Interpreting Cumulative Frequency Diagrams (Cambridge International IGCSE Mathematics)

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 $\frac{3 n}{4} ( \mathrm{i} . \mathrm{e} . 3 \times \frac{n}{4} )$

  • 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^thpercentile is another way of describing the median
  • The 25^th and 75^thpercentiles 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 $\frac{n p}{100} ( \mathrm{i} . \mathrm{e} . \frac{n}{100} \times p )$
    • So for the 10^th percentile ($p=10$) with 60 data values ($n=60$)
    • The position is $\frac{10}{100}\times60=\frac{1}{10}\times60=6$

  • 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

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.

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

Answer:

There are 100 pieces of data, so $n=100$

$\begin{gathered} \frac{n}{2} & = \frac{100}{2} = 50 \\ \frac{n}{4} & = \frac{100}{4} = 25 \\ \frac{3 n}{4} & = 3 \times 25 = 75 \end{gathered}$

So the median is the 50^th value The lower quartile is the 25^th value The upper quartile is the 75^th 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

Final answer: 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.

Answer:

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)

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

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