Interpreting Cumulative Frequency Diagrams (Cambridge (CIE) O Level Maths): Revision Note

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 $\frac{n}{2}$

    • 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 $\frac{60}{2}=30$

  • STEP 2
    Draw a horizontal line from $\frac{n}{2}$ 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 $\frac{n}{4}$

  • STEP 2
    Draw a horizontal line from $\frac{n}{4}$ 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 $\frac{3n}{4}\left(\mathrm{i}.\mathrm{e}.3\times \frac{n}{4}\right)$

  • STEP 2
    Draw a horizontal line from $\frac{3n}{4}$ 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 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 $n$ data values, this will be $\frac{np}{100}\left(\mathrm{i}.\mathrm{e}.\frac{n}{100}\times p\right)$

    • So for the 10^th percentile ($p=10$) with 60 data values ($n=60$)

    • The position is $\frac{10}{100}\times 60=\frac{1}{10}\times 60=6$

  • STEP 2
    Draw a horizontal line from $\frac{np}{100}$ 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