How to work out the cumulative frequencies and draw the cumulative frequency graph.

If you start with the first frequency in the table, and then keep adding on the next frequency onto this start frequency until you reach the final frequency, then you will have the worked out the cumulative frequencies. It's referred to as a running total.

Once the c.f's have been worked out you can draw out the graph. Make sure you plot the upper limits (end points) of each group - you must not plot the midpoints of the groups.

In examination questions you might be asked to work out some key values from the curve:

Lower Quartile (Q1) = a quarter of the way up the curve
Median (Q2) = a half up the the curve
Upper Quartile (Q3) = three quarters up the curve

You can also use the upper quartile and lower quartile to find the inter quartile range:

Inter Quartile Range = Upper Quartile - Lower Quartile

Example 1 - Test Scores

Cumulative Frequency Table On Test Scores

The frequency table shows the test scores of a group of 40 students. Draw a cumulative frequency graph and work out the L.Q, Median, U.Q and the I.Q.R.

First you will need to calculate the cumulative frequencies before the curve can be drawn:

2 people scored up to 20% (2)
15 had a total score of 40% or less (2 + 13)
35 people achieved a score of up to 60% (2 + 13 + 20)
39 scored 80% or less (2 + 13 + 20 + 4)
40 people achieved 100% or less (2 + 13 + 20 + 4 + 1). This is the total amount of students who took part in this test.

Working out the key values from the graph

Once these cumulative frequencies have been calculated you can now draw out the graph. Remember, to plot the end values (upper limits) of each group with the c.f.

So plot your points at (20,2), (40,15), (60,35), (80,39) and (100,40) and join up the points.

Cumulative frequency polygon

All you need to do now is work out the key values:

Lower Quartile = ¼ of 40 (since 40 people took the test) = 10th person up the curve = 32%

Median = ½ of 40 = 20th person up the curve = 45%

Upper Quartile = ¾ of 40 = 30th person up the curve = 55%

cumulative frequency polygon with quartiles

If you found this article useful then check out my other math related pages. You will find many mathematical topics from the fields of algebra, number, geometry and statistics. All of these articles have been written in an easy to understand way. You will be shown easy to follow methods for solving simple equations, calculating averages from real life data, solving basic and advanced number problems involving fractions, percentages and ratio.

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