- Home
- Standard 11
- Mathematics
13.Statistics
hard
નીચે આપેલ આવૃત્તિ-વિતરણ માટે મધ્યક, વિચરણ અને પ્રમાણિત વિચલન શોધો.
| વર્ગ | $30-40$ | $40-50$ | $50-60$ | $60-70$ | $70-80$ | $80-90$ | $90-100$ |
|
આવૃત્તિ |
$3$ | $7$ | $12$ | $15$ | $8$ | $3$ | $2$ |
A
$14.18$
B
$14.18$
C
$14.18$
D
$14.18$
Solution
Let the assumed mean $A =65 .$ Here $h=10$
We obtain the following Table from the given data :
| Class |
Frequency ${f_i}$ |
Mid-point ${x_i}$ |
${y_i} = \frac{{{x_i} – 65}}{{10}}$ | ${y_i}^2$ | ${f_i}{y_i}$ | ${f_i}{y_i}^2$ |
| $30-40$ | $3$ | $35$ | $-3$ | $9$ | $-9$ | $27$ |
| $40-50$ | $7$ | $45$ | $-2$ | $4$ | $-14$ | $28$ |
| $50-60$ | $12$ | $55$ | $-1$ | $1$ | $-12$ | $12$ |
| $60-70$ | $15$ | $65$ | $0$ | $0$ | $0$ | $0$ |
| $70-80$ | $8$ | $75$ | $1$ | $1$ | $8$ | $8$ |
| $80-90$ | $3$ | $85$ | $2$ | $4$ | $6$ | $12$ |
| $90-100$ | $2$ | $95$ | $3$ | $9$ | $6$ | $18$ |
| $N=50$ | $-15$ | $105$ |
Therefore $\bar x = A + \frac{{\sum {{f_i}{y_i}} }}{{50}} \times h = 65 – \frac{{15}}{{50}} \times 10 = 62$
Variance ${\sigma ^2} = \frac{{{h^2}}}{{{N^2}}}\left[ {N{{\sum {{f_i}{y_i}} }^2} – {{\left( {\sum {{f_i}{y_i}} } \right)}^2}} \right]$
$=\frac{(10)^{2}}{(50)^{2}}\left[50 \times 105-(-15)^{2}\right]$
$=\frac{1}{25}[5250-225]=201$
and standard deviation $(\sigma)=\sqrt{201}=14.18$
Standard 11
Mathematics
