Frequency refers to the number of times an event or outcome occurs. The cumulative frequency at a point x is the sum of the individual frequencies up to and at point x. Typically, cumulative frequency gives us the number of times an outcome occurs that is above or below a given value.
Consider the following table:
Number of pizzas eaten in January (x) | Number of persons (frequency) |
0 | 3 |
1 | 1 |
2 | 4 |
3 | 2 |
4 | 0 |
5 | 2 |
The above frequency table can tell us how many people had a certain number of pizzas in January. For example, the number of people who had exactly 2 pizzas in January was 4. Now, suppose we want to know how many people had a maximum of 2 pizzas in January. This would be represented by the cumulative frequency at x = 2, which would be equal to the sum of the number of people who had 0, 1 and 2 pizzas in January ie. 3 +1 +4 = 8.
Cumulative frequency table
A cumulative frequency table is a very useful statistical tool to help us deal with frequencies and cumulative frequencies. To construct a cumulative frequency table from the above example just add another column for the cumulative frequency. The cumulative frequency for each value of x is equal to the sum of the frequency for that value of x and the cumulative frequency for the previous value of x (the cumulative frequency for the first value of x will be the same as the frequency).
Thus, we get the following cumulative frequency table:
Number of pizzas eaten in January (x) | Number of persons (frequency) | Cumulative frequency |
0 | 3 | 3 |
1 | 1 | 3 + 1 = 4 |
2 | 
| 4 + 4 = 8 |
3 | 2 | 8 + 2 = 10 |
| 0 | 10 + 0 = 10 |
5 | 2 | 10 + 2 = 12 |
Cumulative frequency graph
Another very commonly used tool to deal with cumulative frequencies is a cumulative frequency graph.
Let's draw the cumulative frequency graph for the above example. The value of x is represented on the x-axis and the cumulative frequency on the y-axis.
Example cumulative frequency graph, Nilabhro Datta - StudySmarter Originals
Cumulative frequency for grouped frequency distribution
In statistics, data are very often grouped into classes which represent a continuous range of values. This is a very common practice in the case of frequency distribution.
For example, consider the following frequency distribution table:
Restaurant ratings (x) | Number of restaurants (frequency) |
0.0 - 1.0 | 
|
1.0 - 2.0 | 28 |
2.0 - 3.0 | 45 |
3.0 - 4.0 | 40 |
4.0 - 5.0 | 35 |
To obtain the cumulative frequency table from the above data, we can follow the same steps that we did for the earlier example with discrete values.
Restaurant ratings (x) | Class Mark | Number of restaurants (frequency) | Cumulative frequency (y) |
0.0 - 1.0 | 0.5 | 
| 
|
1.0 - 2.0 | 1.5 | 28 | 40 |
2.0 - 3.0 | 2.5 | 45 | 85 |
3.0 - 4.0 | 3.5 | 40 | 125 |
4.0 - 5.0 | 4.5 | 35 | 160 |
Now, to create the cumulative frequency graph, we need to use the class mark for each class. The class mark is the middle value of each class. Therefore, the class mark for the class, 1.0 - 2.0 will be (1.0 + 2.0) / 2 = 1.5. Similarly, the class mark for the class, 4.0 - 5.0 will be 4.5.
Thus the cumulative frequency graph obtained will be the following:
The cumulative frequency curve for the given frequency distribution, Nilabhro Datta - Study Smart Originals
As you can see, the graph has been plotted using the respective class mark of each class (0.5, 1.5, 2.5 ...). Note that the lowest possible value is 0, therefore the graph starts from (0, 0)
For the following frequency table showing the mass of mangoes in grams, construct the cumulative frequency table and cumulative frequency graph.
Mass in grams (x) | Frequency |
50 ≤ x < 70 | 22 |
70 ≤ x < 90 | 23 |
90 ≤ x < 110 | 47 |
110 ≤ x < 130 | 18 |
130 ≤ x < 150 | 7 |
Solution
Create the resultant cumulative frequency table.
Mass in grams (x) | Class Mark | Frequency | Cumulative frequency |
50 ≤ x < 70 | 60 | 22 | 22 |
70 ≤ x < 90 | 80 | 23 | 45 |
90 ≤ x < 110 | 100 | 47 | 92 |
110 ≤ x < 130 | 120 | 18 | 110 |
130 ≤ x < 150 | 140 | 7 | 117 |
Now you can draw the corresponding cumulative frequency graph.
The cumulative frequency curve for the given grouped frequency distribution, Nilabhro Datta - Study Smarter Originals
Estimating medians, quartiles, and percentiles using cumulative frequency
In the case of grouped frequency distribution, it is usually not possible to calculate the exact values of medians, quartiles and percentiles. Using cumulative frequency graphs, it is possible to estimate these values.
Tip: The values obtained are approximations and are not going to be exact values.
Here is a rough outline of the process that you can follow to obtain the value of medians, quartiles and percentiles from a grouped frequency distribution using cumulative frequency graphs.
Steps:
1) Given a grouped frequency distribution table, obtain the cumulative frequency table.
2) On the graph, plot the points obtained from the cumulative frequency table using the upper class boundary (not the class mark), and the corresponding cumulative frequency.
3) Draw an approximate best fit curve through the plotted points.
4) Estimate the required median/quartile/percentile values from the graph. For example, in a graph plotted from a frequency distribution with 200 noted outcomes:
200/2 = 100th value is the media
200/4 = 50th value is the lower quartile and 200 × 3/4 = 150th value is the upper quartile
200 × 90/100 = 180th value is the 90th percentile and 200 × 30/100 = 60th value is the 30th percentile.
Consider the following frequency table showing the mass of mangoes in grams, construct the cumulative frequency table and cumulative frequency graph.
Estimate the value(s) of the a) median b) upper and lower quartiles c) 43rd percentile d) 85th percentile
Mass in grams (x) | Frequency |
50 ≤ x < 70 | 17 |
70 ≤ x < 90 | 23 |
90 ≤ x < 110 | 30 |
110 ≤ x < 130 | 18 |
130 ≤ x < 150 | 12 |
Solution
Create the resultant cumulative frequency table.
Mass in grams (x) | Frequency | Cumulative frequency |
50 ≤ x < 70 | 17 | 17 |
70 ≤ x < 90 | 23 | 40 |
90 ≤ x < 110 | 30 | 70 |
110 ≤ x < 130 | 18 | 88 |
130 ≤ x < 150 | 12 | 100 |
Now plot the points on a graph taking mass along the X-axis and cumulative frequency along the Y-axis, and draw the best fit curve through those points.
Example best fit cumulative frequency curve to estimate median, quartiles and percentiles, Nilabhro Datta - StudySmarter Originals
From the above graph, we can obtain our estimates for the necessary median, quartiles and percentiles.
Median = value of (100/2 = 50)th data point = 95.78
Upper quartile = value of (100 × 3/4 = 75)th data point = 115.53 Lower quartile = value of (100 × 1/4 = 25)th data point = 77.88
Value of 43rd percentile = value of (100 × 43/100 = 43)rd data point = 90.87
Value of 85th percentile = value of (100 × 85/100 = 85)th data point = 125.95
Cumulative Frequency - Key takeaways
Frequency refers to the number of times an event or outcome occurs. The cumulative frequency at a point x is the sum of the individual frequencies up to and at point x.
Two commonly used methods for representing cumulative frequency information are cumulative frequency graphs and cumulative frequency tables.
For grouped frequency distribution, it is usually not possible to calculate the exact values of medians, quartiles and percentiles. Using cumulative frequency graphs, it is possible to estimate these values.
Medians, quartile and percentile values obtained from cumulative frequency graphs are usually best approximations and not exact values.
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