Range, median, mean and mode: 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 build-a-set challenge cards, which-measure 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 at 100%, one set per pair: the build-a-set cards (Day 3) and the which-measure cards (Day 5).
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

Four numbers summarise a data set. The mean shares the total equally; the median is the middle value once the data is in order; the mode is the value that occurs most often; and the range, the largest value take away the smallest, shows how far the data stretches. Working them out is quick. The real skill is choosing: one far-off value drags the mean but hardly moves the median, and a question about what is most common is answered by the mode.

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 work this week

A discrete variable is counted (people, goals, pets); a continuous variable is measured (lengths, times, masses), so its values are rounded. Means are given to one decimal place unless they come out exactly, and a mean like 3.75 people is fine: it is a centre, not a person. A data set can have one mode, more than one, or none. Calculators are fine for totals once students can explain the steps.

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

Days 1 and 2 calculate the four measures, from data the class collects itself; Day 3 shows why the mean balances and works backwards from it; Day 4 reads the measures from tables and dot plots; and Day 5 chooses and justifies the measure that tells the truest story. It builds on the mode and range of Year 6 (AC9M6ST01), and it leads into describing and comparing distributions (AC9M7ST02) and running statistical investigations (AC9M7ST03).

Materials for the week (one trip)

• From the classroom: pencils, rulers to measure hand spans (Day 1), calculators, this pack printed at 100%.
• Cut-out sheet 1, one set per pair: 12 build-a-set challenge cards (Day 3).
• Cut-out sheet 2, one set per pair: 12 which-measure cards (Day 5).

Words for the word wall

data, variable, discrete, continuous, mean, median, mode, range, ordered, outlier, frequency, frequency table, dot plot, centre, spread, typical, measure of central tendency

Dear families

This week in maths, Year 7 is finding the mean, median, mode and range of data sets, and deciding which one gives the fairest picture, such as why a typical house price is usually reported as a median, not a mean.

Try this at home

• Time five trips to the shops or to school. What are the mean and the median? Which describes a usual trip better?
• Look for “average” in a news story or an advertisement. Does it say which average it means?
• Count the people in each car in a car park. What is the mode, and why might a car maker want to know it?
• Ask: can the average number of people in a car be 1.6? What does it mean?

Numbers this week

Fill one row a day with something you counted or measured, and one of its measures.

Printed from the free seegongsik Range, median, mean and mode teaching pack · seegongsik.com/au/y7/statistics/AC9M7ST01/pack

The mean and the range

The mean shares the total equally; the range shows how far the data stretches. Students sort variables into discrete and continuous, collect two class data sets, and find the mean and range of each.

We are learning to

• tell a discrete variable (counted) from a continuous one (measured),
• calculate the mean and the range of a data set,
• say what a mean that is not a possible value, such as 3.75 people, still tells us.

Success criteria

• I can say whether a variable is counted or measured, and why.
• I can find the mean and the range and say what each one tells me.

You need

The worksheet, rulers for measuring hand spans, and calculators.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Dividing by the wrong count, or leaving out a value of 0.
• Giving the largest value as the range. The range is the largest take away the smallest.
• Rounding a mean like 3.75 to 4 to make it a “real” number of people.

Answers

• Part A: (1) discrete, (2) continuous, (3) discrete, (4) continuous, (5) continuous, (6) discrete, (7) continuous, (8) discrete.
• Part B: answers depend on the class; check each mean is the total divided by 10, and each range uses the largest and smallest values.
• Part C: hand spans mean 227 ÷ 12 ≈ 18.9 cm, range 21 − 17 = 4 cm; people at home mean 45 ÷ 12 = 3.75, range 4.
• Part D: mean 252 ÷ 13 ≈ 19.4 cm, range 8 cm. The range doubles; the mean moves by 0.5 cm.

The mean and the range

A. Counted or measured?

Write discrete (counted) or continuous (measured) for each variable.

B. Our class

Measure your hand span to the nearest half centimetre. Record ten classmates, then find the mean and the range of each column.

C. Another class

Twelve students in another Year 7 class measured their hand spans (cm) and counted the people at home.

1. Find the mean and the range of the hand spans.   ______________________
2. Find the mean and the range of the people at home.   ______________________
3. Your mean in question 2 is not a number of people any household could have. What does it still tell you?

D. Stretch

A thirteenth student with a hand span of 25 cm joins the class. Find the new mean and range. Which changed more, and why?

The median and the mode

The median is the middle value once the data is in order; the mode is the value that occurs most often. Students order data sets with odd and even counts, and meet data with one mode, two modes and none.

We are learning to

• order a data set and find the median, with an odd or an even number of values,
• find the mode, including data with two modes or none.

Success criteria

• I order the data before I look for the middle, every time.
• I can find the median of an even count as the mean of the two middle values.

You need

The worksheet.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Taking the middle of the list as written, before ordering it.
• Giving how often the mode occurs instead of the value itself.
• Saying a set with no repeated value has a mode of 0.

Answers

• Part A: 9, 10, 11, 12, 12, 12, 14, 15, 16: median 12, mode 12; 15, 20, 25, 30, 35, 40, 40: median 30, mode 40.
• Part B: 9, 11, 12, 15, 15, 20: median 13.5, mode 15; 9, 11, 13, 14, 16, 18, 18, 20: median 15, mode 18.
• Part C: (1) no mode; (2) 2 and 6; (3) 7.
• Part D: for example 1, 4, 6, 8, 8: two 8s for the mode, 6 in the middle, and 8 − 1 = 7.

The median and the mode

A. Odd counts

Write each set in order, then find the median and the mode.

B. Even counts

With an even count, the median is the mean of the two middle values.

C. One mode, two or none?

D. Stretch

Write a set of five whole numbers with a median of 6, a mode of 8 and a range of 7. Explain how you built it.

The mean as a balance point

The values below the mean and the values above it are exactly the same total distance from it, so the mean is where the data balances. Students check the balance, work backwards from a mean to a missing value, and build sets to fit given measures.

We are learning to

• explain why the distances below and above the mean balance,
• use the mean times the count to find a total, and from it a missing value,
• build data sets that fit a given mean, median, mode or range.

Success criteria

• I can find a missing value when I know the mean and the other values.
• I can build a set that fits a card, and check it by working out every measure the card names.

You need

The worksheet and the build-a-set challenge cards (one set per pair).

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Expecting the mean to be in the middle of the values on the line, like the median.
• Averaging the known values instead of starting from the total.
• Building a set that fits one condition on a card but not the other.

Answers

• Part A: 3, 5, 6, 10: mean 6, below 4, above 4; 1, 2, 2, 4, 11: mean 4, below 7, above 7.
• Part B: (1) 5 × 7 − 28 = 7; (2) 4 × 12 − 34 = 14; (3) 6 × 4 − 5 × 3 = 9 goals.
• Part C: sets vary; one for each card: 1 (3, 4, 5), 2 (2, 4, 6, 9), 3 (2, 2, 3, 5, 8), 4 (3, 4, 5, 8, 10), 5 (1, 4, 6, 9), 6 (4, 6, 6, 6, 8), 7 (1, 3, 7, 8, 10, 12), 8 (1, 1, 2, 2, 10), 9 (2, 2, 5, 5), 10 (1, 2, 3, 5, 10), 11 (5, 5, 5), 12 (0, 5, 5, 5, 10).
• Part D: no: the distances below and above must balance, so unless every value is the same, some values sit above the mean.

The mean as a balance point

A. Does it balance?

Find the mean. Then write how far each value is from the mean, and total the distances below and above.

B. The missing value

1. Five quiz scores have a mean of 7. Four of them are 6, 9, 5, 8. What is the fifth?   ________
2. The mean of four numbers is 12. Three of them are 10, 15, 9. Find the fourth.   ________
3. A player’s mean was 3 goals a game after 5 games. After the next game it was 4. How many goals in that game?   ________

C. Build a set

Turn over four challenge cards. Write a set that fits each, and show your check.

D. Stretch

Can the mean of a data set be bigger than every value in it? Use the idea of balance to explain.

Data in tables and dot plots

Data often comes already counted, in a frequency table or a dot plot. Students find all four measures without writing out every value, and see what one far value does to each.

We are learning to

• find the mean, median, mode and range from a frequency table,
• read the same measures from a dot plot,
• explain which measures one far value changes, and which it leaves alone.

Success criteria

• I can total a frequency table by multiplying each value by its frequency.
• I can find the median of a table or dot plot by counting to the middle position.

You need

The worksheet and calculators.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Dividing by the number of rows in the table (5) instead of the number of households (20).
• Giving the biggest frequency as the mode instead of its value.
• Finding the middle row of the table instead of the middle household.

Answers

• Part A: total 0 × 4 + 1 × 7 + 2 × 5 + 3 × 3 + 4 × 1 = 30; mean 30 ÷ 20 = 1.5; median 1 (the 10th and 11th households); mode 1; range 4.
• Part B: mean 265 ÷ 15 ≈ 17.7 s, median 16 s (the 8th time), mode 15 s, range 19 s.
• Part C: the median: the 31 s time pulls the mean above most of the times.
• Part D: only the mean (255 ÷ 15 = 17 s) and the range (12 s) change; the median and the mode stay the same.

Data in tables and dot plots

A. From a frequency table

Twenty households were asked how many pets they have. Fill in the last column, then find the four measures.

B. From a dot plot

Fifteen students timed how long they took to solve a puzzle, to the nearest second.

C. Which describes it best?

Which measure best describes a typical time for this class? Explain using the dot plot.

D. Stretch

1. The 31 s time was written down wrongly; it should be 21 s. Which of the four measures change, and which stay the same? Work out any that change.

Choosing a measure

Each measure answers a different question. Students sort situations by the measure that answers them, compare measures on real-looking data, and spot a misleading average. Then the quiz.

We are learning to

• choose the measure of centre that answers a question best, and justify it,
• explain how an outlier or the shape of the data affects each measure.

Success criteria

• I can say which measure I would report for a situation, and give a reason.
• I can show with numbers when the mean gives a misleading picture.

You need

The worksheet, the which-measure cards, calculators, and the unit’s self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after problem 2; start the second with problem 3.

Watch for these ideas

• Always reporting the mean because it “uses all the data”, even when one value dominates.
• Choosing the mode for a question about a typical amount when every value occurs once.
• Comparing two sets by their means alone, without their spread.

Answers with working

1. Mean C, F, I, L; median A, E, H, J; mode B, D, G, K. The mean for an equal share or values close together; the median when one far value would pull the mean, or for a half-way split; the mode for “most often”.
2. Mean 550 000 ÷ 6 ≈ $91 667; median (50 000 + 52 000) ÷ 2 = $51 000. The median: the mean is above every wage but the owner’s.
3. Size 8, the mode: it sold 4 times.
4. Both means 6.8; ranges 2 and 11. Player A is more consistent: the same centre, much less spread.
5. False: with an even count it can fall between two values, as 13.5 did on Day 2.
6. No: the mean is a balance point; few students, or none, may have scored exactly that.
7. Sets vary; for example 1, 1, 3, 6, 9: mean 4, median 3, mode 1. The mode describes only two of the five values, so report the median or the mean.
8. (10 × 8 − 26) ÷ 9 = 6.

Choosing a measure

Give a reason with every answer.

Problems (7 and 8 are the stretch)

1. Sort the which-measure cards. Write each card’s letter under the measure that answers it best.

   Mean	Median	Mode

2. A small business pays $45 000, $48 000, $50 000, $52 000, $55 000 and the owner $300 000. Find the mean and the median wage. Which is the fairer “typical” wage?
3. A shoe shop sold these sizes in a day: 9, 7, 8, 8, 10, 8, 9, 11, 8, 7. Which size should it reorder most of? Which measure did you use?   ______________
4. In a game, player A scored 6, 6, 7, 7, 8 and player B scored 1, 2, 7, 12, 12. Compare their means and ranges. Who is more consistent?
5. True or false: the median is always one of the values in the data. Explain.   ______________________
6. A class’s mean mark is 62. Does that mean most students scored about 62? Explain.
7. Stretch. Write a set of five whole numbers whose mean, median and mode are all different. Which would you report, and why?
8. Stretch. The mean of 10 numbers is 8. The number 26 is removed. What is the mean of the rest?   ________

Build-a-set challenge cards

Day 3, one set per pair. Cut out the 12 cards and shuffle them face down. Turn one over, write a set of numbers that fits it, and check it by working out every measure the card names. There is more than one answer to every card.

Which measure? cards

Day 5, one set per pair. Cut out the 12 cards. Sort them into three piles, mean, median and mode, by the measure that answers the question best, and be ready to say why.

What we know: Range, median, mean and mode

Work on your own. Show your working.

1. Find the mean of 4, 9, 6, 5, 11.   ________
2. Find the median of 8, 3, 10, 5, 7, 12.   ________
3. Find the mode and the range of 6, 2, 9, 6, 4, 6, 3.   ______________
4. Discrete or continuous? (a) the number of letters in a name   ________   (b) the time to walk to school   ________

5. Five numbers have a mean of 10. Four of them are 8, 12, 7, 9. Find the fifth.   ________
6. Ten students counted their siblings: 3 have 0, 5 have 1, 2 have 2. Find the mean and the median.   ______________
7. A canteen sells sandwiches for $5 to $9 and one catering platter for $45. Which measure gives the typical price of an item, and why?
8. Explain why one very large value changes the mean much more than the median.

Answers and marking guide

Answers with working

1. 35 ÷ 5 = 7.
2. In order 3, 5, 7, 8, 10, 12; the two middle values 7 and 8 give 7.5.
3. Mode 6 (three times); range 9 − 2 = 7.
4. (a) discrete: letters are counted; (b) continuous: time is measured.
5. 5 × 10 − 36 = 14.
6. Mean 9 ÷ 10 = 0.9; median 1, as the 5th and 6th values are both 1.
7. The median: the $45 platter would pull the mean above almost every price, while the median stays among the sandwiches.
8. The mean uses the size of every value, so a huge value adds a lot to the total; the median only depends on which value is in the middle.

A quick three-level guide

Eight questions, four ideas. A student at standard calculates all four measures, from a list or a table, tells counted data from measured data, and chooses the measure that describes the data honestly.

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 mean and range, the median and mode, the balance point, tables and dot plots, and choosing a measure.