Which data set has the smaller standard deviation? A: 2, 6, 10, 14… Step by Step

Step by step

The question

Question: Which data set has the smaller standard deviation?

A: 2, 6, 10, 14.

B: 6, 7, 9, 10.

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 32.

32 divided by 4 is 8.

mean of A = 2 + 6 + 10 + 144 = 324 = 8

SD = how far the numbers usually are from the mean

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

The total is 32.

32 divided by 4 is 8.

mean of B = 6 + 7 + 9 + 104 = 324 = 8

2 Distances from the mean

Under each number of A, we write how far it is from the mean, 8.

A : 2, 6, 10, 14

how far = bigger − smaller

We add the distances: 6 plus 2 plus 2 plus 6 is 16.

A total = 6 + 2 + 2 + 6 = 16

Now the same for B.

Its mean is 8.

B : 6, 7, 9, 10

We add the distances: 2 plus 1 plus 1 plus 2 is 6.

B total = 2 + 1 + 1 + 2 = 6

3 Compare

Both sets have 4 numbers.

So we can just compare the totals.

4 numbers in each: compare the totals

A's numbers are further from the mean, 16 against 6.

= A: 16 > 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 B has the smaller standard deviation.

= B has the smaller SD

4 Check

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

A: 12.

B: 4.

A is spread wider too.

So it is right!

range A: 14 − 2 = 12, B: 10 − 6 = 4

Let's look back.

First: find each mean.

Then: distances from the mean.

Then: compare.

Last: check.

Answer