Cumulative Frequency Charts (Edexcel GCSE Statistics)

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

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 ( seconds)	Frequency
    	3
    	8
    	17
    	12
    	7
    	3
    Total	50

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

    Time taken ( seconds)	Frequency	Cumulative Frequency
    	3	3
    	8	3 + 8 = 11
    	17	11 + 17 = 28
    	12	28 + 12 = 40
    	7	40 + 7 = 47
    	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 ()

    • 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

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  data values, this will be 

    • 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

• STEP 2
  Draw a horizontal line from  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 data values this will be 

• STEP 2
  Draw a horizontal line from  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 data values this will be 

• STEP 2
  Draw a horizontal line from  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 50^th percentile is another way of describing the median

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

To find the p^th  percentile

• STEP 1
  Find the position of the p^th percentile

  • For  data values, this will be 

    • So for the 10^th percentile () with 60 data values ()

    • the position is

• STEP 2
  Draw a horizontal line from  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 p^th percentile