Cumulative Frequency
In a school examination, 500 students appeared for a math test. Their scores are presented in the form of a grouped frequency distribution table:
| Score Range (Marks) | Number of Students |
|---|---|
| 0 – 20 | 20 |
| 20 – 40 | 50 |
| 40 – 60 | 100 |
| 60 – 80 | 180 |
| 80 – 100 | 150 |
Two students, Rohan and Meera, are analyzing the table:
Rohan: How many students took the exam?
Meera: The total is 500 students.
Rohan: How many students scored between 60 and 80?
Meera: There are 180 students in that range.
Rohan: If the passing mark is 60, how many students passed?
Meera: The students who scored 60 and above are:
60 – 80 range:
80 – 100 range:
So, the total number of students who passed is 180 + 150 =
Rohan: What if only the top 200 students qualify for a scholarship? What should be the cut-off score?
Meera: We need to find how many students scored more than 80:
80 – 100 range: 150 students
Since only 150 students scored above 80, we need 50 more from the 60 – 80 range to reach 200. Thus, the cut-off should be somewhere within the
So, we have:
Students scoring 80 and above:
Students scoring 60 and above:
Students scoring 40 and above:
This method of adding up values progressively is called cumulative frequency analysis, which helps in determining cut-offs and distributions in exams or other data sets.
Cumulative frequency is calculated by progressively summing up frequencies either from the first or last class interval to a particular class. There are two types:
Greater Than Cumulative Frequency (GCF) – Calculated by adding up frequencies from the last class to the lower boundary of a particular class.
Less Than Cumulative Frequency (LCF) – Calculated by adding up frequencies from the first class to the upper boundary of a particular class.
Le's take the following example:
Exam Scores of 50 Students
| Score Range (Marks) | Frequency (No. of Students) |
|---|---|
| 0 – 10 | 5 |
| 10 – 20 | 8 |
| 20 – 30 | 12 |
| 30 – 40 | 15 |
| 40 – 50 | 10 |
To calculate GCF, start from the last class and sum the frequencies downward:
| Score Range (Marks) | Frequency | Greater Than Cumulative Frequency |
|---|---|---|
| 0 – 10 | 5 | 50 (5+8+12+15+10) |
| 10 – 20 | 8 | |
| 20 – 30 | 12 | |
| 30 – 40 | 15 | |
| 40 – 50 | 10 |
This helps in determining how many students scored more than a given mark.
To calculate LCF, start from the first class and sum the frequencies upward:
| Score Range (Marks) | Frequency | Less Than Cumulative Frequency |
|---|---|---|
| 0 – 10 | 5 | |
| 10 – 20 | 8 | |
| 20 – 30 | 12 | |
| 30 – 40 | 15 | |
| 40 – 50 | 10 |
This helps in determining how many students scored less than or equal to a given mark.