**Q) **The median of the following data is 50. Find the values of ‘p” and ‘q’, if the sum of all frequencies is 90. Also find the mode of the data.

**Ans: **Let’s re-organize the data in the frequency table and calculate the values:

The sum of all frequencies, By summing up the families, we get:

78 + p + q = 90

p + q = 12 …….. (i)

**(i) Values of p and q:**

Let’s start from the median of the data. We know that to calculate median of the grouped data following are the steps:

- First, we need to find the cumulative frequency in the frequency table to find the median. Its shown in last column.
- Next, the median class is the class where the cumulative frequency crosses 50% of the half the total number of families. Here in the table, cumulative frequency of 90 is crossing 50% of frequencies i.e. 45, at class “50-60”. Hence, our Median class = 50-60.
- Next, To find the median, we use the formula:

Median = L+x h

Here:

L = Lower boundary of the median class = 50

n = Total number of students = 90 (given)

= Cumulative frequency of the class before the median class = 40 + p

f = Frequency of the median class = 60 + p

h = Class width = 60 – 50 = 10

Hence, the Median = 50 + x 10 = 50 (given)

⇒ 50 + 10 x = 50

⇒ 10 x = 0

⇒ = 0

⇒ 5 – p = 0 ⇒ **p = 5 **

from equation (i), q = 12 – p = 12 – 5 ⇒ **q ****= 7 **

**(ii) Mode of the data:**

Since we know that, the modal class is the class with the highest frequency.

Now that we have values of p and q, in the given data (check above table), class “40 – 50” has the highest frequency of 25 (check column f)

**Hence, class “40 – 50” is the modal class.**

Now mode of the grouped data is calculated by:

Mode = L + x h

Here,

L = lower class limit of modal class, 40

_{ }= frequency of modal class, 25

_{ }= frequency of class proceeding to modal class,15

_{ }= frequency of class succeeding to modal class, 20

h = class size, 10

Let’s put values, we get

Mode = 40 + x 10

= 40 + x 10 **= 46.67**