Question

Consider the following distribution:

Marks	0 or more	10 or more	20 or more	30 or more	40 or more	50 or more
Number of students	63	58	55	51	48	42

(a) Calculate the frequency of class 30-40.
(b) Calculate the class mark of the class 10-20

Answer 1

Hint: In this question, we are given cumulative frequency series and for solving the given two parts, we have to change the given series to simple frequency series. For converting to simple frequency series, we will take classes by taking given numbers as the lower limit and succeeding numbers as the upper limit. For distributing frequency in the classes, we will take the difference of frequency of the first two rows as the frequency of first-class and similarly for other classes. For the last class, the frequency, at last, will remain the same. After this, we will check the frequency of class 30-40 and for (b) we will find class mark using the formula:
\[\text{Class mark}=\dfrac{\text{Upper limit}+\text{Lower limit}}{2}\].

Complete step-by-step solution:
Here, we are given a series in terms of cumulative frequency series. Let us first change it to a simple frequency series. For this, we take classes as 0-10, 10-20, 20-30, 30-40, 40-50 and 50-60.
We have taken these classes by taking lower limits as the number given and upper limits as succeeding numbers.
For distributing frequencies between classes, we will take the first two differences and write it as the frequency of the first class.
For 0-10, we will make the difference between 63 and 58 that is the frequency for 0-10 becomes 63-58 = 5.
For 10-20, we will make the difference between 58 and 55 that is frequency for 10-20 becomes
58-55 = 3.
For 20-30, we will make the difference between 63 and 58 that is frequency for 20-30 becomes
63-58 = 5.
For 30-40, we will make the difference between 51 and 48 that is frequency for 30-40 becomes
51-48 = 3.
For 40-50, we will make the difference between 48 and 42 that is frequency for 40-50 becomes
48-42 = 6.
For 50-60, frequency is taken as the last frequency that frequency for 50-60 becomes 42.
Therefore, our simple frequency table becomes:

Class	Frequency
0-10	5
10-20	3
20-30	4
30-40	3
40-50	6
50-60	42

Now, let us solve both parts:
(a) We have to calculate the frequency of 30-40.
As calculated earlier, the frequency of 30-40 is 3.
(b) We have to calculate the class mark of class 10-20.
As we know, the class mark is the mid value of the class given by $\dfrac{\text{Upper limit}+\text{Lower limit}}{2}$.
Hence, \[\text{Class mark}=\dfrac{10+20}{2}=15\]

Note: Students should note that for the last class we have 50-60, just to make the width of all classes the same. When counting frequency for a particular class, take differences properly and don't get confused if we have to keep the first frequency the same or the last frequency the same. Students can do (b) part directly without tables also.