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 $(c)=6.12$
The mean and the standard deviation $(s.d.)$ of five observations are $9$ and $0,$ respectively. If one of the observations is changed such that the mean of the new set of five observations becomes $10,$ then their $s.d.$ is?
If $\sum_{i=1}^{5}(x_i-10)=5$ and $\sum_{i=1}^{5}(x_i-10)^2=5$ then standard deviation of observations $2x_1 + 7, 2x_2 + 7, 2x_3 + 7, 2x_4 + 7$ and $2x_5 + 7$ is equal to-
The mean and standard deviation of marks obtained by $50$ students of a class in three subjects, Mathematics, Physics and Chemistry are given below:
Subject | Mathematics | Physics | Chemistty |
Mean | $42$ | $32$ | $40.9$ |
Standard deviation | $12$ | $15$ | $20$ |
Which of the three subjects shows the highest variability in marks and which shows the lowest?
Let in a series of $2 n$ observations, half of them are equal to $a$ and remaining half are equal to $-a.$ Also by adding a constant $b$ in each of these observations, the mean and standard deviation of new set become $5$ and $20 ,$ respectively. Then the value of $a^{2}+b^{2}$ is equal to ....... .
Suppose values taken by a variable $x$ are such that $a \le {x_i} \le b$, where ${x_i}$ denotes the value of $x$ in the $i^{th}$ case for $i = 1, 2, ...n.$ Then..