Cumulative frequency is defined as a running total of frequencies. The frequency of an element in a set refers to how many of that element there are in the set. Cumulative frequency can also defined as the sum of all previous frequencies up to the current point.
The cumulative frequency is important when analyzing data, where the value of the cumulative frequency indicates the number of elements in the data set that lie below the current value. The cumulative frequency is also useful when representing data using diagrams like histograms.
Cumulative Frequency Table
The cumulative frequency is usually observed by constructing a cumulative frequency table. The cumulative frequency table takes the form as in the example below.
Example 1: The set of data below shows the ages of participants in a certain summer camp. Draw a cumulative frequency table for the data.
Solution: The cumulative frequency at a certain point is found by adding the frequency at the present point to the cumulative frequency of the previous point.
The cumulative frequency for the first data point is the same as its frequency since there is no cumulative frequency before it.
Cumulative Frequency Graph (Ogive):-
A cumulative frequency graph, also known as an Ogive, is a curve showing the cumulative frequency for a given set of data. The cumulative frequency is plotted on the y-axis against the data which is on the x-axis for un-grouped data. When dealing with grouped data, the Ogive is formed by plotting the cumulative frequency against the upper boundary of the class. An Ogive is used to study the growth rate of data as it shows the accumulation of frequency and hence its growth rate.
Example 2: Plot the cumulative frequency curve for the data set below
Percentiles:- A percentile is a certain percentage of a set of data. Percentiles are used to observe how many of a given set of data fall within a certain percentage range; for example; a thirtieth percentile indicates data that lies the 13% mark of the entire data set.
Let designate a percentile as Pm where m represents the percentile we're finding, for example for the tenth percentile, m} would be 10. Given that the total number of elements in the data set is N
Quartiles:- The term quartile is derived from the word quarter which means one fourth of something. Thus a quartile is a certain fourth of a data set. When you arrange a date set increasing order from the lowest to the highest, then you divide this data into groups of four, you end up with quartiles. There are three quartiles that are studied in statistics.
Calculating the Different Quartiles
The different quartiles can be calculated using the same method as with the median.
Calculating Quartiles from Cumulative Frequency
As mentioned above, we can obtain the different quartiles from the Ogive, which means that we use the cumulative frequency to calculate the quartile.
Given that the cumulative frequency for the last element in the data set is given as fc, the quartiles can be calculated as follows:
The quartile is then located by matching up which element has the cumulative frequency corresponding to the position obtained above.
Example 3: Find the First, Second and Third Quartiles of the data set below using the cumulative frequency curve.
From the Ogive, we can see the positions where the quartiles lie and thus can approximate them as follows
Q1 = 11.5
Q2 = 14.5
Q3 = 15.5
Interquartile Range:- The interquartile range is the difference between the third quartile and the first quartile.
Interquartile range = Q3 - Q1