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Maths · Further statistics

Comparing and describing distributions

Compare data sets using averages and spread, use cumulative frequency curves and box plots to compare, and describe the shape (skew) of a distribution.

  • 6 key terms
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Warm-up

Answer each one, then check.

  1. 1

    Name a measure of average.

    Show answerHide answer

    Mean, median or mode

  2. 2

    Name a measure of spread.

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    Range or interquartile range

  3. 3

    What is an outlier?

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    A value far from the others

  4. 4

    Does an outlier affect the median much?

    Show answerHide answer

    No

  5. 5

    Does an outlier affect the mean much?

    Show answerHide answer

    Yes

Learning Objectives

  1. 1Compare two distributions using an average and a measure of spread.
  2. 2Choose the best average and spread.
  3. 3Describe skew.
  4. 4Write comparisons in context.

COMPARING

To compare distributions, compare one average and one measure of spread, using numbers and context.

A higher average means the values are generally bigger. A larger spread means the values are less consistent.

Choosing Averages and Spreads

Averages

  • Mean uses every value, but is affected by outliers.
  • Median is not affected by outliers.
  • Mode is the most common value.

Spreads

  • Range uses the largest and smallest values only.
  • IQR is the spread of the middle 50%.
  • IQR is not affected by outliers.

Comparing Two Data Sets

Boys: median 12.4 s, IQR 1.2 s. Girls: median 13.1 s, IQR 2.0 s. Compare the times for the 100 m sprint.

Show the solutionHide the solution
  1. 1 Average Boys' median 12.4 is lower than girls' 13.1
  2. 2 Meaning On average the boys were faster
  3. 3 Spread Boys' IQR 1.2 is smaller than girls' 2.0
  4. 4 Meaning The boys' times were more consistent

AnswerOn average the boys were faster (median 12.4 s against 13.1 s) and their times were more consistent (IQR 1.2 s against 2.0 s).

Comparing Cumulative Frequency Curves

Two cumulative frequency curves show the times of Group A and Group B (60 people each). Group A: median 28, IQR 16. Group B: median 22, IQR 17. Compare the groups.

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  1. 1 Median B's median is lower
  2. 2 Spread The IQRs are very close
  3. 3 Conclusion Group B were quicker on average, with a similar spread

AnswerGroup B were quicker on average (median 22 against 28) and both groups had a similar spread (IQR 17 against 16).

Describing Skew

The mean is 30 and the median is 25. Describe the skew.

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  1. 1 Compare The mean is greater than the median
  2. 2 Shape The tail is to the right

AnswerThe distribution has positive skew.

Writing Good Comparisons

A full-mark answer has four parts.

  • An average

    Compare medians or means, quoting numbers.

  • A spread

    Compare IQRs or ranges, quoting numbers.

  • Context

    Write in terms of the situation.

  • A conclusion

    For example "so Group B were faster and more consistent".

Write the Comparison

Set A: mean 24, range 10. Set B: mean 21, range 25. Write a comparison of the two sets.

1. Compare averages.

2. Compare spreads.

A good answer shows: On average Set A has higher values (mean 24 against 21) and Set A is more consistent (range 10 against 25).

Can I...?

  1. 1Compare two medians.
  2. 2Compare two IQRs.
  3. 3Compare means and ranges.
  4. 4Read a cumulative frequency graph.
  5. 5Describe skew.
  6. 6Write in context.
  7. 7Use numbers.
  8. 8Explain the choice of average.

Summary & Exam Focus

  • Compare one average and one spread.
  • Median and IQR are best if there are outliers.
  • Positive skew: long tail to the right, mean above median.
  • Give numbers and a conclusion.

Exam focus

The cumulative frequency graph shows the times taken by two groups. Compare the times. (4 marks) (4 marks)

Compare an average and a spread, and use the words of the question.

Key terms

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

Distribution
How values are spread across a data set.
Skew
A lack of symmetry; the tail shows the direction.
Symmetrical
The same on both sides of the middle.
Outlier
A value much larger or smaller than the rest.
Consistent
Little variation.
In context
Written using the real situation.

Practice questions

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

  1. Question 1 Compare 4 marks

    The cumulative frequency graphs show the times taken by 60 people in Group A and 60 people in Group B to complete a puzzle. Compare the times taken by the two groups.

    Two cumulative frequency curves for Group A and Group B on the same grid.
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    Model answer

    The median for B (about 22) is lower than for A (about 28), so B were quicker on average. The IQRs are similar (B about 17, A about 16), so the spread is similar.

    Mark scheme

    • Median for each read from the graph — M1
    • Comment on average in context — A1
    • IQR for each read from the graph — M1
    • Comment on spread in context — A1
  2. Question 2 Compare 2 marks

    Boys: median 12.4 s, interquartile range 1.2 s. Girls: median 13.1 s, interquartile range 2.0 s. Write down two comparisons.

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    Model answer

    On average the boys were faster (12.4 s against 13.1 s). The boys' times were more consistent (IQR 1.2 s against 2.0 s).

    Mark scheme

    • Average compared in context — B1
    • Spread compared in context — B1
  3. Question 3 Compare 2 marks

    Set A has mean 24 and range 10. Set B has mean 21 and range 25. Compare the two sets.

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    Model answer

    Set A has a higher mean (24 against 21). Set A has a smaller range (10 against 25), so it is more consistent.

    Mark scheme

    • Compares means — B1
    • Compares ranges — B1
  4. Question 4 Describe 2 marks

    A distribution has a mean of 30 and a median of 25. Describe the skew of the distribution.

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    Model answer

    The mean is greater than the median, so the distribution has positive skew (a long tail to the right).

    Mark scheme

    • Mean above median — M1
    • Positive skew — A1
  5. Question 5 Explain 2 marks

    A data set has one very large outlier. Explain why the median is a better average than the mean.

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    Model answer

    The outlier pulls the mean up but has almost no effect on the median, so the median better represents the typical value.

    Mark scheme

    • Mean affected by outlier — M1
    • Median not affected — C1
  6. Question 6 Explain 2 marks

    The mean time for pupils in a London school is lower than in a school in Leeds. Ben says every pupil in the London school must be faster than every pupil in Leeds. Explain why Ben is wrong.

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    Model answer

    The mean is only an average. Some pupils in the London school could be slower than some in Leeds; the data may be spread widely.

    Mark scheme

    • Mean is an average only — M1
    • Individual values may overlap — C1

Quick check

  1. To compare two data sets you should compare...

    1. AOnly the mean
    2. BOnly the range
    3. CAn average and a measure of spread
    4. DThe highest values
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    C: An average and a measure of spread

    An average and a measure of spread.

  2. Which measure of spread is not affected by outliers?

    1. ARange
    2. BInterquartile range
    3. CMaximum
    4. DMinimum
    Show answerHide answer

    B: Interquartile range

    The interquartile range uses only the middle half.

  3. A smaller IQR means the data are...

    1. AMore consistent
    2. BLess consistent
    3. CLarger
    4. DSkewed
    Show answerHide answer

    A: More consistent

    More consistent.

  4. Mean 30 and median 25 suggests...

    1. ASymmetry
    2. BNegative skew
    3. CNo skew
    4. DPositive skew
    Show answerHide answer

    D: Positive skew

    Positive skew.

  5. A symmetrical distribution has...

    1. AMean above median
    2. BMean below median
    3. CMean equal to median
    4. DNo median
    Show answerHide answer

    C: Mean equal to median

    The mean equal to the median.

  6. In a comparison, "in context" means...

    1. AUsing only numbers
    2. BReferring to the real situation
    3. CUsing long sentences
    4. DAdding a graph
    Show answerHide answer

    B: Referring to the real situation

    Using the real situation in your wording, such as "the boys were faster".

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