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Year 10 · Mathematics · Statistics

Comparing data distributions

Compare two sets of data using back-to-back stem-and-leaf plots and histograms, describing the shape, centre and spread of each.

Awaiting educator review. This material was drafted by our content team with AI assistance and is waiting for review by a qualified educator. Please check it suits your child before using it. Created 10 October 2026. How we review

These lessons were written around Australian Curriculum strands and matched to UK year groups by age (Year 1 is the year a child turns 6 in England). They are not mapped to the National Curriculum in England or the curricula of Scotland, Wales or Northern Ireland, so check them against your own plans.

Learning goals

Learners will

  • Construct and read back-to-back stem-and-leaf plots.
  • Interpret histograms with grouped data.
  • Compare two data sets by describing their shape, centre and spread.

Success looks like

  • I can read the values from both sides of a back-to-back stem-and-leaf plot.
  • I can describe a distribution as symmetric, positively skewed, negatively skewed or bimodal.
  • I can write a comparison of two data sets that mentions shape, centre (mean or median) and spread (range).

The big idea

When we compare two sets of data, we look at three features: shape (what the distribution looks like), centre (a typical value — the mean or median) and spread (how varied the values are — the range).

A stem-and-leaf plot splits each value into a stem (for example, the tens digit) and a leaf (the units digit). A back-to-back plot puts two data sets either side of the same stems. The leaves on the right are written from smallest to largest moving outwards from the stem, and the leaves on the left are also written from smallest to largest moving outwards — so on the left you read them from right to left. Always include a key, such as “5 | 2 = 52”.

For example, a left-side row that reads “8 5 2 | 4 |” with stems in tens means the values 42, 45 and 48.

A histogram shows grouped data, such as times from 10 up to (but not including) 20 seconds. The columns touch because the data is continuous. The tallest column is the modal class.

Common shapes of distributions:

  • Symmetric: a central peak with both sides about the same.
  • Positively skewed: most values are low, with a long tail stretching towards the high values.
  • Negatively skewed: most values are high, with a long tail stretching towards the low values.
  • Bimodal: two separate peaks, which can mean two different groups are mixed together.

A good comparison uses numbers and context, for example: “Class B's median score (73.5) was higher than Class A's (66), but Class B's scores were more spread out (range 45 compared with 29).” For a skewed distribution, the median is usually a better measure of centre than the mean, because extreme values pull the mean towards the tail.

Worked examples

Building a back-to-back plot

Test scores. Class A: 52, 58, 61, 64, 65, 67, 70, 72, 75, 81. Class B: 45, 49, 53, 60, 71, 76, 78, 84, 88, 90. Draw a back-to-back stem-and-leaf plot.

  1. Use the tens digits 4 to 9 as stems in the middle. Key: 5 | 2 = 52.
  2. Stem 4: A has no leaves; B has 5 9.
  3. Stem 5: A has 8 2 (read outwards: 52, 58); B has 3.
  4. Stem 6: A has 7 5 4 1; B has 0.
  5. Stem 7: A has 5 2 0; B has 1 6 8.
  6. Stem 8: A has 1; B has 4 8.
  7. Stem 9: A has no leaves; B has 0.

Answer: Class A's leaves cluster around stems 6 and 7; Class B's leaves are spread from stem 4 to stem 9.

Comparing the two classes

Compare the centre and spread of Class A and Class B from the previous example.

  1. Each class has 10 scores, so the median is the average of the 5th and 6th values.
  2. Class A median = (65 + 67) ÷ 2 = 66. Range = 81 − 52 = 29.
  3. Class B median = (71 + 76) ÷ 2 = 73.5. Range = 90 − 45 = 45.
  4. Class A is clustered and roughly symmetric; Class B is much more spread out.

Answer: Class B has a higher median (73.5 vs 66) but a much larger spread (range 45 vs 29), so Class A's results were more consistent.

Describing a histogram

Times (seconds) to solve a puzzle: 0–<10: 2 people; 10–<20: 5; 20–<30: 9; 30–<40: 4; 40–<50: 1. Describe the distribution.

  1. Total = 2 + 5 + 9 + 4 + 1 = 21 people.
  2. The tallest column is 20–<30 seconds, so that is the modal class.
  3. The columns rise to a single peak in the middle and fall away on both sides by similar amounts.

Answer: A single-peaked, roughly symmetric distribution with modal class 20–<30 seconds.

Practice check

Have a go, then check your answers. Each answer comes with an explanation.

  1. 1.In a back-to-back stem-and-leaf plot, where are the stems?
  2. 2.Most values in a data set are low, with a long tail towards the high values. The distribution is:
  3. 3.Which of these is a measure of spread?
  4. 4.A histogram has two separate peaks. Its shape is described as:
  5. 5.On the left side of a back-to-back stem-and-leaf plot, one row reads “8 5 2 | 4 |”. The key is 4 | 2 = 42. List the three values in order from smallest to largest.
  6. 6.Find the median of 12, 15, 15, 18, 20, 23, 31.
  7. 7.Find the range of 12, 15, 15, 18, 20, 23, 31.
  8. 8.Team A's points per game have a median of 24 and a range of 10. Team B's have a median of 24 and a range of 30. Which team scores more consistently? Answer A or B.
Answer guide for parents

  1. In a back-to-back stem-and-leaf plot, where are the stems?

    In the middle, shared by both data sets — The stems sit in a central column, with one data set's leaves on each side.

  2. Most values in a data set are low, with a long tail towards the high values. The distribution is:

    positively skewed — A tail towards the high (positive) end is positive skew.

  3. Which of these is a measure of spread?

    Range — The range (highest − lowest) measures spread. The others measure centre or the most common value.

  4. A histogram has two separate peaks. Its shape is described as:

    bimodal — Two peaks (two modes) means bimodal.

  5. On the left side of a back-to-back stem-and-leaf plot, one row reads “8 5 2 | 4 |”. The key is 4 | 2 = 42. List the three values in order from smallest to largest.

    42, 45, 48 — Left-side leaves are read outwards from the stem: 2, 5, 8, giving 42, 45 and 48.

  6. Find the median of 12, 15, 15, 18, 20, 23, 31.

    18 — There are 7 values in order, so the median is the 4th value: 18.

  7. Find the range of 12, 15, 15, 18, 20, 23, 31.

    19 — 31 − 12 = 19.

  8. Team A's points per game have a median of 24 and a range of 10. Team B's have a median of 24 and a range of 30. Which team scores more consistently? Answer A or B.

    A — The medians are equal, but Team A's range is much smaller, so its scores vary less.

Watch out for

  • Reading left-side leaves the wrong way, so “8 5 2 | 4” is read as 48, 45, 42 in the wrong order or as 84, 54, 24. Leaves always attach to the stem, and grow outwards from smallest to largest.
  • Calling a distribution with a tail on the right “negatively skewed” because the peak is on the left. The skew is named after the direction of the tail.
  • Comparing only averages. Two data sets can have the same median but very different spreads — always comment on shape, centre and spread.

Tips for parents

  • Collect real data together, such as reaction times for two family members over 15 tries, and compare them with a back-to-back plot.
  • Ask “What would a typical value be?” and “How consistent is it?” to get the learner talking about centre and spread.
  • Encourage full-sentence comparisons that include numbers, such as “The median for A was 66, which is lower than B's median of 73.5.”

Go further

  • Time how long it takes family members to do a task with their writing hand and with their other hand, and compare the two distributions.
  • Record the daily maximum temperature for a month and draw a histogram. Describe its shape and suggest reasons for it.
  • Find a data set with a bimodal shape (such as heights of a mixed group of adults and children) and explain what causes the two peaks.

About 50 minutes

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You'll need

  • Grid paper, ruler and pencil
  • Calculator
  • A stopwatch (to collect reaction-time or task-time data)

Key words

Distribution
The way data values are spread out across their possible values.
Back-to-back stem-and-leaf plot
A display of two data sets sharing a central column of stems, with one set's leaves on the left and the other's on the right.
Histogram
A column graph for grouped numerical data, where the columns touch because the class intervals are continuous.
Symmetric
A distribution whose left and right sides are roughly mirror images.
Skewed
A distribution with a long tail on one side: positively skewed if the tail is towards the high values, negatively skewed if the tail is towards the low values.
Bimodal
A distribution with two clear peaks.
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