Cumulative Frequency is a running total of values (frequencies) through the classes in a frequency distribution.
It helps us to generate cumulative graphs.
It is useful in finding Median, Quartiles and Percentiles.
Watch: Cumulative Frequency
How to Draw a Cumulative Frequency Diagram
Solved Example
Following is a table showing the marks of students from a school, in a test, draw the cumulative frequency diagram.
SOLUTION
1
Work out the Cumulative Frequency of each class:
2
Now, we need to draw the graph. So, add scales on the graph and show the Cumulative frequency on the vertical axis.
3
Plot value on graph at the end point of each interval.
4
Join the points with straight line or smooth curve.
How to Find Median and Quartiles from a Cumulative Frequency Diagram?
Solved Example
The following is a Cumulative frequency diagram. Find the median and the Upper and lower quartiles.
SOLUTION
We know median is the value at half of the total, thus we find the half and plot the value on graph to find the median.
In a similar way we do find the Upper Quartile and Lower Quartile’s values.
Median = $\frac{12}{2} =$ 6th value
Lower Quartile = $\frac{12}{4} =$ 3rd value
Upper Quartile = $3\left(\frac{12}{4}\right) =$ 9th value
Final Answer: Median = $3.2$, Lower Quartile = $2.6$, Upper Quartile = $4$
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The Frequency table shows the masses of 50 cats copy and complete the frequency table:
Mass (kg)
Frequency
$3 < m \le 4$
4
$4 < m \le 5$
12
$5 < m \le 6$
17
$6 < m \le 7$
10
$7 < m \le 8$
7
SOLUTION
Mass (kg)
CF
$3 < m \le 4$
4
$4 < m \le 5$
$4 + 12 = 16$
$5 < m \le 6$
$16 + 17 = 33$
$6 < m \le 7$
$33 + 10 = 43$
$7 < m \le 8$
$43 + 7 = 50$
Solved Example
The frequency table shows the amount of pocket money for 40 teenagers:
Pocket Money $x$ (£)
CF
$0 < x \le 1$
2
$0 < x \le 2$
8
$0 < x \le 3$
19
$0 < x \le 4$
32
$0 < x \le 5$
39
$0 < x \le 6$
40
(a) Draw a Cumulative Frequency diagram for the table.
(b) Use the Cumulative frequency diagram to estimate the median pocket money –
(c) Estimate the range from the cumulative frequency diagram
SOLUTION
(a) To draw a Cumulative frequency diagram for the table, we select a scale and plot the CF on the Horizontal axis.
(b) Use the Cumulative frequency diagram to estimate the median pocket money –
Total Students = $40$
Half of total = $\frac{40}{2} = 20$
Median = 20th value
Now, plot the half on the graph to get the median of pocket money
As we can see on the graph that it points to CF 3, and hence the median of pocket money the teenagers get is 3£.
Median = 3£
(c) Estimate the range from the cumulative frequency diagram
Highest Possible Value = 6£
Lowest Possible Value = 0£
Range = 6£ – 0£
Final Answer: Range = 6£
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