Cumulative frequency and relative frequency are two ways to analyze data in statistics. Cumulative frequency shows the running total of frequencies up to a certain point in a data set, while relative frequency indicates how often a particular value occurs compared to the total number of observations. These concepts help in understanding the distribution and proportions within a data set.
What is Cumulative Frequency?
Cumulative frequency is the total count of all data points up to a certain value in a data set. It shows how many observations fall within or below a particular category or value.
Formula for Cumulative Frequency
Cumulative Frequency to calculate the cumulative frequency for a particular value in a data set is:
Cumulative Frequency=
\sum_{i=1}^{k} f_i
Where fi is the frequency of the ith value, and k is the value you are calculating the cumulative frequency for.
Note: Cumulative Frequency for the kth value = Cumulative Frequency for the (k−1)th value + fk
What is Relative Frequency?
Relative frequency is the proportion or percentage of the total number of observations that fall into a specific category or value. It shows how often a particular outcome occurs compared to the overall number of observations.
Relative Frequency is Frequency of that value divided by the total number of observations.
Relative Frequency = Frequency of the specific event or value/Total number of observations
OR
Relative Frequency= f/n
Where f is the frequency of the value, and n is the total number of observations.
Calculation of Cumulative Frequency and Relative Frequency
Suppose given set of data has test scores as: 45, 50, 55, 60, 65, 70, 75, 80, 85, 90
Frequency Table
Score | Frequency |
|---|---|
45 | 1 |
50 | 2 |
55 | 3 |
60 | 2 |
65 | 1 |
70 | 1 |
75 | 1 |
80 | 1 |
85 | 1 |
90 | 1 |
Cumulative Frequency Table
In order to find the cumulative frequency, we add the frequencies sequentially:
Score | Frequency | Cumulative Frequency |
|---|---|---|
45 | 1 | 1 |
50 | 2 | 1 + 2 = 3 |
55 | 3 | 3 + 3 = 6 |
60 | 2 | 6 + 2 = 8 |
65 | 1 | 8 + 1 = 9 |
70 | 1 | 9 + 1 = 10 |
75 | 1 | 10 + 1 = 11 |
80 | 1 | 11 + 1 = 12 |
85 | 1 | 12 + 1 = 13 |
90 | 1 | 13 + 1 = 14 |
The cumulative frequency table shows how many scores fall up to and including each score.
Relative Frequency Table
To find the relative frequency, divide each frequency by the total number of observations. The total number of observations is 14.
Score | Frequency | Relative Frequency |
|---|---|---|
45 | 1 | 1/14 = 0.071 (7.1%) |
50 | 2 | 2/14 = 0.143 (14.3%) |
55 | 3 | 3/14 = 0.214 (21.4%) |
60 | 2 | 2/14 = 0.143 (14.3%) |
65 | 1 | 1/14 = 0.071 (7.1%) |
70 | 1 | 1/14 = 0.071 (7.1%) |
75 | 1 | 1/14 = 0.071 (7.1%) |
80 | 1 | 1/14 = 0.071 (7.1%) |
85 | 1 | 1/14 = 0.071 (7.1%) |
90 | 1 | 1/14 = 0.071 (7.1%) |
The relative frequency table shows the proportion of each score in the total data set, both as a fraction and a percentage.
Cumulative Frequency v/s Relative Frequency
Given below is a table of comparison between cumulative frequency and relative frequency based on different aspects.
Aspects | Cumulative Frequency | Relative Frequency |
|---|---|---|
Definition | The total count of all data points up to a certain value. | The proportion or percentage of the total number of observations for a specific value. |
Formula | f/n | |
Value Range | It increases or stays the same, never decreases. | It ranges between 0 and 1 (or 0% and 100%). |
Find value | Equals the total number of observations in the data set. | The sum of all relative frequencies equals 1 (or 100%). |
Purpose | It shows how many observations are below or at a certain value. | It shows how often a particular outcome occurs compared to the total observations. |
Use Case | It is useful for understanding the running total of frequencies. | It is useful for comparing different data sets or understanding proportions within a data set. |
Graphical Representation | It represented by a cumulative frequency curve or ogive. | Often represented as a bar chart or pie chart |
Calculation Process | Add frequencies sequentially to get cumulative values. | Divide each frequency by the total number of observations. |
Sensitivity to Sample Size | Does not change with the sample size; it's a running total. | Changes with sample size; a larger sample might lead to smaller relative frequencies. |
Conclusion
Cumulative frequency and relative frequency offer complementary insights into data distribution. Cumulative frequency highlights the running total of observations up to a certain point, while relative frequency shows the proportion of each value within the entire dataset. Both methods are essential for a comprehensive understanding of how data is distributed and compared.
