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Statistics

Statistics

Grouped data and comparing

Mode Median Mean- Range Table Extended 6 min read

On this page
  1. In this note
  2. Formula bank
  3. The mean of grouped data
  4. Comparing two sets of data
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 9.2 and 9.3: grouped data and comparing
  • 9.3Estimate the mean of grouped data using midpoints.
  • 9.3Find the modal class and the class that contains the median.
  • 9.2Compare two sets of data with an average and a measure of spread.

Before you start: Averages and range. Next: Charts and scatter diagrams.

Formula bank

MidpointLearn it\(x=\dfrac{\text{lower}+\text{upper}}{2}\)the value that stands for the whole class.Use it when grouped data.
Estimated meanLearn it\(\dfrac{\sum fx}{\sum f}\)an ESTIMATE: we don't know the exact values.Use it when 'calculate an estimate of the mean'.
Modal classLearn itthe class with the highest frequencyUse it when 'modal class'.
Median classLearn itthe class holding the \(\tfrac{n+1}{2}\)th valueUse it when 'which class contains the median'.

The mean of grouped data

When data are grouped (\(10<t\le20\)) you don't know the exact values. Pretend every value in a class sits at its midpoint; then the mean is only an estimate.

Time t (min)Frequency fMidpoint xf × x
\(0<t\le10\)5525
\(10<t\le20\)1215180
\(20<t\le30\)1825450
\(30<t\le40\)1035350
\(40<t\le50\)545225

General methodHow to estimate the mean of grouped data

  1. Add a midpoint column: (lower + upper) ÷ 2.
  2. Add an f × x column.
  3. Estimated mean \(=\sum fx\div\sum f\). Check: it must lie inside the range of the data.

Example Estimate the mean time from the table.

Midpoints 5, 15, 25, 35, 45.

  1. \(\sum fx=25+180+450+350+225=1230\)
  2. \(\sum f=50\)
  3. \(1230\div50\)

Answer 24.6 minutes

The modal class is \(20<t\le30\) (frequency 18). The median is the 25.5th value: the running total is 5, 17, 35…, so it lies in \(20<t\le30\).

Try it yourself

1
Classes \(0<m\le20\), \(20<m\le40\), \(40<m\le60\), \(60<m\le80\) with frequencies 6, 14, 20, 10. Estimated mean?
Show answer

Answer 43.6

2
Scores 1-5, 6-10, 11-15, 16-20 with frequencies 3, 8, 6, 3. Estimated mean (2 d.p.)?
Hint
Midpoints 3, 8, 13, 18.
Show answer

Answer 10.25

Comparing two sets of data

To compare two groups, give TWO comments: one about an average (which is bigger, so on average higher) and one about the spread (range or IQR: smaller spread = more consistent). Always say it in context.

Example Class A: mean mark 62, range 30. Class B: mean mark 58, range 15. Compare the marks.

Average, then spread, in context.

  1. A's mean is higher, so A did better on average.
  2. B's range is smaller, so B's marks were more consistent.

Answer two comments, one each

Ways they test it

Type 1 Estimate the mean

"Calculate an estimate of the mean." (4 marks)

Spot it a grouped table.

  1. Midpoints.
  2. \(\sum fx\div\sum f\).

For example 5, 12, 18, 10, 5 in 10s

  1. \(1230\div50\)

Answer 24.6

Type 2 Modal / median class

"Write down the modal class." (1 mark)

Spot it grouped data.

  1. Highest frequency.
  2. Median: running total to \(\tfrac{n+1}{2}\).

Type 3 Why an estimate?

"Explain why your answer is an estimate." (1 mark)

Spot it 'explain'.

  1. The exact values are not known; midpoints were used.

Type 4 Compare

"Make two comparisons between the two groups." (2 marks)

Spot it two data sets with averages and ranges.

  1. Average comment.
  2. Spread comment, in context.

Mistakes to avoid

  • ✗ Using the class width or upper bound instead of the midpoint✓ midpoint
  • ✗ Dividing \(\sum fx\) by the number of classes✓ ÷ \(\sum f\)
  • ✗ 'The range of B is smaller' with no meaning✓ '…so B's marks are more consistent'The mark needs the interpretation.
  • ✗ Modal class written as a frequency✓ write the CLASS: \(20<t\le30\)

Summary

  • Grouped mean (estimate) \(=\sum fx\div\sum f\) with midpoints.
  • Modal class = highest frequency; median class from the running total.
  • Compare = average + spread, in context.

If you see …

  • classes like 10 < t ≤ 20midpoints.
  • 'compare'two comments.

Practice

Practice: easy, medium, hard

Easy

1
Midpoint of \(10<t\le20\)?
Show answer

Answer 15

2
Midpoint of the class 11-15?
Show answer

Answer 13

3
Frequencies 5, 12, 18, 10, 5. How many values in total?
Show answer

Answer 50

Medium

1
Times \(0<t\le10\) … \(40<t\le50\) with frequencies 5, 12, 18, 10, 5. Estimated mean?
Guide me

What is it asking? a mean

What do we know? \(\sum fx=1230\)

Plan \(\div50\)

Show working
  1. 24.6

Answer 24.6

2
Same table: \(\sum fx\)?
Guide me

What is it asking? a total

What do we know? midpoint × frequency each row

Plan add

Show working
  1. 1230

Answer 1230

3
Frequencies 6, 14, 20, 10 in classes of 20 from 0. Which class holds the median? Type its upper bound.
Guide me

What is it asking? a class

What do we know? 25.5th value

Plan running total 6, 20, 40

Show working
  1. \(40<m\le60\)

Answer 60

Hard

1
Classes of 20 from 0 to 80, frequencies 6, 14, 20, 10. Estimated mean?
Guide me

What is it asking? a mean

What do we know? midpoints 10, 30, 50, 70

Plan \(2180\div50\)

Show working
  1. 43.6

Answer 43.6

PartAnswer / workingMarksGuidance
midpointsM1
\(\sum fx=2180\)M1
÷ 50M1
43.6A1
Total: 4 marks
2
Scores 1-5, 6-10, 11-15, 16-20; frequencies 3, 8, 6, 3. Estimated mean (2 d.p.)?
Guide me

What is it asking? a mean

What do we know? midpoints 3, 8, 13, 18

Plan \(205\div20\)

Show working
  1. 10.25

Answer 10.25

PartAnswer / workingMarksGuidance
\(\sum fx=205\)M2
10.25A1
Total: 3 marks
3
Times table (5, 12, 18, 10, 5). Modal class: type its lower bound.
Guide me

What is it asking? a class

What do we know? highest frequency 18

Plan \(20<t\le30\)

Show working
  1. \(20<t\le30\)

Answer 20

PartAnswer / workingMarksGuidance
\(20<t\le30\)B1
Total: 1 mark

Can I do this?

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

Quiz

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