Comparing distributions with boxplots: a week of ready-to-teach maths

Five lessons for Year 10 Statistics. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus twelve boxplot cards to match, twelve shape cards to sort, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets, one copy per pair. Cut the 12 boxplot cards before Day 1 and the 12 shape cards before Day 3, and keep each set in an envelope.
4. Rulers and calculators are used every day. A spreadsheet or graphing tool can draw boxplots on Day 5 if you have computers.
5. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.

The idea in plain words

A boxplot draws five numbers: the minimum, the lower quartile, the median, the upper quartile and the maximum. The box holds the middle half of the data, so its width, the interquartile range, measures spread without being thrown by extreme values. Values more than 1.5 interquartile ranges beyond the box are flagged as outliers. Drawn on one scale, boxplots compare groups at a glance, and every comparison is told in the context of the data: centre, spread, shape and outliers, in the units of what was measured.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write these

Quartiles are the medians of the lower and upper halves; with an odd number of values, the median itself is left out of both halves. Some calculators and spreadsheets use other methods and can give slightly different quartiles, which is worth a mention.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 builds the boxplot; Day 2 measures spread and finds outliers; Day 3 reads shape; Day 4 compares two groups; and Day 5 chooses displays and writes a full comparison. It builds on Year 9, where students compared data with histograms and summary statistics, and it prepares the data analysis of senior mathematics.

Materials for the week (one trip)

• From the classroom: rulers, calculators, scissors, envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per pair: 12 boxplot cards (Day 1).
• Cut-out sheet 2, one per pair: 12 shape cards (Day 3).
• If you have them: computers with a spreadsheet or graphing tool (Day 5). No other equipment to buy.

Words for the word wall

continuous, distribution, minimum, maximum, median, lower quartile, upper quartile, five-number summary, boxplot, whisker, range, interquartile range, outlier, symmetric, positively skewed, negatively skewed, centre, spread, histogram

Dear families

This week in maths, Year 10 is learning to compare sets of measurements, such as travel times or reaction times, with boxplots, and to describe what the data shows about the typical value, the spread, the shape and any unusual values.

Try this at home

• Time your trip to school each day this week. What are the shortest, the longest and the middle time?
• Ask: which tells you more about a typical trip, the average time or the middle time, if one day there was a traffic jam?
• Ask: two buses both average 20 minutes. Why might you still prefer one? (One might be far more consistent.)
• Ask: what is an outlier, and should it always be thrown away? (No: first ask why it happened.)

My measurements week

Fill one row a day. Write something you measured or timed, the value, and whether it was typical or unusual.

The five-number summary

A boxplot is drawn from five numbers. Students find the five-number summary of an odd and an even set of continuous data, draw a boxplot on a scale, and match six small data sets to their boxplots.

We are learning to

• order data and find the minimum, quartiles, median and maximum,
• handle both an odd and an even number of values,
• draw a boxplot on a numbered scale.

Success criteria

• I can find the five-number summary of 11 travel times.
• I can draw the boxplot and check each part against the summary.

You need

Rulers; the boxplot cards (cut-out sheet 1), one set per pair; the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Finding quartiles before putting the data in order.
• Counting the median in both halves when the number of values is odd.
• Drawing the line in the box at the mean instead of the median.

Answers

• Part A: 6, 9, 14, 20, 38 minutes (minimum, lower quartile, median, upper quartile, maximum).
• Part B: 3.2, 4.3, 5.15, 6, 7.4 cm; the median is halfway between the 6th and 7th values.
• Part C: the card key is with the mini-check answers.
• Part D: for example 2, 4, 5, 7, 8, 9, 12: the 2nd, 4th and 6th values set the quartiles and median.

The five-number summary

A. Travel times (odd count)

Minutes that 11 students take to travel to school: 6, 8, 9, 11, 12, 14, 15, 17, 20, 24, 38.

Draw the boxplot on the scale. The minimum is done for you in the table.

B. Seedling heights (even count)

Heights in cm of 12 seedlings after two weeks: 3.2, 3.8, 4.1, 4.5, 4.9, 5.0, 5.3, 5.6, 5.8, 6.2, 6.7, 7.4.

C. Match the cards

Match each data card to its boxplot card. Write the letters.

D. Stretch: work backwards

Spread and outliers

The range uses only the two extreme values; the interquartile range measures the spread of the middle half and is far less thrown by one unusual value. Students find both, test values against the 1.5 × IQR fences, and draw a boxplot that shows an outlier apart.

We are learning to

• find the range and the interquartile range,
• use the fences Q1 − 1.5 × IQR and Q3 + 1.5 × IQR to flag outliers,
• draw a boxplot with an outlier apart, and say what the outlier means in context.

Success criteria

• I can show that 38 minutes is an outlier among the travel times.
• I can explain why the IQR describes this spread better than the range.

You need

Calculators; rulers; the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Calling any large value an outlier: test it against the fence.
• Using the range in the fence instead of the IQR.
• Deleting an outlier without asking why it happened.

Answers

• Part A: range 38 − 6 = 32; IQR 20 − 9 = 11; fences −7.5 and 36.5; outlier 38.
• Part B: 2.5, 9.45, 10.6, 11.6, 12.4 hours; IQR 2.15; fences 6.225 and 14.825; 2.5 is an outlier, so the whisker stops at 9.1.
• Part C: the 2.5-hour phone may have a faulty battery, and the 38-minute trip may be a student who lives far away: report outliers and look into them rather than delete them.
• Part D: without it the range falls from 32 to 18 but the IQR only from 11 to 8: the range rests on one value.

Spread and outliers

A. The travel times again

From Day 1: 6, 8, 9, 11, 12, 14, 15, 17, 20, 24, 38 minutes, with five-number summary 6, 9, 14, 20, 38. Fill in the table. The first entry is done for you.

B. Battery life

Hours of battery life of nine phones of one model: 2.5, 9.1, 9.8, 10.2, 10.6, 11, 11.3, 11.9, 12.4.

Draw the boxplot, with the whisker stopping at the last value inside the fences and the outlier as a separate dot.

C. What should we do with an outlier?

D. Stretch: range or IQR?

Shape

A distribution can be symmetric or skewed, and its shape shows in a histogram and in a boxplot alike. Students describe shapes, link them to the mean and the median, and sort twelve histograms by shape.

We are learning to

• describe a distribution as symmetric, positively skewed or negatively skewed,
• read shape from a boxplot’s whiskers and the position of its median,
• predict from the shape whether the mean sits above or below the median.

Success criteria

• I can explain what a long right whisker tells me about the data.
• I can say why the mean of positively skewed data sits above its median.

You need

The shape cards (cut-out sheet 2), one set per pair; the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Naming the skew by the peak: it is named by the tail.
• Expecting equal-width quarters: they hold equal counts, not equal widths.
• Thinking the mean and median always agree.

Answers

• Part A: travel times positively skewed (left whisker 3, right whisker 18 minutes); seedlings roughly symmetric (median 0.85 above the lower quartile and 0.85 below the upper).
• Part B: travel mean 15.82 against median 14: the long tail pulls the mean up. Seedling mean 5.21 against median 5.15: close, as the shape is symmetric.
• Part C: the card key is with the mini-check answers.
• Part D: a long left whisker and the median nearer the upper quartile; most values are high and a few stretch low.

Shape

A. Read the boxplots

B. Mean against median

C. Sort the cards by shape

D. Stretch: draw a shape

Comparing two groups

Parallel boxplots on one scale compare two groups at a glance. Students draw boxplots of two classes’ reaction times, then compare their centre, spread, shape and outliers, always in the context of the data.

We are learning to

• draw parallel boxplots on one scale,
• compare medians and IQRs, shapes and outliers,
• write a comparison in context, with units.

Success criteria

• I can say which class reacted faster, and by how much, using the medians.
• I can say which class was more consistent, using the IQRs.

You need

Calculators; rulers; the worksheet; an online reaction test on the board if you want to try one.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Comparing only the fastest or slowest student.
• Leaving out units and context: “B is smaller” says nothing.
• Ignoring the size of a difference: say how much faster.

Answers

• Part A: class A 0.21, 0.25, 0.29, 0.33, 0.42 s; class B 0.18, 0.21, 0.23, 0.26, 0.3 s.
• Part B: class B’s median is 0.06 s lower, so B typically reacted faster; B’s IQR is 0.05 s against 0.08 s, so B was more consistent; A is positively skewed (long right whisker), B roughly symmetric; no outliers (fences 0.13 to 0.45 and 0.135 to 0.335).
• Part C: a paragraph covering all four features, with numbers, units and the context of a reaction test.
• Part D: the ranges overlap: class A’s fastest (0.21 s) beat class B’s median (0.23 s), so not everyone in B was faster.

Comparing two groups

A. Two classes

Reaction times, in seconds, of ten students in each class on the same online test.

Draw both boxplots on the scale below.

B. Compare

C. Write it up

D. Stretch: everyone faster?

Choosing a display

Each display shows some things and hides others: a dot plot shows every value, a histogram the shape of grouped data, and a boxplot compares groups but can hide gaps and clusters. Students draw a histogram, choose displays for purposes, find what a boxplot hides, and play the self-check quiz.

We are learning to

• draw a histogram of grouped continuous data and read its modal class,
• choose a display that suits a purpose,
• recognise what a boxplot can hide.

Success criteria

• I can draw the histogram of apple masses and name the modal class and the class holding the median.
• I can show two data sets with the same boxplot but different shapes.

You need

Rulers; computers with a spreadsheet or graphing tool if you have them; the worksheet; the self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part B; start the next with Part C.

Watch for these ideas

• Gaps between histogram bars: continuous data has touching bars.
• Mixing up the modal class and the class holding the median.
• Choosing a boxplot for a few values, where every value matters.

Answers

• Part A: bars 2, 5, 9, 8, 4, 2; modal class 140 to under 150 g; the median (the 15th and 16th of 30) is in 140 to under 150 g.
• Part B: 1 parallel boxplots; 2 a histogram; 3 a dot plot.
• Part C: both have summary 1, 2, 7, 12, 13. The first has two clusters with a gap and one value between; the second is spread evenly. A boxplot hides clusters and gaps.
• Part D: in classes of 20 g the counts are 7, 17, 6: one tall middle bar hides how the masses rise and fall. Very narrow classes look ragged instead.

Choosing a display

A. Apple masses

B. Which display?

C. Same boxplot?

D. Stretch: class width

Data sets and their boxplots

For Day 1, one set per pair. Six cards list seven values; six show a boxplot. Match each data set to its boxplot.

Teacher note: the key is with the mini-check answers.

What shape is it?

For Day 3, one set per pair. Sort the histograms into three piles: symmetric, positively skewed (tail to the right) and negatively skewed (tail to the left).

Teacher note: the key is with the mini-check answers.

What we know: Comparing distributions

Work on your own. A calculator is allowed. Show your working.

1. Find the five-number summary of 3, 5, 6, 8, 9, 11, 12, 14, 20. ______________________
2. Find the IQR and the upper fence for the same data. Is 20 an outlier? ______________________
3. A boxplot has minimum 10, lower quartile 12, median 14, upper quartile 17 and maximum 35. Describe its shape. Is the maximum an outlier? ______________________
4. Group P has summary 20, 25, 30, 34, 40; group Q has 18, 27, 31, 33, 45. Compare their medians, IQRs and ranges. ______________________
5. For positively skewed data, is the mean usually above or below the median? Why? ______________________
6. The box of a boxplot holds, circle one:   the middle 50% of the values  ·  half of the range  ·  every value above the mean
7. Times grouped as 0 to under 10 minutes: 3; 10 to under 20 minutes: 7; 20 to under 30 minutes: 5; 30 to under 40 minutes: 2. Which is the modal class? ______________________
8. Two schools’ median travel times are 18 and 25 minutes, with IQRs 6 and 14 minutes. Write one sentence comparing them. ______________________

Answers and marking guide

Answers with working

1. 3, 5.5, 9, 13, 20: the quartiles are the medians of the four values each side of the median.
2. IQR 7.5; upper fence 13 + 1.5 × 7.5 = 24.25; 20 is not an outlier.
3. Positively skewed (right whisker 18, left 2); the upper fence is 24.5, so 35 is an outlier.
4. Medians 30 and 31: Q slightly higher. IQRs 9 and 6: Q’s middle half is less spread. Ranges 20 and 27: Q has the more extreme values.
5. Above: the few large values in the tail pull the mean up, while the median stays with the middle value.
6. The first: the middle 50% of the values.
7. 10 to under 20 minutes.
8. For example: the first school’s students typically travel 7 minutes less, and their times are far more consistent.

A quick three-level guide

Eight questions, four ideas. A student at standard compares centre, spread, shape and outliers, with numbers and units, every time.

Card keys

Boxplot cards (Day 1): G with A; D with J; E with L; B with H; K with F; I with C. Shape cards (Day 3): symmetric: N, Q, T, W; positively skewed: M, P, R, U, X; negatively skewed: O, S, V.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: the five-number summary, spread and outliers, shape, comparing, and displays.