Which data set has the smaller standard deviation? A: 40, 45, 50… Step by Step

Step by step

The question

Question: Which data set has the smaller standard deviation?

A: 40, 45, 50, 55, 60.

B: 30, 50, 50, 50, 70.

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 250.

250 divided by 5 is 50.

mean of A = 40 + 45 + 50 + 55 + 605 = 2505 = 50

SD = how far the numbers usually are from the mean

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

The total is 250.

250 divided by 5 is 50.

mean of B = 30 + 50 + 50 + 50 + 705 = 2505 = 50

2 Distances from the mean

Under each number of A, we write how far it is from the mean, 50.

A : 40, 45, 50, 55, 60

how far = bigger − smaller

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

A total = 10 + 5 + 0 + 5 + 10 = 30

Now the same for B.

Its mean is 50.

B : 30, 50, 50, 50, 70

We add the distances: 20 plus 0 plus 0 plus 0 plus 20 is 40.

B total = 20 + 0 + 0 + 0 + 20 = 40

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, 40 against 30.

= A: 30 < B: 40

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 smaller standard deviation.

= A has the smaller SD

4 Check

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

A: 20.

B: 40.

B is spread wider too.

So it is right!

range A: 60 − 40 = 20, B: 70 − 30 = 40

Let's look back.

First: find each mean.

Then: distances from the mean.

Then: compare.

Last: check.

Answer