Which data set has the bigger standard deviation? A: 15, 30, 45. B… Step by Step

Step by step

The question

Question: Which data set has the bigger standard deviation?

A: 15, 30, 45.

B: 25, 30, 35, 40, 20.

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 90.

90 divided by 3 is 30.

mean of A = 15 + 30 + 453 = 903 = 30

SD = how far the numbers usually are from the mean

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

The total is 150.

150 divided by 5 is 30.

mean of B = 25 + 30 + 35 + 40 + 205 = 1505 = 30

2 Distances from the mean

Under each number of A, we write how far it is from the mean, 30.

A : 15, 30, 45

how far = bigger − smaller

We add the distances: 15 plus 0 plus 15 is 30.

A total = 15 + 0 + 15 = 30

Now the same for B.

Its mean is 30.

B : 25, 30, 35, 40, 20

We add the distances: 5 plus 0 plus 5 plus 10 plus 10 is 30.

B total = 5 + 0 + 5 + 10 + 10 = 30

3 Compare

The sets have 3 and 5 numbers.

More numbers make a bigger total, so the totals are not fair to compare.

We share each total by its count: the usual distance.

30 divided by 3 is 10.

A usual distance = 303 = 10

30 divided by 5 is 6.

B usual distance = 305 = 6

A's numbers are further from the mean, 10 against 6.

= A: 10 > B: 6

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 A has the bigger standard deviation.

= A has the bigger SD

4 Check

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

A: 30.

B: 20.

A is spread wider too.

So it is right!

range A: 45 − 15 = 30, B: 40 − 20 = 20

Let's look back.

First: find each mean.

Then: distances from the mean.

Then: compare.

Last: check.

Answer