Cumulative Frequency

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:

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 i.e. \(3 +1 +4 = 8\).

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:

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:

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.

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 \(\frac{1.0 + 2.0}{2} = 1.5\). 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)

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.

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:

• \(\frac{200}{2}\) = 100^th value is the media

• \(\frac{200}{4}\) = 50^th value is the lower quartile and \(200 \cdot \frac{3}{4}\) = 150^th value is the upper quartile

• \(200 \cdot \frac{90}{100}\) = 180^th value is the 90th percentile and \(200 \cdot \frac{30}{100}\) = 60^th value is the 30^th percentile.