Cumulative Frequency Distribution

Cumulative frequency is a statistical concept used to determine the total number of observations that fall below or at a given value in a dataset. It helps in understanding the distribution of data and is essential for constructing cumulative frequency graphs such as ogives.

In this blog, we will explore:

1. Definition of cumulative frequency

2. How to calculate the cumulative frequency

3. Types of cumulative frequency

4. Numerical examples

5. Applications in real-world scenarios

   

1. Definition of Cumulative Frequency

Cumulative frequency is the running total of frequencies as you move from the lowest to the highest value in a dataset. It helps in understanding the spread of data and identifying percentiles, medians, and quartiles.

2. How to Calculate Cumulative Frequency

To calculate cumulative frequency:

• Start with the first class frequency as it is.

• Then add the next frequency to the previous cumulative total.

• Continue adding each frequency to the cumulative sum.

  
  

3. Types of Cumulative Frequency

There are two types of cumulative frequency:

i) Less than Cumulative Frequency

• This represents the total number of values less than or equal to a specific value.

• It is obtained by adding each frequency successively.

ii) More than Cumulative Frequency

• This represents the total number of values greater than or equal to a specific value.

• It is obtained by subtracting the cumulative total from the total frequency.

  
  

4. Numerical Example

Consider the following dataset  (Table 1) showing the marks obtained by students in an exam:

Table 1: Hypothetical Data of Age Distribution

Class Interval (Marks)	Frequency (Number of Students)
0 - 10	5
10 - 20	8
20 - 30	12
30 - 40	15
40 - 50	10

To calculate the "Less than" cumulative frequency, we add up the values. For the "More than" cumulative frequency, we subtract from the total frequency (or N = 50 in our example) (see, Table 2):

Table 2: Calculation of Less than" and "More than" Cumulative Frequency

Class Boundaries (of Marks)	Frequency	Calculation of CF Less than Type	CF Less than Type	Calculation of CF Greater than Type	CF Greater than Type
0	0			50
9.5	5	0+5 =	5	50 - 5 =	45
19.5	8	5 + 8 =	13	45 - 8 =	37
29.5	12	13 + 12 =	25	37 - 12 =	25
39.5	15	25 + 15 =	40	25 - 15 =	10
50.5	10	40 + 10 =	50	10 - 10 =	0

5. Graphical Representation: Cumulative Frequency Curve (Ogive)

The cumulative frequency can be plotted on a graph using an ogive.

• The "less than" ogive is an increasing curve.

• The "more than" ogive is a decreasing curve.

• The intersection of both ogives helps determine the median.

  

  Figure 1: Cumulative Frequency Distributions

6. Applications of Cumulative Frequency

Cumulative frequency is widely used in:

• Business & Economics: Income distribution, demand forecasting

• Education: Student performance analysis

• Healthcare: Patient age distribution in hospitals

• Finance: Stock market analysis

7. Final Thoughts

Cumulative frequency is an essential statistical tool for understanding data distribution. Whether analyzing student grades, stock market trends, or customer purchases, cumulative frequency provides meaningful insights.

This blog covered the basics, calculations, numerical examples, and applications of cumulative frequency. Understanding this concept helps in making better data-driven decisions in various fields.