Week 11 · Center and Spread

Thursday

Comparing two groups

// One number for the middle, one number for the scatter

⏱ about 20 min

Thursday: Comparing Two Groups

Raven lays two score sheets side by side on the Numbers Desk. "Sack hop times. Younger kids on the left, older kids on the right."

Rocket scans them. "The older kids are faster. Obviously. Let me build them a longer course."

"How much faster?" Nova asks. "And is the difference big compared with how spread out each group is?"

Rocket stacks two dot plots on the same number line. "The groups barely touch. Older kids sit around 12, younger around 16."

"What do you notice about the spread?" Raven asks.

"Both groups are tight," Rocket says. "Each one is about a second wide."

"Then a four second gap between the centers is huge," Raven says. "Five times the spread."

Nova brightens. "Now that is a fair comparison."

Two groups, one number line

At practice, 10 younger kids and 10 older kids each hopped the sack course once. Raven timed them in whole seconds.

To compare two groups, draw both on the same number line. Then compare the centers, and compare the spreads.

Group	Times in seconds, in order
Younger kids	14, 15, 15, 16, 16, 16, 16, 17, 17, 18
Older kids	10, 11, 11, 12, 12, 12, 12, 13, 13, 14

$A dot plot of the younger kids' seconds from 10 to 18, clustered around 16.$

$A dot plot of the older kids' seconds from 10 to 18, clustered around 12.$

Centers and spreads

Younger kids: the times add to 160, so the mean is 16. The median is 16. The MAD is 0.8.

Older kids: the times add to 120, so the mean is 12. The median is 12. The MAD is 0.8.

The difference between the means is 16 - 12 = 4 seconds.

Each group has a MAD of 0.8. A difference of 4 is 4 ÷ 0.8 = 5 MADs.

When the gap between centers is many MADs wide, the two groups hardly overlap. The difference is real, not just noise.

If the gap were less than one MAD, the dot plots would overlap a lot. Then you could not say one group is clearly faster.

	Younger kids	Older kids
Count	10	10
Mean	16	12
Median	16	12
Range	4	4
MAD	0.8	0.8

COMPARE THE GROUPS

Read the question.
Tap your answer.

$A dot plot of the younger kids' seconds$ What is the mean time of the younger kids, in seconds?

How many seconds separate the two means?

The gap between the means is 4 seconds and the MAD is 0.8. How many MADs is the gap?

The gap between the centers is about five MADs. How much do the two dot plots overlap?

How many older kids hopped the course in exactly 12 seconds? Type the number.

WHY THIS EXERCISEThe tallest stack in each group sits right at its center, which is why the two means are so easy to see.

MEDIANS TOO

Read the question.
Tap your answer.

What is the median time of the older kids?

How many seconds separate the two medians?

Rocket wants a longer course for the older kids. Does the data support him?

Statement	True or false?
Both groups have 10 kids.	?
The younger kids' mean time is 16 seconds.	?
A gap of five MADs between centers means the groups overlap a lot.	?
The two groups have the same MAD.	?
If the gap were less than one MAD, you could still say one group is clearly faster.	?

WHY THIS EXERCISEThe size of the gap, measured in spreads, decides whether a difference between groups means anything.

Try it

Say the comparison in one sentence: which group is faster, by how many seconds, and by how many MADs.
Then say what would change if both groups had a MAD of 4 instead.

Numbers Notebook: write a four-line report comparing the two groups. Count, center, spread, and what the gap means.Think of two groups at home you could compare the same way, such as morning and evening reading minutes. Write the question.Rocket wanted a longer course for the older kids. Write one sentence for and one sentence against, using the data.

You compared two groups fairly, using their spread as the ruler. Tomorrow you decide which measures fit which shapes.

Grown-up checked my answers

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