Which data set has the bigger standard deviation? A: 2, 8, 8, 8, 14… Step by Step

Step by step

The question

Question: Which data set has the bigger standard deviation?

A: 2, 8, 8, 8, 14.

B: 5, 6, 8, 10, 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 40.

40 divided by 5 is 8.

mean of A = 2 + 8 + 8 + 8 + 145 = 405 = 8

SD = how far the numbers usually are from the mean

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

The total is 40.

40 divided by 5 is 8.

mean of B = 5 + 6 + 8 + 10 + 115 = 405 = 8

2 Distances from the mean

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

A : 2, 8, 8, 8, 14

how far = bigger − smaller

We add the distances: 6 plus 0 plus 0 plus 0 plus 6 is 12.

A total = 6 + 0 + 0 + 0 + 6 = 12

Now the same for B.

Its mean is 8.

B : 5, 6, 8, 10, 11

We add the distances: 3 plus 2 plus 0 plus 2 plus 3 is 10.

B total = 3 + 2 + 0 + 2 + 3 = 10

3 Compare

Both sets have 5 numbers.

So we can just compare the totals.

5 numbers in each: compare the totals

A's numbers are further from the mean, 12 against 10.

= A: 12 > B: 10

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: 12.

B: 6.

A is spread wider too.

So it is right!

range A: 14 − 2 = 12, B: 11 − 5 = 6

Let's look back.

First: find each mean.

Then: distances from the mean.

Then: compare.

Last: check.

Answer