Drawing Cumulative Frequency Diagrams

Drawing Cumulative Frequency Diagramsvideo

Drawing cumulative frequency diagrams

What is a cumulative frequency diagram?

• A cumulative frequency diagram is a way of representing grouped continuous data

• A cumulative frequency diagram can be used to estimate other statistical values

  • For example the median, quartiles or percentiles

How do I draw a cumulative frequency diagram?

• This is best explained with an example

  • The times taken to complete a short general knowledge quiz taken by 50 students are shown in the table below:

    Time taken ($s$ seconds)	Frequency
    $25\le s<30$	3
    $30\le s<35$	8
    $35\le s<40$	17
    $40\le s<45$	12
    $45\le s<50$	7
    $50\le s<55$	3
    Total	50

  • Then the cumulative frequency is the running total of the frequencies

    Time taken ($s$ seconds)	Frequency	Cumulative Frequency
    $25\le s<30$	3	3
    $30\le s<35$	8	3 + 8 = 11
    $35\le s<40$	17	11 + 17 = 28
    $40\le s<45$	12	28 + 12 = 40
    $45\le s<50$	7	40 + 7 = 47
    $50\le s<55$	3	47 + 3 = 50
    Total	50

• We can now draw the cumulative frequency diagram

  • The most important part is that cumulative frequency is plotted against the end (upper bound) of the class interval

    • The end of the class interval is the x-coordinate

    • The cumulative frequency is the y-coordinate

    • For the above example the first two points to plot would be (30, 3) and (35, 11)

  • To explain this, consider the second row ($30\le s<35$)

    • The 8 students in this group could have taken any time between 30 and 35 seconds

    • They cannot all be guaranteed to have been accounted for until we reach 35 seconds

  • Once all points from the table are plotted, a point for the start needs to be added

    • This will be at the lowest time from the table

    • i.e. at 25 seconds with a cumulative frequency of 0

    • Plot the point (25, 0)

  • Join points up with a smooth curve (this takes some practice)

    • Make sure it goes through all of the marked points

  • In general a cumulative frequency diagram has a stretched-S-shape appearance

    • A cumulative frequency diagram will never come back towards the x-axis

• Here is the final cumulative frequency diagram for the quiz times