Question

The following table gives the literacy rate (in \[\% \]) of 25 cities. Find the median class and the modal class.

Literacy rate (in percent)	50 - 60	60 - 70	70 - 80	80 - 90
Number of cities	9	6	8	2

Answer 1

Hint: Here we will use the concept of the median and modal classes. We will calculate the cumulative frequency and then based on that we will get our median class. Based on the definition of the modal class, we will obtain it.

Formula Used: We will use the following formula of \[{\rm{median}} = \dfrac{N}{2}\], where \[N\] is the cumulative frequency.

Complete step-by-step answer:
The median class is the class where the cumulative frequency reaches half the sum of the absolute frequencies.
The cumulative frequencies are obtained by adding the frequency to its predecessor. Cumulative frequencies are found for the grouped data.
Let us prepare a cumulative frequency table now.

Literacy rate (in percent)	Number of cities (frequency)	Cumulative frequency
50 – 60	9	9
60 – 70	6	\[9 + 6 = 15\]
70 – 80	8	\[15 + 8 = 23\]
80 – 90	2	\[23 + 2 = 25\]

From the table, we have found the cumulative frequency as 25.
Now we will find the median class using the formula.
Median class = \[ = \dfrac{N}{2} = \dfrac{{25}}{2} = 12.5\]
Now the frequency that is the closest to \[12.5\] is 15.
The class corresponding to frequency 15 is 60 – 70.
So, our median class will be 60 – 70.
Now we will find the modal class.
The modal class has the highest frequency. To find the modal class we check all the frequencies and then find the greatest amongst them. Then we will check the class corresponding to the highest frequency. That particular class is known as our modal class.
In this case, the highest frequency is 9. The class corresponding to frequency 9 is 50 – 60.
So, our modal class is 50 – 60.
Hence, the median class is 60 – 70 and the modal class is 50 – 60.
Note: Median is basically the central tendency of the data. It helps us in determining how much the extremes deviate from the mean. We might calculate the median class, by taking half of the difference of the extremes of the data table. This will be done as follows –
\[90 - 50 = 40\]
Then we might take half of it and based on that we would find the median class. However, this method is wrong. To calculate the median class of the grouped data the cumulative frequency is found and half of it is considered.