The diameters of circles (in mm) drawn in a design are given below:

Diameters $33-36$ $37-40$ $41-44$ $45-48$ $49-52$
No. of circles $15$ $17$ $21$ $22$ $25$

Calculate the standard deviation and mean diameter of the circles.

[ Hint : First make the data continuous by making the classes as $32.5-36.5,36.5-40.5,$ $40.5-44.5,44.5-48.5,48.5-52.5 $ and then proceed.]

Vedclass pdf generator app on play store
Vedclass iOS app on app store
Class Interval

Frequency

${f_i}$ 

Mid=point

${x_i}$

${y_i} = \frac{{{x_i} - 42.5}}{4}$ ${f_i}^2$ ${f_i}{y_i}$ ${f_i}{y_i}^2$
$33-36$ $15$ $34.5$ $-2$ $4$ $-30$ $60$
$37-40$ $17$ $38.5$ $-1$ $1$ $-17$ $17$
$41-44$ $21$ $42.5$ $0$ $0$ $0$ $0$
$45-48$ $22$ $46.5$ $1$ $1$ $22$ $22$
$49-52$ $25$ $50.5$ $2$ $4$ $50$ $100$
  $100$       $25$ $199$

here, $N=100,$ $h=4$

Let the assumed mean, $A,$ be $42.5$

Mean,   $\bar x = A + \frac{{\sum\limits_{i = 1}^5 {{f_i}{y_i}} }}{N} \times h$

$ = 42.5 + \frac{{25}}{{100}} \times 4 = 43.5$

Variance,  $\left( {{\sigma ^2}} \right) = \frac{{{h^2}}}{{{N^2}}}\left[ {N\sum\limits_{i = 1}^5 {{f_i}{y_i}^2 - {{\left( {\sum\limits_{i = 1}^5 {{f_i}{y_i}} } \right)}^2}} } \right]$

$=\frac{16}{10000}\left[100 \times 199-(25)^{2}\right]$

$=\frac{16}{10000}[19900-625]$

$=\frac{16}{10000} \times 19275$

$=30.84$

$\therefore$ Standard deviation $(\sigma)=5.55$

Similar Questions

Find the mean and variance of the frequency distribution given below:

$\begin{array}{|l|l|l|l|l|} \hline x & 1 \leq x<3 & 3 \leq x<5 & 5 \leq x<7 & 7 \leq x<10 \\ \hline f & 6 & 4 & 5 & 1 \\ \hline \end{array}$

The variance $\sigma^2$ of the data is $ . . . . . .$

$x_i$ $0$ $1$ $5$ $6$ $10$ $12$ $17$
$f_i$ $3$ $2$ $3$ $2$ $6$ $3$ $3$

  • [JEE MAIN 2024]

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]

For a frequency distribution, standard deviation is computed by

If the mean deviation about the mean of the numbers $1,2,3, \ldots ., n$, where $n$ is odd, is $\frac{5(n+1)}{n}$, then $n$ is equal to

  • [JEE MAIN 2022]