Which data set has the smaller standard deviation? A: 50, 52, 54… Step by Step

Step by step

The question

Question: Which data set has the smaller standard deviation?

A: 50, 52, 54, 56, 58.

B: 1, 2, 3, 4, 5.

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 270.

270 divided by 5 is 54.

mean of A = 50 + 52 + 54 + 56 + 585 = 2705 = 54

SD = how far the numbers usually are from the mean

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

The total is 15.

15 divided by 5 is 3.

mean of B = 1 + 2 + 3 + 4 + 55 = 155 = 3

2 Distances from the mean

Under each number of A, we write how far it is from the mean, 54.

A : 50, 52, 54, 56, 58

how far = bigger − smaller

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

A total = 4 + 2 + 0 + 2 + 4 = 12

Now the same for B.

Its mean is 3.

B : 1, 2, 3, 4, 5

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

B total = 2 + 1 + 0 + 1 + 2 = 6

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 6.

= A: 12 > 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

So B has the smaller standard deviation.

= B has the smaller SD

4 Check

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

A: 8.

B: 4.

A is spread wider too.

So it is right!

range A: 58 − 50 = 8, B: 5 − 1 = 4

Let's look back.

First: find each mean.

Then: distances from the mean.

Then: compare.

Last: check.

Answer