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

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The question

Question: Which data set has the bigger standard deviation?

A: 3, 5, 7.

B: 103, 105, 107.

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 15.

15 divided by 3 is 5.

mean of A = 3 + 5 + 73 = 153 = 5

SD = how far the numbers usually are from the mean

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

The total is 315.

315 divided by 3 is 105.

mean of B = 103 + 105 + 1073 = 3153 = 105

2 Distances from the mean

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

A : 3, 5, 7

how far = bigger − smaller

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

A total = 2 + 0 + 2 = 4

Now the same for B.

Its mean is 105.

B : 103, 105, 107

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

B total = 2 + 0 + 2 = 4

3 Compare

Both sets have 3 numbers.

So we can just compare the totals.

3 numbers in each: compare the totals

The distances are the same: 4 and 4.

The numbers are just as far from their mean.

= A: 4 = B: 4

The same distances from the mean means the same standard deviation.

= same distances → same SD

Careful!

Bigger numbers do not mean more spread.

The SD only looks at how far each number is from its own mean.

Careful: not bigger numbers → bigger SD ✗

So the two sets have the same standard deviation.

= the same SD

4 Check

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

A: 4.

B: 4.

The same width too.

So it is right!

range A: 7 − 3 = 4, B: 107 − 103 = 4

Let's look back.

First: find each mean.

Then: distances from the mean.

Then: compare.

Last: check.

Answer