D. Data · Data Literacy and Probability

Comparing Two or More Data Sets

D1.5 D1.6 A1About 50 minutes

Learning goals

I can use the mean, median, mode and range to compare two or more data sets.
I can explain why comparing means is fairer than comparing totals when groups are different sizes.
I can use my comparison to make an informed decision and explain it.

Words to know

compare
To look at how two or more things are alike and different.

consistent
Staying about the same each time. Consistent data has a small range.

informed decision
A choice made by looking carefully at the evidence (the data).

measure of spread
A number that tells how spread out data is. The range is one measure of spread.

Remember

In the last lesson you found the mean, median, mode and range. Today you'll use them like a detective's tools to compare data sets and make decisions.

Which airplane is better?

Jasmine folded two paper airplane designs. She threw each one 5 times and measured the distances in metres.

Design A: 4.2, 5.0, 4.8, 5.1, 4.9
Design B: 2.5, 6.8, 7.0, 3.1, 4.6

Which design is better? Let's find out with the measures.

Each dot is one flight, placed to scale. Dots that are very close are stacked so you can see them all. Design A's flights are bunched between 4.2 and 5.1 m. Design B's flights are spread from 2.5 to 7.0 m.

Measure	Design A	Design B
Mean	24.0 ÷ 5 = 4.8 m	24.0 ÷ 5 = 4.8 m
Median	4.2, 4.8, 4.9, 5.0, 5.1 → 4.9 m	2.5, 3.1, 4.6, 6.8, 7.0 → 4.6 m
Mode	none	none
Range	5.1 − 4.2 = 0.9 m	7.0 − 2.5 = 4.5 m

The two designs have the same mean, but very different ranges.

What the measures tell us

The means are the same (4.8 m). On average, both designs fly the same distance.
Design A's median is a bit higher (4.9 m vs 4.6 m).
Design A's range is much smaller (0.9 m vs 4.5 m). Design A is consistent. Design B is unpredictable: sometimes great, sometimes poor.

So which is "better"? It depends on the goal:

In a contest where the single longest flight wins, choose Design B. It has the two longest flights.
In a contest where you must land in a zone about 5 m away, choose Design A. It lands near 5 m almost every time.

That is an informed decision: a choice backed by data and a clear reason.

Worked example

Example 1: Groups of different sizes

Two reading groups recorded minutes read on Saturday.

Group A (4 students): 30, 40, 50, 40
Group B (6 students): 20, 30, 40, 50, 60, 70

Group B read 270 minutes in total and Group A read only 160. Can we say Group B's students read more?

The totals aren't fair to compare: Group B has more students, so its total is bigger just because there are more people.
Compare the means, which are like a fair share for each student. A: 160 ÷ 4 = 40 min. B: 270 ÷ 6 = 45 min.
Compare the medians. A: 30, 40, 40, 50 → (40 + 40) ÷ 2 = 40 min. B: 20, 30, 40, 50, 60, 70 → (40 + 50) ÷ 2 = 45 min.
Compare the ranges. A: 50 − 30 = 20 min. B: 70 − 20 = 50 min.

Yes, but for the right reason: a typical Group B student read a bit more (mean and median of 45 vs 40 minutes). Group B's reading times were also much more spread out (range 50 vs 20).

Worked example

Example 2: Make an informed decision

Two routes to school. Ethan timed each route on 5 mornings (minutes).

Route 1: 12, 13, 12, 14, 19
Route 2: 14, 15, 14, 15, 14

He must be at school within 16 minutes of leaving home. Which route should he take?

Route 1: mean 70 ÷ 5 = 14, median 13, range 19 − 12 = 7.
Route 2: mean 72 ÷ 5 = 14.4, median 14, range 15 − 14 = 1.
Route 1 is usually a little faster (lower median), but once it took 19 minutes. Maybe there was a long wait at a crosswalk.
Route 2 is consistent. It never took more than 15 minutes.

Route 2. Its small range shows it is reliable, and it was always under 16 minutes. Route 1 is often faster but sometimes too slow.

Route 1's dots are spread out (range 7), and one walk crosses the 16-minute line. Route 2's dots are bunched together (range 1), all safely left of the line.

Different conclusions, respectfully

Two people can look at the same data and make different choices because they have different goals. That's okay! Listen to the other person's reason, share yours with data, and ask questions like "What made you choose that?" Being open to other ideas makes everyone's thinking stronger.

Just for fun

Design B is the kind of paper airplane that flies 7 m when nobody is watching... and 2.5 m the moment you say "Hey, look at this!"

Hands-on activity

Ruler-drop reaction test

You need: a 30 cm ruler and a partner.

Your partner holds the ruler at the 30 cm end so it hangs down. Put your thumb and finger open around the 0 cm end, without touching it.
Your partner lets go without warning. Catch it as fast as you can! Read the number where you caught it (for example 14.5 cm). A smaller number means a faster reaction.
Do 5 catches with your writing hand and 5 with your other hand.
Find the mean, median and range for each hand.
Compare: Which hand is faster? Which is more consistent? Write two sentences using your measures.

Option: if you did the paper airplane activity, fold a second design and compare the two designs the same way.

Your turn

Two basketball players' points in 6 games:

Jordan: 12, 14, 13, 15, 12, 12 Priya: 4, 22, 18, 6, 20, 8

Find each player's mean.

Find each player's median.

Find each player's range.

Which player is more consistent? How do you know?

The coach needs someone who will score about 12 to 15 points every game. Who should she choose? Explain.

Give a reason someone might choose Priya instead.

Does Jordan's mode (12) help you compare the players? Explain.

More practice: warm-up

Set A has a range of 5. Set B has a range of 20. Which set is more consistent?

Why is it unfair to compare the total minutes read by a group of 4 and a group of 6?

Airplane table: which design had the higher median?

Two sets have the same mean. Does that mean they are the same?

Route example: which route had the smaller range?

More practice: on your own

Science fair: Team Red's bean plants grew 8, 9, 9, 10 cm. Team Blue's grew 5, 7, 11, 12, 15 cm. Compare the means and ranges.

In the last question, which team's plants had the tallest single plant?

Week 1 noon temperatures had a mean of −3°C and a range of 4°C. Week 2 had a mean of −3°C and a range of 15°C. Describe the difference.

Two classes' spelling scores out of 20: Class A median 16, range 6. Class B median 14, range 12. Make two comparison statements.

Why is it a good idea to use more than one measure when comparing data sets?

Challenge

Three skiers' times (s) on the same hill: Ana 42, 44, 43. Ben 39, 50, 40. Chloe 45, 45, 45. Who is fastest on average, who is most consistent, and who would you pick for one important race? Explain.

Group A (5 students) has a mean of 12 books read. Group B (10 students) has a mean of 9. Find the mean for all 15 students together.

Make up two data sets of 4 values each that have the same median but different ranges, and say which is more consistent.

Check your understanding

Quiz

Two data sets have the same mean. What else should you check to see how they differ?

Which set is more consistent: A (range 2) or B (range 14)?

Group A has 3 students and Group B has 8. To compare how much a typical student read, compare their:

Design A: mean 4.8 m, range 0.9 m. Design B: mean 4.8 m, range 4.5 m. For landing in a target zone 5 m away, which is the better choice?

Route 1 takes 12 to 19 minutes. Route 2 takes 14 to 15 minutes. If you must never take more than 16 minutes, which route?

Data: Team X scores 3, 4, 5. Team Y scores 1, 4, 7. Which statement is true?

For parents and teachers

What this lesson teaches: using the mean, median, mode and range to compare two or more data sets (D1.5), then drawing conclusions and making informed decisions (D1.6). Discussing different conclusions respectfully builds Strand A skills (A1).

How to help at home:

Compare real things: two routes to the store, two brands of batteries, two players' scores. Ask "Which is faster on average? Which is more reliable?"
Ask your child to back every choice with a measure: "I chose Route 2 because its range is only 1 minute."
Point out that the best choice depends on the goal.

Common difficulties:

Comparing totals when the groups are different sizes.
Looking only at the mean and ignoring the spread.
Thinking there is always one "right" answer. Good answers use data and match a stated goal.