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Interpreting Cumulative Frequency Diagrams (Cambridge (CIE) IGCSE International Maths)

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Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Cambridge (CIE) International Maths: ExtendedEdexcel Maths A (Modular): Higher Unit 2Edexcel Maths A: HigherCambridge (CIE) Maths: Extended

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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

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

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

  • Finding the 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

  • Finding the 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

  • Finding the 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

    • 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.

Cumulative frequency diagram for the length of phone calls.

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
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

Cumulative frequency diagram for the length of phone calls with the lower quartile, median and upper quartile marked on.

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)

Cumulative frequency diagram for the length of phone calls with a vertical line from the x-axis at x=12 leading to a value on the y axis of y=90.

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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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths

    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

    Author: Dan Finlay

    Expertise: Maths Lead

    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.