The mean of the following frequency distribution of 100 observations is 148. Find the missing frequencies f₁ and f₂ :

This is a grouped frequency distribution.

To find f₁ and f₂, we’ll need to find mean of the following distribution by direct method and equate it to the given mean of the following distribution, 148.

This given data is in inclusive type, and we need not convert it into exclusive since it won’t make a difference in the calculation.

So, let’s construct a table finding midpoints and stating frequencies.

So now, we have

∑x_if_i = 8869 + 124.5f₁ + 274.5f₂

And ∑f_i = 62 + f₁ + f₂

Mean is given by

⇒ [given, mean = 148 and using the values from the table]

⇒ 148 × (62 + f₁ + f₂) = 8869 + 124.5f₁ + 274.5f₂

⇒ 9176 + 148f₁ + 148f₂ = 8869 + 124.5f₁ + 274.5f₂

⇒ 274.5f₂ – 148f₂ + 124.5f₁ – 148f₁ = 9176 – 8869

⇒ 126.5f₂ – 23.5f₁ = 307

⇒

⇒

⇒ 253f₂ – 47f₁ = 307 × 2

⇒ 253f₂ – 47f₁ = 614 …(i)

Since, there are 100 observations.

⇒ Frequency = 100 [Frequencies depict total number of observation]

⇒ ∑f_i = 100

⇒ 62 + f₁ + f₂ = 100

⇒ f₁ + f₂ = 100 – 62 = 38

⇒ f₁ + f₂ = 38 …(ii)

Multiplying 47 by equation (ii), we get

f₁ + f₂ = 38 [× 47

⇒ 47f₁ + 47f₂ = 1786 …(iii)

Solving equations (i) and (iii), we get

253f₂ – 47f₁ = 614

47f₂ + 47f₁ = 1786

300f₂ + 0 = 2400

⇒ 300f₂ = 2400

⇒

⇒ f₂ = 8

Putting f₂ = 8 in equation (ii), we get

f₁ + 8 = 38

⇒ f₁ = 38 – 8

⇒ f₁ = 30

Thus, the missing frequencies are f₁ = 30 and f₂ = 8.