of the first $n$ natural numbers is $10$ and the variance of the first m even natural numbers is $16$, then $m + n$ is equal to
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If each of the observation $x_{1}, x_{2}, \ldots ., x_{n}$ is increased by $'a'$ where $a$ is a negative or positive number, show that the variance remains unchanged.