Describing data distributions: a week of ready-to-teach maths
Five lessons for Year 7 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 dot-plot shape cards, display-and-description match cards, a mini-check and every answer with its working.
Start here: five minutes to Monday
- Skim the week at a glance on the next page.
- Print the five days. Each day is two A4 sheets: a plan and a worksheet.
- Print the two cut-out sheets at 100%, one set per pair: the shape cards (Day 1) and the match cards (Day 5).
- Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
- Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.
The idea in plain words
A display turns a list of numbers into a picture of its distribution. Dot plots show every value; stem-and-leaf plots keep every value while grouping them; histograms group measured data into intervals. A good description of a distribution covers three things: its shape (symmetric, skewed, two peaks), its centre (the median or the mean) and its spread (the range), and it names any outlier, a value far from the rest that can pull the mean towards it.
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 describe shapes
Skewed right means the tail stretches to the right, towards the larger values, with most of the data bunched at the low end; skewed left is the mirror image. A stem-and-leaf plot always has a key and ordered leaves. If computers are available, Day 5 shows the spreadsheet formulas for the four measures and how to make a column chart; the pack works without them.
The week at a glance
One lesson a day for a week. Each day stands on the day before, so run them in order.
| Day | Lesson | Students learn and do | On screen and on the desk |
|---|---|---|---|
| 1 | Dot plots and shape | Draw a dot plot, sort dot plots by shape, and see where the mean and median sit when data is skewed | The shape of a distribution; shape cards |
| 2 | Stem-and-leaf plots | Build an ordered stem-and-leaf plot with a key, and read the median, mode, range and shape from one | Stem-and-leaf plot |
| 3 | Comparing two groups | Read and build back-to-back stem-and-leaf plots, and compare two groups by centre and spread | Centre and spread |
| 4 | Histograms and outliers | Group measured data into intervals and draw a histogram, find an outlier, and decide whether to keep it | How an outlier pulls the mean |
| 5 | Describing a distribution | Match displays with descriptions, choose a display for a purpose, use spreadsheet formulas, and describe and compare data in full | Mean, median and mode; match cards; self-check quiz as a class game |
How the week builds
Day 1 reads shape from dot plots; Days 2 and 3 build stem-and-leaf plots and use them to compare two groups; Day 4 groups measured data into a histogram and deals with outliers; and Day 5 puts shape, centre and spread together into a full description. It uses the mean, median, mode and range from AC9M7ST01, and it leads into planning and reporting statistical investigations (AC9M7ST03).
Materials for the week (one trip)
- From the classroom: pencils, rulers, calculators, this pack printed at 100%; computers with a spreadsheet if you have them (Day 5).
- Cut-out sheet 1, one set per pair: 12 dot-plot shape cards (Day 1).
- Cut-out sheet 2, one set per pair: 6 display cards and 6 description cards (Day 5).
Words for the word wall
distribution, dot plot, stem-and-leaf plot, stem, leaf, key, back-to-back, histogram, interval, frequency, shape, symmetric, skewed, two peaks, centre, spread, range, outlier, cluster
Dear families
This week in maths, Year 7 is drawing data displays, including stem-and-leaf plots, and describing what they show: the shape of the data, its centre and spread, and any unusual values.
Try this at home
- Write down the ages of everyone at a family gathering and make a stem-and-leaf plot together.
- Look at a graph in a newspaper or on the news. Is the data bunched up, spread out or lopsided?
- Time ten tasks of the same kind, such as brushing teeth. Is one time very different from the rest? Why?
- Ask: if one person in a group is much older than the others, what happens to the average age?
Data this week
Fill one row a day with something you counted or measured, and one thing its display showed.
| Day | What we counted or measured | What the display showed |
|---|---|---|
| Monday | ||
| Tuesday | ||
| Wednesday | ||
| Thursday | ||
| Friday |
Printed from the free seegongsik Describing data distributions teaching pack · seegongsik.com/au/y7/statistics/AC9M7ST02/pack
Dot plots and shape
A dot plot shows every value and the shape of the data at a glance. Students draw a dot plot, sort dot plots by shape, and see that a long tail pulls the mean towards it while the median stays with the bulk of the data.
We are learning to
- draw a dot plot with an even scale and one dot for each value,
- describe its shape as symmetric, skewed right, skewed left or two peaks,
- explain where the mean and median sit in skewed data.
Success criteria
- I can name a dot plot’s shape and say which way its tail points.
- I can explain why the mean of skewed data sits towards the tail.
You need
The worksheet, rulers, calculators, and the shape cards (one set per pair).
Lesson flow (about 60 minutes)
| 10 min | Shapes on the board Show the board (see the box). Ask: ““Skewed right” has most of its dots on the left. Why is it named for the right?” |
| 25 min | Draw and sort Part A on their own, keeping the dots the same size and evenly stacked; then pairs sort the shape cards for Part B. |
| 25 min | Mean and median Part C, then compare answers as a class; early finishers do Part D. Ask: “Three students sent 6 or more messages. Which measure do they pull, and which do they leave alone?” |
Two shorter sessions instead? Split after Part A; start the second with Part B.
Watch for these ideas
- Naming the skew by where the data bunches instead of where the tail stretches.
- Dots of different sizes or uneven gaps, which change the shape the plot shows.
- Calling any lopsided plot skewed when it has two peaks.
Answers
- Part A: roughly symmetric, peaking at 5; median 5, mode 5, range 6.
- Part B: symmetric C, G, L; skewed right A, D, J; skewed left E, H, K; two peaks B, F, I.
- Part C: mean 44 ÷ 15 ≈ 2.9, median 2; the mean is larger, pulled right by the few high values in the tail.
- Part D: plots vary, with the tail on the left; the mean sits below the median, pulled left by the small values.
Dot plots and shape
A. Draw a dot plot
Hours of sport played last week by 20 students: 3, 5, 4, 6, 2, 5, 7, 4, 5, 3, 6, 5, 4, 8, 5, 6, 4, 3, 5, 6
B. Sort the shape cards
Write each card’s letter under its shape.
| symmetric | skewed right | skewed left | two peaks |
|---|---|---|---|
C. Mean and median in skewed data
Text messages sent in one hour by 15 students.
D. Stretch
Sketch a dot plot of 12 values that is skewed left. Where is its mean compared with its median? Explain.
Stem-and-leaf plots
A stem-and-leaf plot groups data like a column graph while keeping every value. Students build an ordered plot with a key, and read the median, mode, range and shape from one.
We are learning to
- build an ordered stem-and-leaf plot, with a key,
- read the median, mode, range and mean from a stem-and-leaf plot,
- describe its shape by turning it on its side.
Success criteria
- I can split each value into a stem and a leaf, and order the leaves.
- I can find the median by counting leaves to the middle position.
You need
The worksheet and calculators.
Lesson flow (about 60 minutes)
| 10 min | Stems and leaves Show the board (see the box). Ask: “Why must the leaves be in order before you look for the median?” |
| 25 min | Build Part A: an unordered plot first, copying each value in turn, then a second, ordered plot. Model the key. |
| 25 min | Read Part B on their own; early finishers do Part C. Ask: “Turn the plot a quarter turn anticlockwise. What shape do the rows make?” |
Two shorter sessions instead? Split after Part A; start the second with Part B.
Watch for these ideas
- Leaving leaves unordered, or leaving out a stem with no leaves.
- Leaving off the key, so nobody knows whether 4 | 2 means 42 or 4.2.
- Counting rows instead of leaves to find the middle.
Answers
- Part A: 5 | 5 8; 6 | 2 4 7 8 9; 7 | 1 2 3 4 5 6 7 9; 8 | 1 5 8; 9 | 0 3; median (73 + 74) ÷ 2 = 73.5, range 38, no mode; roughly symmetric.
- Part B: n = 15; median 28 (the 8th age), mode 22, range 40, mean 457 ÷ 15 ≈ 30.5; skewed right, as most people are in their 20s and 30s with a tail of older players.
- Part C: 12 | 2 4 8 9; 13 | 1 1 5 8; 14 | 0 6 (key 12 | 4 means 12.4 s); median 13.1 s.
Stem-and-leaf plots
A. Build a plot
Quiz scores out of 100 for 20 students, in the order they were marked: 67, 72, 58, 81, 75, 64, 90, 73, 69, 77, 85, 62, 71, 79, 55, 88, 74, 68, 93, 76
B. Read a plot
The ages of 15 people at a beach volleyball game.
| 1 | 8 9 |
| 2 | 1 2 2 5 6 8 |
| 3 | 0 1 3 7 |
| 4 | 2 5 |
| 5 | 8 |
- How many people are there? ________
- Find the median, the mode and the range. ______________
- Find the mean, to one decimal place. ________
- Describe the shape. ______________________
C. Stretch
Make a stem-and-leaf plot of these times, using whole seconds as stems and tenths as leaves. Write its key and find the median.
12.4, 13.1, 12.8, 14.0, 13.5, 12.2, 13.8, 13.1, 14.6, 12.9 seconds
Comparing two groups
A back-to-back stem-and-leaf plot puts two groups on one set of stems. Students read and build them, and compare the groups on centre, spread and shape, not on one number alone.
We are learning to
- read a back-to-back stem-and-leaf plot, with the left leaves read outwards from the stem,
- compare two groups using a measure of centre and a measure of spread,
- write a comparison that names both groups and uses numbers.
Success criteria
- I can read a value on either side of a back-to-back plot.
- I can compare two groups with a sentence about centre and a sentence about spread.
You need
The worksheet and calculators.
Lesson flow (about 60 minutes)
| 10 min | Same centre, different spread Show the board (see the box). Ask: “All three sets have a mean of 6. Would you call them the same? What else would you report?” |
| 25 min | Read a back-to-back plot Part A. Model reading 4 | 3 on the left as 34, not 43. |
| 25 min | Build and compare Parts B and C; early finishers do Part D. Ask: “Which tells you more about the two groups: their medians, their ranges, or both?” |
Two shorter sessions instead? Split after Part A; start the second with Part B.
Watch for these ideas
- Reading the left side in the wrong order: its leaves grow outwards from the stem.
- Comparing only the largest values, or only the means.
- Statements with no numbers, such as “7B did better”.
Answers
- Part A: 7A median 37, range 24, mean 438 ÷ 12 = 36.5; 7B median 38.5, range 35, mean 432 ÷ 12 = 36. Similar centres; 7B’s marks are far more spread out.
- Part B: Year 7 in order 12, 18, 21, 24, 25, 27, 30, 33, 35, 41, median 26, range 29; Year 9 in order 22, 26, 29, 31, 34, 36, 38, 40, 44, 49, median 35, range 27.
- Part C: for example, Year 9 did more push-ups, with a median of 35 against 26, while the spreads are similar (ranges 27 and 29).
- Part D: either, with a reason: 7A is the safer, more consistent pick; 7B has the higher top marks.
Comparing two groups
A. Read a back-to-back plot
Marks out of 50 for two classes of 12 students.
| 7A | 7B | |
|---|---|---|
| 1 | 5 9 | |
| 8 3 | 2 | 6 |
| 9 8 6 5 4 1 | 3 | 0 3 7 |
| 7 4 2 1 | 4 | 0 3 5 6 8 |
| 5 | 0 |
- Find each class’s median. 7A ______ 7B ______
- Find each class’s range. 7A ______ 7B ______
- Find each class’s mean. 7A ______ 7B ______
- Which class is more consistent? Why? ______________
B. Build a back-to-back plot
Push-ups in a minute, Year 9: 31, 26, 44, 22, 38, 34, 49, 29, 40, 36
C. Compare
Write two sentences comparing Year 7 and Year 9: one about centre and one about spread. Use numbers.
D. Stretch
You need four students from 7A or 7B for a quiz team. Use the plot in Part A to argue for each class in turn.
Histograms and outliers
Measured data rarely repeats exactly, so it is grouped into equal intervals and drawn as a histogram, with columns that touch. Students draw one, then meet an outlier and decide what it does to each measure and whether it belongs.
We are learning to
- group measured data into equal intervals and draw a histogram,
- find an outlier’s effect on the mean and the median, and decide whether to keep it.
Success criteria
- I can tally data into intervals and draw a histogram with touching columns.
- I can explain why an outlier moves the mean much more than the median.
You need
The worksheet, rulers and calculators.
Lesson flow (about 60 minutes)
| 10 min | One far value Show the board (see the box). Ask: “The mean moved and the median did not. Which would you report for these values, and why?” |
| 25 min | Group and draw Part A: tally the heights into the table first, crossing each off as it is counted, then draw the histogram. |
| 25 min | Outliers Parts B and C; early finishers do Part D. Ask: “Is the 48 mm day a mistake? What would you do with it?” |
Two shorter sessions instead? Split after Part A; start the second with Part B.
Watch for these ideas
- Gaps between the columns, as in a column graph: intervals of measured data meet, so the columns touch.
- Counting a value such as 150 in two intervals, or in none.
- Deleting an outlier just because it is inconvenient, when it is a real value.
Answers
- Part A: 140–144: 1, 145–149: 4, 150–154: 8, 155–159: 7, 160–164: 3, 165–169: 1; roughly symmetric, peaking at 150–154 cm.
- Part B: with the 48 mm day, mean 60 ÷ 12 = 5 mm and median 1 mm; without it, mean 12 ÷ 11 ≈ 1.1 mm and median 1 mm.
- Part C: (1) correct or remove it, as no one reacts in 0.02 s; (2) keep it, as the storm was real; the median gives a typical day.
- Part D: 140–149: 5, 150–159: 15, 160–169: 4; the peak and the shape inside each interval are lost.
Histograms and outliers
A. Group and draw
Heights of 24 students (cm): 148, 152, 156, 143, 151, 159, 162, 147, 153, 155, 150, 158, 146, 154, 161, 157, 152, 149, 166, 155, 151, 157, 153, 160
| Height (cm) | Frequency |
|---|---|
| 140–144 | |
| 145–149 | |
| 150–154 | |
| 155–159 | |
| 160–164 | |
| 165–169 |
B. An outlier
Rainfall (mm) at a school on 12 days: 0, 2, 1, 0, 3, 1, 48, 2, 0, 1, 2, 0
- Which value is the outlier? ________
- Find the mean and the median with it. ______________
- Find the mean and the median without it. Which measure changed more? ______________
C. Keep it or not?
- In a reaction-time test, one result is 0.02 s, recorded when a student pressed before the light came on. ______________
- The 48 mm day in Part B was a real storm. ______________
D. Stretch
Group the heights into intervals of 10 cm (140–149, 150–159, 160–169). What does the new histogram hide?
Describing a distribution
A full description covers shape, centre and spread, and names any outlier. Students match displays to descriptions, compare two teams, choose displays and formulas, then play the quiz.
We are learning to
- describe a distribution by its shape, centre and spread, naming any outlier,
- choose a display for a purpose, and the spreadsheet formulas for the four measures.
Success criteria
- I can describe a display in one sentence that covers shape, centre and spread.
- I can compare two groups and back the comparison with numbers.
You need
The worksheet, the match cards, calculators, and the unit’s self-check quiz on the board.
Lesson flow (about 60 minutes)
| 10 min | Three measures Show “Mean, median and mode” (it opens on Set A); press “Set C”, then “show median”. Ask: “In Set C the mean is 8 but the median is 6. What in the data causes the gap?” |
| 30 min | Cards and problems Pairs match the cards (problem 1), then problems 2 to 6; 7 and 8 are the stretch. |
| 20 min | Class quiz game Run the self-check quiz as a class game (see the box), then wrap up. |
Two shorter sessions instead? Split after problem 2; start the second with problem 3.
Watch for these ideas
- Describing a distribution with one number. Shape, centre and spread each say something different.
- Matching by shape alone, when two displays share a shape and only the median and range tell them apart.
- Typing a spreadsheet range as A1-A20 instead of A1:A20.
Answers with working
- A 3, B 5, C 1, D 6, E 4, F 2.
- Team P: symmetric, median 4.5, range 5, mean 54 ÷ 12 = 4.5. Team Q: skewed right, median 2, range 10, mean 38 ÷ 12 ≈ 3.2. P usually scores more, and more consistently.
- (a) a histogram; (b) a stem-and-leaf plot or a dot plot; (c) a dot plot.
- =AVERAGE(A1:A20), =MEDIAN(A1:A20), =MODE(A1:A20), =MAX(A1:A20)-MIN(A1:A20).
- No: the mean is a balance point, and few students, perhaps none, are exactly that height.
- For example 1, 20: 1, 4, 5, 5, 6, 7, 20 has median 5 and mean 48 ÷ 7 ≈ 6.86, up from 5.4.
- For example 2, 4, 5, 6, 6, 6, 6, 7, 8, 10: total 60, so the mean is 6; 10 − 2 = 8.
- Plots vary: for example two peaks against one central peak, with the same middle and ends.
Describing a distribution
Problems (7 and 8 are the stretch)
- Match each display card with its description card.
A → B → C → D → E → F → - Goals scored in 12 games by two teams. Compare them: shape, centre and spread.Team PTeam Q
- Which display would you choose?
(a) the heights of 300 students ________ (b) 15 test scores ________
(c) the number of siblings of each student in a class ________ - Data is in cells A1 to A20 of a spreadsheet. Write the formulas for the mean, the median, the mode and the range.
- “The mean height is 154 cm, so most students are 154 cm tall.” Do you agree? Explain. ______________
- Add two values to 4, 5, 5, 6, 7 so that the median stays the same but the mean goes up. ______________
- Stretch. Write a symmetric data set of 10 whole numbers with a mean of 6 and a range of 8.
- Stretch. Sketch two dot plots with the same median and range but different shapes.
Dot-plot shape cards
Day 1, one set per pair. Cut out the 12 cards and sort them by shape: symmetric, skewed right, skewed left, or two peaks.
Display and description cards
Day 5, one set per pair. Cut out the 12 cards. Match each display (A to F) with its description (1 to 6).
What we know: Describing data distributions
Work on your own. Show your working.
- Describe the shape of this dot plot and find its median. ______________________
- Find the median, the mode and the range from this plot. ______________________
3 1 4 8 4 2 5 5 7 5 1 3 8 6 6 Key: 4 | 2 means 42 - Make an ordered stem-and-leaf plot, with a key, of 27, 13, 35, 21, 18, 29, 32, 24.
- In a back-to-back plot of Class A (left) and Class B (right), what does 7 | 4 | 2 mean? ______________________
- Which value in 12, 14, 15, 15, 16, 41 is an outlier? How much does it raise the mean? ______________
- Why do the columns of a histogram touch? ______________________________
- Group A has a median of 30 and a range of 12; group B has a median of 30 and a range of 40. Describe the difference. ______________
- A data set’s mean is much larger than its median. What shape is it likely to have? Explain. ______________
How did this week feel? Circle one: I can teach it · I am getting there · I need more help
Answers and marking guide
Answers with working
- skewed left: the data bunches at 6 and 7 with a tail towards the small values; median 6 (20 values, the 10th and 11th).
- Median 45 (the 6th of 11), mode 45, range 66 − 31 = 35.
- 1 | 3 8; 2 | 1 4 7 9; 3 | 2 5, with a key such as 1 | 3 means 13.
- 47 for Class A and 42 for Class B.
- 41: the mean is 113 ÷ 6 ≈ 18.83 with it and 72 ÷ 5 = 14.4 without, about 4.4 higher.
- The intervals of measured data meet, with no gaps between them, so the columns meet too.
- The same centre, but group B is far more spread out.
- Skewed right: a tail of large values pulls the mean above the median.
A quick three-level guide
| Idea | Working towards | At standard | Beyond |
|---|---|---|---|
| Shape (Q1, Q8) | names the side where the data bunches | names symmetric and skewed shapes by their tails | links the shape to where the mean sits |
| Stem-and-leaf (Q2, Q3) | leaves unordered, or no key | builds an ordered plot with a key and reads the median | reads a back-to-back plot both ways |
| Outliers and displays (Q5, Q6) | drops a value without a reason | finds an outlier’s effect, and draws touching columns | decides when to keep an outlier |
| Comparing (Q4, Q7) | compares one number only | compares centre and spread with numbers | uses shape as well |
Eight questions, four ideas. A student at standard draws and reads dot plots, stem-and-leaf plots and histograms, describes shape, centre and spread, and explains what an outlier does.
Weekly class record
Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.
| Name | Dot plots | Stem-leaf | Comparing | Histograms | Describing |
|---|---|---|---|---|---|
The five columns are the five days: dot plots and shape, stem-and-leaf plots, comparing two groups, histograms and outliers, and describing.