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Maths · Further statistics

Cumulative frequency

Build cumulative frequency tables, plot cumulative frequency graphs at the upper class boundaries, and read off the median, quartiles and interquartile range.

  • 6 key terms
  • All boards
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Warm-up

Answer each one, then check.

  1. 1

    What is the median?

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    The middle value when the data are in order

  2. 2

    What is the range?

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    Largest minus smallest

  3. 3

    Running total of 4, 11, 20?

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    \(4,\ 15,\ 35\)

  4. 4

    What is a grouped frequency table?

    Show answerHide answer

    A table showing how many values fall in each class

  5. 5

    Write the class \(20 < t \le 30\) in words.

    Show answerHide answer

    More than 20 and up to and including 30

Learning Objectives

  1. 1Work out cumulative frequencies from a grouped table.
  2. 2Plot a cumulative frequency graph.
  3. 3Estimate the median, lower quartile and upper quartile.
  4. 4Find the interquartile range and answer questions from the graph.

CUMULATIVE FREQUENCY

Cumulative frequency is a running total of the frequencies. Plot each total against the upper class boundary.

Points are joined with a smooth curve that always rises.

Building the Table

Time taken (t minutes) to complete a puzzle, 60 students.

  • \(0 < t \le 10\)

    Frequency: \(4\). Cumulative frequency: \(4\)

  • \(10 < t \le 20\)

    Frequency: \(11\). Cumulative frequency: \(15\)

  • \(20 < t \le 30\)

    Frequency: \(20\). Cumulative frequency: \(35\)

  • \(30 < t \le 40\)

    Frequency: \(16\). Cumulative frequency: \(51\)

  • \(40 < t \le 50\)

    Frequency: \(7\). Cumulative frequency: \(58\)

  • \(50 < t \le 60\)

    Frequency: \(2\). Cumulative frequency: \(60\)

Reading Quartiles

Use the total frequency \(n\).

  1. 1 Median

    Go across at \(\tfrac{n}{2}\), then down to the x-axis
  2. 2 Lower quartile

    Go across at \(\tfrac{n}{4}\), then down
  3. 3 Upper quartile

    Go across at \(\tfrac{3n}{4}\), then down
  4. 4 Interquartile range

    \(\text{IQR} = \text{UQ} - \text{LQ}\)

Median and Quartiles

For the 60 students, estimate the median, the quartiles and the interquartile range.

Show the solutionHide the solution
  1. 1 Total \(n = 60\)
  2. 2 Median At 30: about \(27.5\) minutes
  3. 3 Lower quartile At 15: \(20\) minutes
  4. 4 Upper quartile At 45: about \(36\) minutes
  5. 5 Interquartile range \(36 - 20 = 16\) minutes

AnswerMedian \(\approx 27.5\) minutes, IQR \(\approx 16\) minutes.

How Many More Than...?

Use the graph to estimate how many of the 60 students took more than 40 minutes.

Show the solutionHide the solution
  1. 1 Cumulative frequency at 40 \(51\) students took 40 minutes or less
  2. 2 Subtract from the total \(60 - 51 = 9\)

AnswerAbout 9 students took more than 40 minutes.

Common Mistakes

Watch for these.

  • Wrong boundary

    Plot at the upper class boundary, not the middle of the class.

  • Straight lines

    The graph is a smooth curve, not a polygon (unless the question says so).

  • Reading frequency, not value

    Go across from the cumulative frequency, then down to the data value.

  • Forgetting the start

    The curve starts at the lowest boundary with a cumulative frequency of 0.

Build a Cumulative Frequency Table

Heights (h cm) of 40 plants: \(0 < h \le 10\): 3, \(10 < h \le 20\): 9, \(20 < h \le 30\): 16, \(30 < h \le 40\): 8, \(40 < h \le 50\): 4. (a) Complete the cumulative frequency column. (b) At what value of \(n\) do you read the median?

1. Add each frequency to the running total.

2. The last total equals the total frequency.

A good answer shows: (a) 3, 12, 28, 36, 40. (b) \(\tfrac{40}{2} = 20\), which lies in the class \(20 < h \le 30\).

Can I...?

  1. 1Find cumulative frequencies.
  2. 2Plot at upper class boundaries.
  3. 3Draw a smooth curve.
  4. 4Read the median.
  5. 5Read the quartiles.
  6. 6Find the IQR.
  7. 7Find how many are above or below a value.
  8. 8Avoid common mistakes.

Summary & Exam Focus

  • Plot cumulative frequency against upper class boundaries.
  • Median at \(\tfrac{n}{2}\), LQ at \(\tfrac{n}{4}\), UQ at \(\tfrac{3n}{4}\).
  • \(\text{IQR} = \text{UQ} - \text{LQ}\).
  • Values read from a graph are estimates.

Exam focus

The cumulative frequency graph shows the times taken by 80 students. Use it to estimate the median and the interquartile range. (4 marks) (4 marks)

Show your dashed lines on the graph and write the values down.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Cumulative frequency
A running total of frequencies.
Upper class boundary
The largest value in a class.
Median
The middle value; the value at \(\tfrac{n}{2}\).
Lower quartile
The value one quarter of the way through the data.
Upper quartile
The value three quarters of the way through the data.
Interquartile range
Upper quartile minus lower quartile.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Use the graph 4 marks

    The cumulative frequency graph shows the times, in minutes, taken by 80 students to complete a puzzle. Use the graph to find an estimate for (a) the median (b) the interquartile range.

    A cumulative frequency curve for 80 students with time up to 60 minutes.
    Show answerHide answer

    Model answer

    (a) Median at 40: about 30 minutes. (b) LQ at 20: about 21; UQ at 60: about 38; IQR about 17. Accept readings within 1 minute.

    Mark scheme

    • Median: reads across at 40 — M1
    • 30 (accept 29 to 31) — A1
    • LQ and UQ read at 20 and 60 — M1
    • IQR about 17 — A1
  2. Question 2 Use the graph 2 marks

    Use the same graph of the times taken by 80 students to estimate how many students took more than 45 minutes.

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    Model answer

    About 72 students took 45 minutes or less, so \(80 - 72 = 8\) took more. Accept 7 to 9.

    Mark scheme

    • Reads about 72 at 45 — M1
    • \(80 - 72 = 8\) — A1
  3. Question 3 Complete 2 marks

    The table shows the times taken by 60 students. Frequencies: \(0 < t \le 10\): 4, \(10 < t \le 20\): 11, \(20 < t \le 30\): 20, \(30 < t \le 40\): 16, \(40 < t \le 50\): 7, \(50 < t \le 60\): 2. Work out the cumulative frequencies.

    Show answerHide answer

    Model answer

    4, 15, 35, 51, 58, 60

    Mark scheme

    • At least 3 correct — M1
    • All correct — A1
  4. Question 4 Explain 2 marks

    Explain why the points on a cumulative frequency graph are plotted at the upper class boundaries.

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    Model answer

    The cumulative frequency counts everyone up to the end of each class, so the running total is only known at the upper boundary.

    Mark scheme

    • Cumulative frequency includes the whole class — M1
    • Only true at the upper boundary — C1
  5. Question 5 Draw 3 marks

    Describe how to draw a cumulative frequency graph from a table of grouped data.

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    Model answer

    Work out the running totals. Plot each total against the upper class boundary, plus a point at 0 for the lowest boundary. Join the points with a smooth curve.

    Mark scheme

    • Running totals — B1
    • Plot at upper boundaries — B1
    • Smooth curve — B1
  6. Question 6 Work out 2 marks

    A cumulative frequency graph shows the lower quartile is 20 and the upper quartile is 36. Work out the interquartile range and explain what it tells you.

    Show answerHide answer

    Model answer

    \(36 - 20 = 16\). The middle half of the data lie within a range of 16.

    Mark scheme

    • 16 — B1
    • The spread of the middle 50% — C1

Quick check

  1. A cumulative frequency graph is plotted at the...

    1. AMiddle of each class
    2. BUpper class boundary
    3. CLower class boundary
    4. DClass width
    Show answerHide answer

    B: Upper class boundary

    Upper class boundary.

  2. For 100 values, the median is found at cumulative frequency...

    1. A25
    2. B75
    3. C50
    4. D100
    Show answerHide answer

    C: 50

    \(\tfrac{100}{2} = 50\).

  3. For 80 values, the lower quartile is found at...

    1. A20
    2. B40
    3. C60
    4. D80
    Show answerHide answer

    A: 20

    \(\tfrac{80}{4} = 20\).

  4. The interquartile range is...

    1. AMedian minus LQ
    2. BMaximum minus minimum
    3. CUQ plus LQ
    4. DUQ minus LQ
    Show answerHide answer

    D: UQ minus LQ

    Upper quartile minus lower quartile.

  5. The last cumulative frequency in a table equals...

    1. AThe median
    2. BThe total frequency
    3. CThe largest class
    4. DZero
    Show answerHide answer

    B: The total frequency

    The total frequency.

  6. The graph of cumulative frequency is always...

    1. AFalling
    2. BA straight line
    3. CRising (or level)
    4. DSymmetrical
    Show answerHide answer

    C: Rising (or level)

    A running total can only stay the same or go up.

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