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Find the missing frequencies in the following frequency distribution if it is known that the mean of the distribution is 50.
$x$ | 10 | 30 | 50 | 70 | 90 |
$f$ | 17 | $f_1$ | 32 | $f_2$ | 19 |
Given:
The arithmetic mean of the given data is 50.
To do:
We have to find the missing frequencies in the given frequency distribution.
Solution:
$x$ | $f$ | $f \times\ x$ |
10 | 17 | 170 |
30 | $f_1$ | $30f_1$ |
50 | 32 | 1600 |
70 | $f_2$ | $70f_2$ |
90 | 19 | 1710 |
Total | $68+f_1+f_2 =120$ | $3480+30f_1+70f_2$ |
$68+f_1+f_2 =120$
$\Rightarrow f_1+f_2=120-68=52$
$\Rightarrow f_1=52-f_2$...........(i)
We know that, Mean$=\frac{\sum fx}{\sum f}$
Mean $50=\frac{3480+30f_1+70f_2}{68+f_1+f_2}$
$50(120)=3480+30f_1+70f_2$
$3480+30f_1+70f_2=6000$
$30(52-f_2)+70f_2=6000-3480$ [From (i)]
$1560-30f_2+70f_2=2520$
$40f_2=2520-1560$
$f_2=\frac{960}{40}$
$f_2=24$
$\Rightarrow f_1=52-24=28$
Therefore, $f_1=28$ and $f_2=24$.
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