The mean of the following frequency distribution is 63.3 and the sum of all the frequencies is 50 compute the missing freqiencies f1 and f2
class 0-20,20-40,40-60,60-80,80-100,100-120
frequency 5 ,f1 ,10 ,f2 , 7 ,8
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The mean of the following frequency distribution is 63.3 and the sum of all the frequencies is 50 compute the missing freqiencies f1 and f2
class 0-20,20-40,40-60,60-80,80-100,100-120
frequency 5 ,f1 ,10 ,f2 , 7 ,8
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[tex]\large\underline\mathfrak\red{Given \: :- }[/tex]
[tex] \large\underline\mathfrak\red{To \: find \: :- }[/tex]
[tex] \large\underline\mathfrak\red{Solution \: :- }[/tex]
We know that,
Sum of frequency is 50.
[tex]\sf\implies{5 + f1 + 10 + f2 + 7 + 8 = 50} \\ \sf\implies{f1 + f2 = 50 - 5 - 10 - 7 - 8} \\ \sf\implies{f1 + f2 = 20} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \sf\implies{3f1 + 3f2 = 60} \: \: \: \: \: ...(1) \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \sf\implies{(multiplying \: both \: side \: by \: 3)} \: \: \: \: \: \: \: \\
[/tex]
Subtract
And also we know that
mean is 62.8
[tex]\sf\implies{ \frac{∑f_1x_1}{N} = 62.8} \\ [/tex]
[tex]\sf\implies{ \frac{30f_1+70f_2 + 2060}{50} = 62.8 } \\ [/tex]
[tex]\sf\implies{30f1 + 70f2 + 2060 = 62.8 \times 50} \\ \sf\implies{30f1 + 70f2 + 2060 = 3140} \: \: \: \: \: \: \: \: \: \\ \sf\implies{30f1 + 70f2 = 3140 - 2060} \: \: \: \: \: \: \: \: \: \\\sf\implies{30f1 + 70f2 = 1080} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \sf\implies{3f1 + 7f2 = 108} \: \: ...(2) \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \sf\implies{(dividing \: both \: side \: by \: 10)} \: \: \: \: \: \: \: \: \: \: \: \: \: \: [/tex]
Subtract equation (1) from equation (2)
[tex]\sf\implies{3f1 + 7f2 - (3f1 + 3f2) = 108 - 60} \\ \sf\implies{3f1 + 7f2 - 3f1 - 3f2 = 48} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \sf\implies{4f2 = 48} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \sf\implies{f2 = \frac{48}{4} = 12} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: [/tex]
Put value of f2 in equation (1)
[tex]\sf\implies{3f1 + 3f2 = 60} \: \: \: \: \\ \sf\implies{3f1 + 3(12) = 60} \\ \sf\implies{3f1 + 36 = 60} \: \: \: \: \: \\ \sf\implies{3f1 = 60 - 36} \: \: \: \: \: \\ \sf\implies{3f1 = 24} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \sf\implies{f1 = \frac{24}{3} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \sf\implies{f1 = 8} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: [/tex]
[tex]{\boxed{\sf{\therefore{f1 = 8 \: and \: f2 = 12 }}}} [/tex]