Cumulative Frequency Charts (Edexcel GCSE Statistics)

Revision Note

Cumulative Frequency

What is cumulative frequency?

  • Cumulative refers to a “running total" or "adding up as you go along”

  • So in a table of grouped data

    • cumulative frequency means all of the frequencies for the different groups totalled up to the end of the group in a given row

  • When working out cumulative frequencies you may see tables presented in two ways

    • A regular grouped data table with an extra column for cumulative frequencies

      • e.g. rows labelled 0 ≤ x < 20, 20 ≤ x < 40, 40 ≤ x < 60, etc

        frequency

        cumulative frequency

        0 ≤ x < 20

        14

        14

        20 ≤ x < 40

        25

        39

        (because 14+25=39)

        40 ≤ x < 60

        29

        68

        (because 14+25+29=68)

        60 ≤ x < 80

        12

        80

        (because 14+25+29+12=80)

    • or a separate table where every group is relabelled as starting at the beginning (often zero)

      • e.g. rows labelled 0 ≤ x < 20, 0 ≤ x < 40, 0 ≤ x < 60, etc.

      • or x < 20, x < 40, x < 60, etc.

        cumulative frequency

        0 ≤ x < 20

        (or x < 20)

        14

        0 ≤ x < 40

        (or x < 40)

        39

        0 ≤ x < 60

        (or x < 60)

        68

        0 ≤ x < 80

        (or x < 80)

        80

    • In the second type of table, you can subtract to find the individual frequencies

      • e.g. the frequency of the 20 ≤ x < 40 class interval is 39-14=25

      • the frequency of the 40 ≤ x < 60 class interval is 68-39=29

      • etc.

Cumulative Frequency Step Polygons

What is a cumulative frequency step polygon?

  • A cumulative frequency step polygon is a way of representing discrete data

    • For grouped continuous data a cumulative frequency diagram would be used instead

How do I draw a cumulative frequency step polygon?

  • This is best explained with an example

    • The numbers of eggs found in each nest in a survey of 50 American alligator nests are shown in the table below:

      Number of eggs

      Frequency

      29

      5

      30

      6

      31

      11

      32

      15

      33

      7

      34

      6

      Total

      50

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

      Number of eggs

      Frequency

      Cumulative Frequency

      29

      5

      5

      30

      6

      5 + 6 = 11

      31

      11

      11 + 11 = 22

      32

      15

      22 + 15 = 37

      33

      7

      37 + 7 = 44

      34

      6

      44 + 6 = 50

      Total

      50

  • We can now draw the cumulative frequency step polygon

    • The cumulative frequency will always go on the vertical axis

      • The values in the data set will appear along the horizontal axis

    • The cumulative frequency is zero until we get to 29 eggs

      • So start at the point (29, 0)

    • Then the cumulative frequency jumps up to 5

      • Draw a vertical line from (29, 0) to (29, 5) to show this jump

    • Nothing changes until we get to 30 eggs

      • Draw a horizontal line from (29, 5) to (30, 5) to show this

    • At 30 eggs the cumulative frequency jumps up to 11

      • Draw a vertical line from (30, 5) to (30, 11) to show this jump

    • Continue the same way through the rest of the values in the table

      • The graph will end at the point (34, 50)

  • Here is the final cumulative frequency diagram for the numbers of eggs

An example of a cumulative frequency step polygon

Cumulative Frequency Diagrams

What is a cumulative frequency diagram?

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

    • For discrete data a cumulative frequency step polygon would be used instead

  • 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 less or equal than s less than 30

      3

      30 less or equal than s less than 35

      8

      35 less or equal than s less than 40

      17

      40 less or equal than s less than 45

      12

      45 less or equal than s less than 50

      7

      50 less or equal than s less than 55

      3

      Total

      50

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

      Time taken (s seconds)

      Frequency

      Cumulative Frequency

      25 less or equal than s less than 30

      3

      3

      30 less or equal than s less than 35

      8

      3 + 8 = 11

      35 less or equal than s less than 40

      17

      11 + 17 = 28

      40 less or equal than s less than 45

      12

      28 + 12 = 40

      45 less or equal than s less than 50

      7

      40 + 7 = 47

      50 less or equal than s less than 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 less or equal than s less than 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

      • so plot the point (25, 0)

    • Join points up with a smooth curve (this takes some practice), or by drawing straight lines from each point to the next one (use a ruler)

      • If you draw a curve, make sure it goes through all of the marked points

      • It is usually easier to draw straight lines

      • You will get full marks for either version

    • 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

Example of a cumulative frequency diagram

Interpreting Cumulative Frequency Diagrams

How do I use and interpret a cumulative frequency diagram?

  • A cumulative frequency diagram provides a way to estimate key facts about the data

    • median 

    • lower and upper quartiles (and interquartile range)

    • percentiles

  • These values will be estimates as the original raw data is unknown

    • Cumulative frequency diagrams are used with grouped data 

    • Points are joined by a smooth curve or by straight lines

      • This means the data is assumed to be smoothly spread out over each interval

  • The median and quartiles are also key features of a box plot

    • It is possible to draw a box plot from a cumulative frequency diagram

    • This can make it easier to compare two data sets

How do I find the median, lower quartile and upper quartile from a cumulative frequency diagram?

  • This is all about understanding how many data values are represented by the cumulative frequency diagram

    • This may be stated in words within the question

    • If not, it will be the highest value on the frequency (y-) axis that the curve on the diagram reaches

      • This should be "top right" of the curve on a cumulative frequency diagram

Median

  • STEP 1
    Find the position of the median

    • For n data values, this will be n over 2

      • This is different from finding the median from a set of data values

      • e.g. for a list of 60 data values the median would be halfway between the 30th and 31st values

      • But for a cumulative frequency diagram it would just be 60 over 2 equals 30

  • STEP 2
    Draw a horizontal line from n over 2 on the cumulative frequency (y-) axis until it hits the curve

  • STEP 3
    Draw a vertical line from that point on the curve down to the horizontal (x-) axis

    • The value where that line hits the horizontal axis will be the median

Lower quartile

  • STEP 1
    Find the position of the lower quartile

    • For n data values this will be n over 4

  • STEP 2
    Draw a horizontal line from n over 4 on the cumulative frequency axis until it hits the curve

  • STEP 3
    Draw a vertical line from that point on the curve down to the horizontal (x-) axis

    • The value where that line hits the horizontal axis will be the lower quartile

Upper quartile

  • STEP 1
    Find the position of the upper quartile

    • For n data values this will be fraction numerator 3 n over denominator 4 end fraction space open parentheses straight i. straight e. space space 3 cross times n over 4 close parentheses

  • STEP 2
    Draw a horizontal line from fraction numerator 3 n over denominator 4 end fraction on the cumulative frequency axis until it hits the curve

  • STEP 3
    Draw a vertical line from that point on the curve to the horizontal (x-) axis

    • The value where that line hits the horizontal axis will be the upper quartile

How do I find a percentile from a cumulative frequency diagram?

  • Percentiles split the data into 100 parts

    • So the 50th percentile is another way of describing the median

    • The 25th and 75th percentiles are the same as the lower and upper quartiles (respectively)

To find the pth  percentile

  • STEP 1
    Find the position of the pth percentile

    • For n data values, this will be fraction numerator n p over denominator 100 end fraction space open parentheses straight i. straight e. space space n over 100 cross times p close parentheses

      • So for the 10th percentile (p equals 10) with 60 data values (n equals 60)

      • the position is 10 over 100 cross times 60 equals 1 over 10 cross times 60 equals 6

  • STEP 2
    Draw a horizontal line from fraction numerator n p over denominator 100 end fraction on the cumulative frequency axis until it hits the curve

  • STEP 3
    Draw a vertical line from that point on the curve down to the horizontal (x-) axis

    • The value where that line hits the horizontal axis will be the pth percentile

Worked Example

A company is investigating the length of telephone calls customers make to its help centre.
The company randomly selects 100 phone calls from a particular day.
The results are displayed in the cumulative frequency diagram below.

CF2 Length of phone calls, IGCSE & GCSE Maths revision notes


(a) Estimate the median, the lower quartile and the upper quartile.

There are 100 pieces of data, so n equals 100.

table row cell n over 2 end cell equals cell 100 over 2 equals 50 end cell row cell n over 4 end cell equals cell 100 over 4 equals 25 end cell row cell fraction numerator 3 n over denominator 4 end fraction end cell equals cell 3 cross times 25 equals 75 end cell end table

So the median is the 50th value, the lower quartile is the 25th value and the upper quartile is the 75th value

Draw horizontal lines from these on the cumulative frequency axis until they hit the curve
Then draw vertical lines down to the time of calls axis and take readings

Q1a-Finding-Q1,Q2,Q3

Median = 6.2 minutes (6 m 12 s)
Lower quartile = 4.2 minutes (4 m 12 s)
Upper quartile = 8.2 minutes (8 m 12 s)

There is no need to convert to minutes and seconds unless the question asks you to
However, writing 6 m 2 s or 6 m 20 s would be incorrect
 

(b) The company is thinking of putting an upper limit of 12 minutes on calls to its help centre.
Estimate the number of these 100 calls that would have been beyond this limit.

Draw a vertical line up from 12 minutes on the time of calls axis until it hits the curve
Then draw a horizontal line across to the cumulative frequency axis and take a reading (in this case, 90)

Q1b 12min Cut Off, IGCSE & GCSE Maths revision notes

This tells us that up to 12 minutes, 90 of the calls had been accounted for

The question wants the number of calls that were greater than 12 minutes so subtract this from the total of 100

100 - 90 = 10

Approximately 10 (out of 100) calls were beyond the 12 minute limit

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Roger B

Author: Roger B

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Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

Dan Finlay

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Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.