Q) Student-teacher ratio expresses the relationship between the number of students enrolled in a school and the number of teachers employed by the school. This ratio is important for a number of reasons. It can be used as a tool to measure teachers’ workload as well as the allocation of resources. A survey was conducted in 100 secondary schools of a state and the following frequency distribution table was prepared :

Based on the above, answer the following questions :
(i) What is the lower limit of the median class ?
(ii) What is the upper limit of the modal class ?
(iii) (a) Find the median of the data.
OR
(iii) (b) Find the modal of the data.

Ans: (i) Median Class:

We know that the median class is the class where the cumulative frequency crosses 50% of the total number of events.

Here in the given table, we need to calculate cumulative frequency and identify the median class and to do that, let’s re-organize the data:

at which it is crossing 50% of frequencies i.e. 45, at class “50-60”. Hence, our Median class

Here Sum of all frequencies (i.e. number of schools) is 100. hence, its 50% is 50. Now, if we look in the last column (cum freq), we can clearly see that the 50% of cumulative frequency (i.e. 50) is crossed at class “35 – 40”. Hence, our median class is 35-40.

Therefore, the lower limit of the Median class is 35.

(ii) Upper limit of Modal Class:

We know that the modal class is the class with the highest frequency.

In the given table of the question, class “35 – 40” has  maximum students (30) i.e. highest frequency of 12. Hence, the modal class is “35 – 40”.

Therefore, the Upper limit of the Modal class is 40.

(iii) (a) Median Value:

We just identified the Median class as ” 35 – 40″.

Next, we know that the Median of a grouped data is given by the formula:

Median = L+x h

Here:

L = Lower boundary of the median class = 35

n = Total number of students = 100 (given)

= Cumulative frequency of the class before the median class = 25

f = Frequency of the median class = 30

h = Class width = 40 – 35 = 5

Hence, the Median = 35 + x 5

= 35 + 5 x

= 35 + 5 x

= 25 + 5 x  = 25 +

= 25 + 4.17 = 29.17

Therefore the median of the data is 29.17

(iii) (b) Modal value:

We just identified the Modal class as ” 35 – 40″.

Next, we know that the Mode of a grouped data is given by the formula:

Mode = L + x h

Here,

L = lower class limit of modal class, 35

= frequency of modal class, 30

= frequency of class proceeding to modal class,25

= frequency of class succeeding to modal class, 15

h = class size, 5

Let’s put values, we get

Mode = 35 + x 5

= 35 + x 4

= 35 + x 4 = 35 + 1 = 36

Therefore the Mode of the data is 36.

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