Question

the mean of the following data is 42 find the missing frequencies X and y if the sum of frequencies is 100​

Answer 1

Question: The mean of the following data is 42. Find the missing frequencies 'x' and 'y' if the sum of the frequencies is 100.

Solution:

Table using the Step Deviation method.

xi = Upper Limit + Lower Limit/2

di = xi - a

ui = di/h

uifi = ui × fi

a = Assumed Mean.

$\begin{tabular}{|c|c|c|c|c|c|}\cline{1-6}\ \sf Class Mark & \sf fi & \sf xi & \sf \ di \ & \sf \ ui \ & \sf uifi\cline{1-6}\cline{1-6}\ 0 - 10&\sf7&\sf5& \sf -20&\sf-2&\sf-14\\\10 - 20 & 10& 15& -10&-1&-10\\\20 - 30 & \sf x & 25 = a & \ 0 & 0 & 0\\\30 - 40&13&35&10&1&13\\\40 - 50& \sf y & 45& 20&2& 2y\\\50 - 60 & 10 & 55 & 30 & 3 & 30\\\60 - 70 & 14 & 65 & 40 & 4 & 56\\\70 - 80 & 9 & 75 & 50 & 5 & 45\\\cline{1-6}& \Sigma \sf fi =63+x+y & & & & \Sigma \sf ui =120+2y\\\cline{1-6}\end{tabular}$

Let us take 25 to be the Assumed Mean [a].

From the table,

∑fi = 63 + x + y

Also,

∑fi = 100

∴ 63 + x + y = 100 → Equation (1)

We know that,

$\Longrightarrow \sf Mean=a + \dfrac{\Sigma uifi}{\Sigma fi} \times h$

Given that Mean = 42.

$\Longrightarrow \sf 42 = 25 + \dfrac{120 + 2y}{100} \times 10$

$\Longrightarrow \sf 42 - 25= \dfrac{120 + 2y}{10} \times 1$

$\Longrightarrow \sf 17 \times 10=120 + 2y$

$\Longrightarrow \sf 170=120 + 2y$

$\Longrightarrow \sf 170 - 120=2y$

$\Longrightarrow \sf 50=2y$

$\Longrightarrow \sf y = \dfrac{50}{2}$

$\Longrightarrow \sf y = 25$

We Know that,

⇒ 63 + x + y = 100 [From equation 1]

⇒ 63 + 25 + x = 100

⇒ 88 + x = 100

⇒ x = 100 - 88

⇒ x = 12

Therefore,

x = 12

y = 25