Which data set is more spread out? A: 5, 5, 6, 7, 7. B: 1, 3, 6, 9… Step by Step

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The question

Question: Which data set is more spread out?

A: 5, 5, 6, 7, 7.

B: 1, 3, 6, 9, 11.

1 Find each mean

The standard deviation, SD, tells how far the numbers usually are from the mean.

So first we need each mean.

A: add them, then divide by how many.

The total is 30.

30 divided by 5 is 6.

mean of A = 5 + 5 + 6 + 7 + 75 = 305 = 6

SD = how far the numbers usually are from the mean

Now B: add them, then divide by how many.

The total is 30.

30 divided by 5 is 6.

mean of B = 1 + 3 + 6 + 9 + 115 = 305 = 6

2 Distances from the mean

Under each number of A, we write how far it is from the mean, 6.

A : 5, 5, 6, 7, 7

how far = bigger − smaller

We add the distances: 1 plus 1 plus 0 plus 1 plus 1 is 4.

A total = 1 + 1 + 0 + 1 + 1 = 4

Now the same for B.

Its mean is 6.

B : 1, 3, 6, 9, 11

We add the distances: 5 plus 3 plus 0 plus 3 plus 5 is 16.

B total = 5 + 3 + 0 + 3 + 5 = 16

3 Compare

Both sets have 5 numbers.

So we can just compare the totals.

5 numbers in each: compare the totals

B's numbers are further from the mean, 16 against 4.

= A: 4 < B: 16

Further from the mean means a bigger standard deviation.

We did not work out the SD itself.

But it grows with these distances, so the bigger distance goes with the bigger SD.

= further from the mean → bigger SD

Careful!

Both means are the same, but the spread is not.

The SD is about the distances.

Careful: not same mean → same SD ✗

So B has the bigger standard deviation.

= B has the bigger SD

4 Check

Let's check with the range, biggest minus smallest.

A: 2.

B: 10.

B is spread wider too.

So it is right!

range A: 7 − 5 = 2, B: 11 − 1 = 10

Let's look back.

First: find each mean.

Then: distances from the mean.

Then: compare.

Last: check.

Answer