Which data set has the bigger standard deviation? A: 3, 4, 5, 6, 7… Step by Step

Step by step

The question

Question: Which data set has the bigger standard deviation?

A: 3, 4, 5, 6, 7.

B: 13, 15, 17, 19, 21.

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 25.

25 divided by 5 is 5.

mean of A = 3 + 4 + 5 + 6 + 75 = 255 = 5

SD = how far the numbers usually are from the mean

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

The total is 85.

85 divided by 5 is 17.

mean of B = 13 + 15 + 17 + 19 + 215 = 855 = 17

2 Distances from the mean

Under each number of A, we write how far it is from the mean, 5.

A : 3, 4, 5, 6, 7

how far = bigger − smaller

We add the distances: 2 plus 1 plus 0 plus 1 plus 2 is 6.

A total = 2 + 1 + 0 + 1 + 2 = 6

Now the same for B.

Its mean is 17.

B : 13, 15, 17, 19, 21

We add the distances: 4 plus 2 plus 0 plus 2 plus 4 is 12.

B total = 4 + 2 + 0 + 2 + 4 = 12

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, 12 against 6.

= A: 6 < B: 12

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

So B has the bigger standard deviation.

= B has the bigger SD

4 Check

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

A: 4.

B: 8.

B is spread wider too.

So it is right!

range A: 7 − 3 = 4, B: 21 − 13 = 8

Let's look back.

First: find each mean.

Then: distances from the mean.

Then: compare.

Last: check.

Answer