Question

median is given 16
total frequency is 70

find the value of x and y​

Answer 1

Given:-

• Median is 16

• Total Frequency is 70

To Find:-

• Value of x and y.

Solution:-

Here, Median ($M_{m}$) = $l + \bigg[h \times \dfrac{\dfrac{N}{2} - cf}{f}\bigg]$

Where, l = Lower limit of median class

h = Class size

N = Total frequency

cf = cumulative frequency of class preceding the median class

f = frequency of median class

Median class :- Class having cumulative frequency is just greater than $\dfrac{N}{2}$.

Now, From the given figure we have,

Median class = $\dfrac{N}{2}$ = $\dfrac{70}{2}$ = 35

From above definition of median class we have,

Class interval = 15-20  and,

Frequency 15

Because Cumulative frequency of class interval 15-20 is just greater than 35.

∵ l = 15

h = 5

N = 70

f = 15

cf = 24 + x

∴ Median =  $15 + \bigg[5 \times \dfrac{\dfrac{70}{2} - (24 + x)}{15}\bigg]$

⇒ 16 = $15 + \bigg[5 \times \dfrac{35 - 24 - x}{15}\bigg]$

⇒ 16 - 15 = $\bigg[5 \times \dfrac{11 - x}{15}\bigg]$

⇒ 1 = $\dfrac{11 - x}{3}$

⇒ 3 = 11 - x

⇒ x = 11 - 3

⇒ x = 8

Since, 55 + x + y = 70              [ total frequency ]

       ⇒ 55 + 8 + y = 70

       ⇒ y = 70 - 63

       ⇒ y = 7

Hence, x and y is 8 and 7 respectively.

Some Important Terms:-

• Mode ($M_{o}$) = $l + \bigg(\dfrac{f_{1} - f_{0}}{2f_{1} - f_{0} - f_{2}}\bigg)$

• Mean ($\bar{x}$) = $\dfrac{ \sum \: fx}{n}$