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Describe the shape of a distribution

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Histogram of completion times in ten-minute intervals Intervals: 0 to less than 10: 8; 10 to less than 20: 5; 20 to less than 30: 3; 30 to less than 40: 2; 40 to less than 50: 1024681001020304050Time (minutes)Frequency

The histogram shows puzzle-completion times. Which way does its long thin tail run?

Year 9, Chapter 9, 4 difficulty levels, No calculatorBuild a worksheet

Distribution shape

Shape describes the overall pattern. State whether you mean the whole distribution or the main body with isolated observations noted separately

  1. Skew direction Positive means a longer thin tail towards larger values. Negative means a tail towards smaller values
  2. Symmetry and peaks A symmetric distribution can have one peak, two peaks or another mirrored pattern
  3. Isolated observations Investigate and report them; do not silently delete them or label the whole distribution symmetric because its main body is symmetric
  4. Centre and purpose The mean uses every magnitude. The median is less sensitive to extremes. Choose for the stated purpose
  5. Compare scales Use the numerical axis values, units and frequencies; equal picture widths do not imply equal data ranges

Worked example

Two groups have values A: 20, 25, 30, 35, 40 and B: 40, 45, 50, 55, 60. Compare centre, spread and shape

\[n_A=n_B=5\]
  1. \(\displaystyle \operatorname{median}_A=30\)The middle of five values is the median
  2. \(\displaystyle \operatorname{median}_B=50\)B is shifted 20 units above A.
  3. \(\displaystyle R_A=40-20=20\)Range is largest minus smallest
  4. \(\displaystyle R_B=60-40=20\)B has the higher centre; ranges agree. Both lists are symmetric about their own centres

Common mistakes

Naming skew from the peak
✕\(\displaystyle \text{high peak}\Rightarrow\text{positive}\)✓\(\displaystyle \text{right tail}\Rightarrow\text{positive}\)
Look at the direction of the longer thin tail, not the location of the tallest stack
Assuming pictures share a scale
✕\(\displaystyle \begin{gathered}\text{same width}\\\Rightarrow\text{same range}\end{gathered}\)✓\(\displaystyle \text{range}=\text{max}-\text{min}\)
Range is maximum minus minimum. Read the numerical axes before comparing spread
Treating centre choice as universal
✕\(\displaystyle \text{skew}\Rightarrow\text{mean invalid}\)✓\(\displaystyle \text{mean}=\text{total}/n\)
The mean remains valid. Choose median for resistance to extremes or mean for a total-based purpose

Difficulty levels

introIdentify clear histogram shapes
standardRead dot plots and compare paired histograms on shared numeric scales
challengeDistinguish a symmetric main body from an isolated point and select a centre for a purpose
examInterpret direct displays and justify a centre choice. Boxplot inference is reserved for later-course work

Questions about this topic

Does positive skew mean positive values?+

No. It means the long tail extends to larger values

Can a symmetric distribution be bimodal?+

Yes. Two mirrored peaks can be symmetric

Is the median always the best centre?+

No. It is useful for resistance to extremes, while the mean is appropriate for total-based averages

Should an outlier be deleted?+

Investigate its accuracy and context first. A genuine unusual observation may be important

Do equal summaries prove equal distributions?+

No. Different shapes and spreads can share a mean or median

Can a boxplot determine the full shape?+

No. It summarises selected positions. Its asymmetry may suggest skew, but different full distributions can share those positions

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 9 Mathematics, chapter 9: Probability and statistics

In this chapter

Finite events, probability assignments and complementsDescribe and order probability values9B Venn diagrams and tables: Venn diagrams, two-way tables and set notation for two eventsComplete and use a two-way probability tableCompound events, the addition rule and mutual exclusivity9C Set notation: Event notation and shaded regionsFinite sets and set operationsBuild two-step sample spaces with and without replacementUse two-step probability treesExperimental probability from observed frequenciesDesign and interpret a probability simulationFind the mean and the median, and choose between themWork backwards from a mean or medianMode, bimodal and no-mode data, and the effect of an outlierConstruct an ordered stem plot from raw dataConstruct and read back-to-back stem-and-leaf plotsDescribe the shape of a distributionGroup numerical data into class intervalsConstruct a histogram from a frequency tableRead values from a histogramFind the range, the quartiles and the IQRProduce a five-number summary and construct a boxplotRead values and percentages from a boxplotCompare a numerical variable across groupsSurveys and statistical investigations