Worked examples
The question
Question: Which data set has the bigger standard deviation?
A: 4, 5, 6, 7, 8.
B: 2, 4, 6, 8, 10.
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 30.
30 divided by 5 is 6.
mean of A = 4 + 5 + 6 + 7 + 85 = 305 = 6
SD = how far the numbers usually are from the mean
Now B: add them, then divide by how many.
The total is 30.
30 divided by 5 is 6.
mean of B = 2 + 4 + 6 + 8 + 105 = 305 = 6
2 Distances from the mean
Under each number of A, we write how far it is from the mean, 6.
A : 4, 5, 6, 7, 8
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 6.
B : 2, 4, 6, 8, 10
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
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: 4.
B: 8.
B is spread wider too.
So it is right!
range A: 8 − 4 = 4, B: 10 − 2 = 8
Let's look back.
First: find each mean.
Then: distances from the mean.
Then: compare.
Last: check.
Answer = B has the bigger SD
▶ Watch it solved
The question
Question: Which data set has the bigger standard deviation?
A: 10, 11, 12, 13, 14.
B: 2, 3, 4, 5, 6.
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 60.
60 divided by 5 is 12.
mean of A = 10 + 11 + 12 + 13 + 145 = 605 = 12
SD = how far the numbers usually are from the mean
Now B: add them, then divide by how many.
The total is 20.
20 divided by 5 is 4.
mean of B = 2 + 3 + 4 + 5 + 65 = 205 = 4
2 Distances from the mean
Under each number of A, we write how far it is from the mean, 12.
A : 10, 11, 12, 13, 14
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 4.
B : 2, 3, 4, 5, 6
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
The distances are the same: 6 and 6.
The numbers are just as far from their mean.
= A: 6 = B: 6
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: 14 − 10 = 4, B: 6 − 2 = 4
Let's look back.
First: find each mean.
Then: distances from the mean.
Then: compare.
Last: check.
▶ Watch it solved
The question
Question: Which data set has the bigger standard deviation?
A: 10, 20, 30, 40, 50.
B: 100, 200, 300, 400, 500.
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 150.
150 divided by 5 is 30.
mean of A = 10 + 20 + 30 + 40 + 505 = 1505 = 30
SD = how far the numbers usually are from the mean
Now B: add them, then divide by how many.
The total is 1500.
1500 divided by 5 is 300.
mean of B = 100 + 200 + 300 + 400 + 5005 = 15005 = 300
2 Distances from the mean
Under each number of A, we write how far it is from the mean, 30.
A : 10, 20, 30, 40, 50
how far = bigger − smaller
We add the distances: 20 plus 10 plus 0 plus 10 plus 20 is 60.
A total = 20 + 10 + 0 + 10 + 20 = 60
Now the same for B.
Its mean is 300.
B : 100, 200, 300, 400, 500
We add the distances: 200 plus 100 plus 0 plus 100 plus 200 is 600.
B total = 200 + 100 + 0 + 100 + 200 = 600
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, 600 against 60.
= A: 60 < B: 600
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: 40.
B: 400.
B is spread wider too.
So it is right!
range A: 50 − 10 = 40, B: 500 − 100 = 400
Let's look back.
First: find each mean.
Then: distances from the mean.
Then: compare.
Last: check.
Answer = B has the bigger SD
▶ Watch it solved
