Question

The cumulative frequency of first group in more than cumulative frequency distribution is always equal to
A. $x$
B. $100$
C. $\sum f$
D. $\sum x$

Answer 1

According to the question given in the question we have to find the cumulative frequency of the first group in more than cumulative frequency distribution is always equal to. So, first of all we have to obtain a table for class marks frequency.
Now, we have to obtain the cumulative frequency for the given frequency but first of all we have to understand about cumulative frequency as explained below:
Cumulative frequency: The cumulative frequency can be calculated or obtained by adding each given frequency from a frequency distribution table to the sum of its predecessors and its last value will be always equal to the total for all observations, since all the frequencies will already have been added to the previous total.
Hence, after obtaining cumulative frequency we have to check for as given in the question that the cumulative frequency of the first group is more than cumulative frequency distribution.

Complete step-by-step solution:
Step 1: First of all we have to draw a table for class marks and frequency as given below:

Class marks	Frequency
$0 - 10$	$4$
$10 - 20$	$5$
$20 - 30$	$12$
$30 - 40$	$11$
$40 - 50$	$8$

Step 2: Now, we have to obtain the cumulative frequency for the table as mentioned in the solution hint.

Class marks	Frequency	Cumulative frequency
$0 - 10$	$4$	$40 = 4 + (5 + 12 + 11 + 18)$
$10 - 20$	$5$	$36 = 5 + (12 + 11 + 8)$
$20 - 30$	$12$	$31 = 12 + (11 + 8)$
$30 - 40$	$11$	$19 = 11 + (8)$
$40 - 50$	$8$	8

Step 3: Now, as we have obtained from the table as in step 2 that cumulative frequency of first group in more than cumulative frequency distribution always equal to the sum of all the frequencies or $\sum f$
Hence, The cumulative frequency of the first group in more than cumulative frequency distribution is always equal to $\sum f$.

Therefore option (C) is correct.

Note: Cumulative frequency can be calculated or obtained by adding each given frequency from a frequency distribution table to the sum of its predecessors and its last value will be always equal to the total for all observations, since all the frequencies will already have been added to the previous total.