Comparing data distributions: a week of ready-to-teach maths

Five lessons for Year 9 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 12 matching cards, 12 shape cards, 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: sheet 1 has the data and box plot matching cards (Day 1), sheet 2 the shape cards (Day 5). Cut them before Day 1.
4. Each student needs a ruler and a calculator; each pair needs a 30 cm ruler for the ruler-drop test on Day 4.
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 distribution is described by its centre, its spread and its shape. The five-number summary (minimum, lower quartile, median, upper quartile, maximum) gives a box plot; the range and the interquartile range (IQR) measure spread. An outlier is a value more than 1.5 IQRs beyond a quartile, and it pulls the mean and the range but hardly moves the median or the IQR. Box plots on one scale compare groups at a glance; the shape of a distribution says whether the mean or the median is the better centre.

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 answers

Quartiles are the medians of the lower and upper halves, leaving the middle value out when there is an odd number of values, as the unit page does. Five-number summaries are listed in order: minimum, lower quartile, median, upper quartile, maximum. Means are given to two decimal places at most.

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 summarises a data set in five numbers and draws its box plot; Day 2 measures spread two ways; Day 3 finds outliers and their effects; Day 4 compares two groups on one scale; and Day 5 reads the shape of a distribution. It builds on Year 8 work with the mean, median, range and stem-and-leaf plots, and on reading surveys earlier in Year 9; it prepares choosing displays and planning investigations.

Materials for the week (one trip)

• From the classroom: rulers, calculators, 30 cm rulers for the ruler-drop test, scissors, envelopes, this pack printed.
• Cut-out sheet 1, one per pair: 6 data cards and 6 box plot cards to match (Day 1).
• Cut-out sheet 2, one per pair: 12 shape cards (Day 5).
• If you have them: a spreadsheet or graphing app that draws box plots, to check hand-drawn ones. No other equipment to buy.

Words for the word wall

distribution, five-number summary, minimum, maximum, lower quartile, upper quartile, median, box plot, whisker, range, interquartile range, outlier, fence, centre, spread, shape, symmetric, skewed, parallel box plots

Dear families

This week in maths, Year 9 is comparing sets of data using box plots, which show the middle, the spread and any unusual values at a glance.

Try this at home

• Time how long the kettle or the toaster takes, five times. Which number is the middle one? How far apart are the fastest and slowest?
• Ask: in a list of house prices, one is ten times the rest. Which is affected more, the average (mean) or the middle value (median)?
• Ask: two teams score the same median number of goals, but one is far more up and down. How would a box plot show that?

My data week

Fill one row a day. Write something you measured or counted, its middle value, and its smallest and largest.

Five numbers and a box

Sort the data, find the median, then the median of each half: the lower quartile (Q1) and the upper quartile (Q3). With the minimum and maximum, these five numbers draw a box plot: a box from Q1 to Q3 with a line at the median, and whiskers out to the extremes. Each part of the plot holds about a quarter of the data.

We are learning to

• find the five-number summary of a data set, odd or even in size,
• draw a box plot on a scale,
• read a box plot back into what the data must look like.

Success criteria

• I can find Q1 and Q3 by splitting the sorted data into halves.
• I can match each data set to its box plot and explain how I knew.

You need

The worksheet; the matching cards (cut-out sheet 1) per pair; a ruler each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with the matching.

Watch for these ideas

• Finding quartiles from unsorted data.
• With an even number of values, splitting into halves of different sizes.
• Thinking a longer whisker or box holds more data: each quarter holds about the same number of values.

Answers

• Part A: minutes of homework, 11 students: 10, 20, 30, 40, 50; push-ups in a minute, 10 students: 18, 22, 28, 33, 41.
• Part B: box plots drawn from those summaries.
• Part C: the card key is with the mini-check answers.
• Part D: many answers; for example 2, 4, 5, 6, 8, 9, 12.

Five numbers and a box

A. Two five-number summaries

B. Draw the box plots

Draw each box plot above its scale.

C. Match the cards

Match each data card (A to F) to its box plot card (1 to 6). Find each summary first, or rule cards out by their minimum and maximum.

D. Stretch: build the data

Write seven whole numbers whose five-number summary is 2, 4, 6, 9, 12.

Range and IQR

The range (maximum minus minimum) uses only the two most extreme values, so one unusual result changes it a lot. The interquartile range (Q3 minus Q1) measures the spread of the middle half and ignores the extremes. Two data sets can share a median and differ completely in spread, and the two measures can even disagree about which is more spread out.

We are learning to

• find the range and the interquartile range,
• explain why the IQR resists extreme values and the range does not,
• compare the spread of two data sets with the same centre.

Success criteria

• I can find both spreads for each player and say which is more consistent.
• I can explain why the range and the IQR disagree here.

You need

The worksheet; a ruler and a calculator each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Giving the range as two numbers (2 to 13) instead of their difference.
• Thinking a bigger range always means the middle of the data is more spread.
• Mixing up the IQR with the box’s position.

Answers

• Part A: Rhea 10, 12.5, 15, 17.5, 20, range 10, IQR 5; Sam 2, 14, 15, 16, 28, range 26, IQR 2.
• Part B: by the range Sam looks more spread; by the IQR Sam’s middle half is tighter. Sam usually scores close to 15 but had two extreme games; Rhea’s scores are spread evenly.
• Part C: changing Sam’s 28 to 18 changes the range from 26 to 16 but leaves the IQR at 2.
• Part D: the IQR, because the middle half of the data is not affected by one very good or very bad game.

Range and IQR

A. Two players

Points scored in nine games. Find each five-number summary, range and IQR.

Draw both box plots on the one scale.

B. Who is more consistent?

Which measure says Sam is more spread out, and which says Sam is more consistent? Who would you pick, and why?

C. Change one game

Sam’s best game of 28 points was really 18. Find the new range and IQR. Which one changed?

D. Stretch: which spread to report?

A coach wants one number for how consistent a player is. Should it be the range or the IQR? Explain.

Outliers

An outlier is a value far from the rest. The 1.5 × IQR rule makes “far” exact: a value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR is an outlier, and a box plot marks it as a separate dot. An outlier pulls the mean and stretches the range, but barely moves the median or the IQR. Whether to leave it out depends on why it is there.

We are learning to

• use the 1.5 × IQR rule to decide whether a value is an outlier,
• measure the effect of an outlier on the mean, median, range and IQR,
• decide whether an outlier belongs in the data.

Success criteria

• I can find the fences for a data set and name its outliers.
• I can say which measures an outlier changes most.

You need

The worksheet; a calculator each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Calling the largest value an outlier just because it is the largest.
• Using 1.5 × IQR from the median instead of from the quartiles.
• Deleting an outlier without a reason: some outliers are the most interesting values.

Answers

• Part A: Q1 = 14.1, Q3 = 15.8, IQR = 1.7; fences 11.55 and 18.35; only 22.5 is beyond them.
• Part B: mean 15.46 with, 14.76 without; median 14.9 with, 14.75 without; range 9.3 with, 2.9 without; iqr 1.7 with, 1.3 without. The range and the mean change most.
• Part C: the walked time does not measure running speed, so leave it out of a report on running times, and say so; a real slow runner would stay in.
• Part D: the whisker stops at 16.1, the largest value inside the fence; 22.5 is drawn as a separate dot.

Outliers

Times (seconds) for 11 students running 100 m: 13.2, 13.8, 14.1, 14.5, 14.6, 14.9, 15.2, 15.4, 15.8, 16.1, 22.5. One student walked the whole way.

A. Find the fences

B. With and without

C. Keep it or not?

A report on how fast the class runs 100 m: should it include the walked time? Explain.

D. Stretch: draw it

Draw the box plot with the outlier marked as a separate dot. Where does the upper whisker stop?

Comparing two groups

Box plots drawn on one scale compare two groups at a glance: compare the medians for centre, the IQRs for spread, and look for outliers. A good comparison says how big the difference in centre is, whether the spreads differ, and how much the boxes overlap. Today the data is a ruler-drop test: the shorter the distance a falling ruler travels before it is caught, the faster the reaction.

We are learning to

• draw parallel box plots on one scale,
• compare two distributions by centre, spread and outliers,
• write a comparison that uses the numbers.

Success criteria

• I can draw both hands’ box plots on one scale and mark any outlier.
• I can write two sentences comparing the hands, with numbers.

You need

The worksheet; a 30 cm ruler per pair; a ruler and a calculator each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with the ruler test.

Watch for these ideas

• Comparing the maximums instead of the medians.
• Drawing the two box plots on different scales.
• Writing “the dominant hand is better” without saying by how much.

Answers

• Part A: dominant hand 9, 12, 15, 18, 29 (IQR 6, upper fence 27, outlier 29); other hand 12, 16, 19, 22, 26 (IQR 6, no outliers).
• Part B: two box plots on the scale, the dominant hand’s 29 drawn as a dot beyond a whisker ending at 21.
• Part C: the dominant hand caught sooner: median 15 cm against 19 cm, 4 cm less. The spreads are similar (IQRs 6 and 6), and one slow dominant catch is an outlier.
• Part D: results vary.

Comparing two groups

Ruler-drop catch distances (cm) for 15 students, each hand:

A. Summaries and outliers

Write the five-number summary, the IQR, the fences and any outliers for each hand.

B. Parallel box plots

C. Compare

Write two or three sentences comparing the hands: centre, spread and outliers, with numbers.

D. Stretch: our own test

Take five catches with each hand. Record the distances, find each median, and compare them with the printed data.

Shape

A symmetric distribution looks the same on both sides of its centre, and its mean and median are close. A right-skewed one has a tail of high values that pulls the mean above the median; a left-skewed one has a tail of low values that pulls the mean below it. When data is skewed or has outliers, the median and the IQR describe it better than the mean and the range.

We are learning to

• describe the shape of a distribution as symmetric, right-skewed or left-skewed,
• predict whether the mean is above or below the median from the shape,
• choose the measures of centre and spread that suit the shape.

Success criteria

• I can sort the twelve shape cards and name the direction of each tail.
• I can say which average to report for skewed data, and why.

You need

The worksheet; the 12 shape cards (cut-out sheet 2) per pair; the unit’s self-check quiz.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the sort; start the second with Part B and the quiz.

Watch for these ideas

• Naming the skew by where the peak is instead of where the tail goes.
• Thinking only a bell shape is symmetric.
• Reporting the mean for strongly skewed data such as incomes.

Answers

• Part A: the card key is with the mini-check answers.
• Part B: right-skewed: mean above the median; left-skewed: mean below it; symmetric: about equal.
• Part C: the median and the IQR, because the long tail and any outliers pull the mean and stretch the range.
• Part D: for example, ages at a primary school: many children and a few adults, a long tail of high ages.

Shape

A. Sort the shape cards

Write each card’s letter in its pile.

B. Mean or median higher?

For each shape, say whether you expect the mean to be above, below or about equal to the median, and why.

C. Which measures?

A report describes weekly pocket money, which is right-skewed with a few very large amounts. Which centre and which spread should it give? Why?

D. Stretch: your own example

Describe a real set of data that you expect to be right-skewed, and one you expect to be left-skewed. Explain each tail.

Data and box plot cards

Day 1, one set per pair. Cut out the six data cards and the six box plot cards. Match each data set to its box plot.

Data cards

Box plot cards

Teacher note: the key is with the mini-check answers, so this sheet can go to pairs whole.

Shape cards

Day 5, one set per pair: 12 cards. Each is a histogram of a data set. Sort them: right-skewed, left-skewed or symmetric.

Teacher note: the key is with the mini-check answers, so this sheet can go to pairs whole.

What we know: Comparing data distributions

Work on your own. Show your working.

1. Find the five-number summary of 4, 7, 8, 10, 12, 13, 15, 18, 21. ______________________
2. Find the range and the IQR of the same data. ______________________
3. Is 30 an outlier in 5, 6, 6, 7, 8, 8, 9, 30? Use the 1.5 × IQR rule. ______________________
4. If 30 is removed from that data, which changes more: the mean or the median? ______________________
5. Group A has median 62 and IQR 10; group B has median 70 and IQR 25. Which has the higher centre? Which is more consistent? ______________________
6. A histogram has a long tail of high values. Is it right-skewed or left-skewed? ______________________
7. For that data, is the mean above or below the median? ______________________
8. Which measure of spread is not affected by one extreme value? ______________________

Answers and marking guide

Answers with working

1. 4, 7.5, 12, 16.5, 21.
2. Range 17, IQR 9.
3. Q1 = 6, Q3 = 8.5, IQR = 2.5; upper fence 12.25, so 30 is an outlier.
4. The mean: 9.88 with, 7 without; the median moves from 7.5 to 7.
5. B has the higher centre (70 against 62); A is more consistent (IQR 10 against 25).
6. Right-skewed: the tail points to the high values.
7. Above: the tail of high values pulls the mean up.
8. The interquartile range.

A quick three-level guide

Eight questions, four ideas. A student at standard summarises, finds outliers and compares two groups by centre and spread.

Card keys

Matching (Day 1): A → 4 (1, 2, 4, 6, 9); B → 6 (0, 3.5, 5, 6.5, 8); C → 5 (0, 0.5, 2, 3, 6); D → 1 (2, 4, 7, 8.5, 10); E → 3 (1, 3, 4, 7, 9); F → 2 (0, 1.5, 3, 6, 7).

Shape cards (Day 5): right-skewed: A, D, F, J, K; left-skewed: B, G, I; symmetric: C, E, H, L.

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: box plots, spread, outliers, comparing two groups, and shape.