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Statistics

Cumulative frequency

Cumulative Freq- Table And Graph Extended 6 min read

On this page
  1. In this note
  2. Formula bank
  3. The cumulative frequency table
  4. Reading from the curve
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 9.6: cumulative frequency
  • 9.6Make a cumulative frequency table and draw the curve.
  • 9.6Read the median, quartiles, interquartile range and percentiles from the curve.
  • 9.6Use the curve to count how many values are above or below a value.

Before you start: Grouped data. Next: Histograms.

Formula bank

Cumulative frequencyLearn itrunning total of the frequenciesplotted at the UPPER end of each class.Use it when 'cumulative frequency'.
MedianLearn itread across at \(\tfrac{n}{2}\)on a curve use n/2, not (n+1)/2.Use it when 'median' from a curve.
QuartilesLearn itLQ at \(\tfrac{n}{4}\), UQ at \(\tfrac{3n}{4}\); IQR = UQ − LQUse it when 'interquartile range'.
PercentileLearn itthe pth percentile is at \(\tfrac{p}{100}\times n\)Use it when '90th percentile'.

The cumulative frequency table

Time t (min)FrequencyCumulative frequency
\(0<t\le10\)55  (\(t\le10\))
\(10<t\le20\)1217  (\(t\le20\))
\(20<t\le30\)1835  (\(t\le30\))
\(30<t\le40\)1045  (\(t\le40\))
\(40<t\le50\)550  (\(t\le50\))

General methodHow to draw a cumulative frequency curve

  1. Keep a running total of the frequencies.
  2. Plot each running total at the UPPER bound of its class: (10, 5), (20, 17), (30, 35)…
  3. Start at the lowest bound with 0: (0, 0).
  4. Join the crosses with a smooth curve (an S shape).

Reading from the curve

Cumulative frequency curve of 50 times, with the median and quartiles read off10203040501020304050tcf
Across from 25 (median), 12.5 (LQ) and 37.5 (UQ), then down.

With n = 50: the median is across from 25, the lower quartile from 12.5, the upper quartile from 37.5. Reading down: median ≈ 24, LQ ≈ 17, UQ ≈ 32, so the IQR ≈ 15. Exam answers allow a small range either side of the exact reading.

General methodHow to read any value from a cumulative frequency curve

  1. Median / quartile / percentile: work out the position (n/2, n/4, 3n/4, p% of n) on the vertical axis.
  2. Go ACROSS to the curve, then DOWN to the horizontal axis. Read the value.
  3. 'How many less than x?': go UP from x to the curve, then ACROSS. 'More than x?': subtract that from n.

Example How many of the 50 times are more than 35 minutes?

Up from 35, across: about 41 are 35 or less.

  1. \(50-41=9\)

Answer about 9

Try it yourself

1
Use the curve: estimate the 90th percentile of the times.
Show answer

Answer 40

2
Use the curve: how many times are 15 minutes or less?
Show answer

Answer about 10

Ways they test it

Type 1 Complete the table

"Complete the cumulative frequency table." (1 mark)

Spot it a frequency table and a cf column.

  1. Running totals.

Type 2 Draw the curve

"Draw the cumulative frequency diagram." (3 marks)

Spot it a grid.

  1. Upper bounds; crosses.
  2. Smooth curve.

Type 3 Median / IQR

"Use your diagram to find the median and the interquartile range." (3 marks)

Spot it 'use your diagram'.

  1. n/2, n/4, 3n/4 across.
  2. Down; IQR = UQ − LQ.

For example n = 50

  1. across from 12.5 and 37.5

Answer IQR ≈ 15

Type 4 Above a value

"Find how many took more than 35 minutes." (2 marks)

Spot it 'more than'.

  1. Up from 35, across.
  2. n − that.

Type 5 Percentile

"Find the 80th percentile." (2 marks)

Spot it 'percentile'.

  1. 80% of n across.
  2. Down.

Mistakes to avoid

  • ✗ Plotting at the midpoint✓ plot at the UPPER bound
  • ✗ 'More than 35' read straight from the curve✓ n − the readingThe curve counts values BELOW.
  • ✗ IQR given as one quartile✓ UQ − LQ
  • ✗ Joining with straight lines when a curve is asked for✓ smooth curve

Summary

  • Running totals plotted at upper bounds, joined by a smooth curve.
  • Median at n/2, LQ at n/4, UQ at 3n/4; IQR = UQ − LQ.
  • Below x: up and across. Above x: n minus that.

If you see …

  • 'use your diagram'across then down (or up then across).

Practice

Practice: easy, medium, hard

Easy

1
Frequencies 5, 12, 18, 10, 5. Third cumulative frequency?
Show answer

Answer 35

2
n = 50. Position of the lower quartile on the cf axis?
Show answer

Answer 12.5

3
n = 80. Position of the 90th percentile?
Show answer

Answer 72

Medium

1
Use the curve: median time (to the nearest minute)?
Guide me

What is it asking? a reading

What do we know? across from 25

Plan down to t

Show working
  1. ≈ 24

Answer 24

2
Use the curve: upper quartile (nearest minute)?
Guide me

What is it asking? a reading

What do we know? across from 37.5

Plan down

Show working
  1. ≈ 32

Answer 32

3
Use the curve: lower quartile (nearest minute)?
Guide me

What is it asking? a reading

What do we know? across from 12.5

Plan down

Show working
  1. ≈ 17

Answer 17

Hard

1
Use the curve: interquartile range?
Guide me

What is it asking? a spread

What do we know? UQ ≈ 32, LQ ≈ 17

Plan subtract

Show working
  1. ≈ 15

Answer about 15

PartAnswer / workingMarksGuidance
UQ and LQ readM1
13 to 17A1
Total: 2 marks
2
Use the curve: how many times are more than 35 minutes?
Guide me

What is it asking? a count

What do we know? up from 35: about 41

Plan 50 − 41

Show working
  1. ≈ 9

Answer about 9

PartAnswer / workingMarksGuidance
reading at 35M1
8 to 10A1
Total: 2 marks
3
Use the curve: 90th percentile?
Guide me

What is it asking? a reading

What do we know? 90% of 50 = 45

Plan across from 45, down

Show working
  1. ≈ 40

Answer 40

PartAnswer / workingMarksGuidance
45 usedM1
39 to 41A1
Total: 2 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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