Which data set has the bigger standard deviation? A: 6, 6, 6, 6. B… Step by Step

Step by step

The question

Question: Which data set has the bigger standard deviation?

A: 6, 6, 6, 6.

B: 5, 6, 6, 7.

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 24.

24 divided by 4 is 6.

mean of A = 6 + 6 + 6 + 64 = 244 = 6

SD = how far the numbers usually are from the mean

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

The total is 24.

24 divided by 4 is 6.

mean of B = 5 + 6 + 6 + 74 = 244 = 6

2 Distances from the mean

Under each number of A, we write how far it is from the mean, 6.

A : 6, 6, 6, 6

how far = bigger − smaller

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

A total = 0 + 0 + 0 + 0 = 0

Now the same for B.

Its mean is 6.

B : 5, 6, 6, 7

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

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

3 Compare

Both sets have 4 numbers.

So we can just compare the totals.

4 numbers in each: compare the totals

B's numbers are further from the mean, 2 against 0.

= A: 0 < B: 2

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 bigger standard deviation.

= B has the bigger SD

4 Check

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

A: 0.

B: 2.

B is spread wider too.

So it is right!

range A: 6 − 6 = 0, B: 7 − 5 = 2

Let's look back.

First: find each mean.

Then: distances from the mean.

Then: compare.

Last: check.

Answer