# Homework Help: How do you calculate the Grand Variance?

1. Jan 18, 2013

### nukeman

1. The problem statement, all variables and given/known data

Im working on this data set, and I cant get the "Grand Variance" calculated correctly, driving me nutz!

So, its 3 groups of data.

Group 1 mean = 2.20
group 2 mean = 3.20
group 3 mean = 5

Group 1 variance = 1.70
Group 2 variance = 1.70
Group 3 variance = 2.50

Now, it asks what is the "Grand Variance"... The correct answer is 3.124 - but am not sure how the book got that value. I keep getting wrong :(

Any pointers?

2. Relevant equations

3. The attempt at a solution

Last edited: Jan 18, 2013
2. Jan 18, 2013

### haruspex

I hope you've been told at some time not to take an average of averages (unless you happen to know that the averages come from sample sets of the same size).
What else do you know about these groups? Their sizes perhaps?

3. Jan 18, 2013

### nukeman

5 people in each group...3 groups

4. Jan 18, 2013

### nukeman

Crap, I corrected my post!!!

It asks for the GRAND VARIANCE!, not grand mean. Grand Variance is 3.124

5. Jan 18, 2013

### haruspex

Same question would have arisen, so no harm done.
Think about the last step that would have been involved in calculating the mean of each group. What can you work backwards from the info you have to determine? Then do the same for the variances.

6. Jan 18, 2013

### nukeman

All I did to get the Grand mean was calculate the mean, of the mean's of each group...

7. Jan 18, 2013

### nukeman

I keep getting wrong answer. I don't know what Im doing wrong. :(

Here is the data:

8. Jan 19, 2013

### haruspex

Since the groups all happen to be the same size, that should be fine (but I hope you would not have done that otherwise). And you get 3.467, right?
But you can't do that with the variances because the formula used there has an n/(n-1) term, where n is the number of data values. So for each group that gives 5/4, but for the grand variance it will be 15/14.
Have you tried to calculate the group variances yourself? Do you get the values in the table?

9. Jan 19, 2013

### I like Serena

Hi nukeman!

The grand variance is
$$\text{Grand Variance} = {SST \over N-1}$$
where:
$SST$ is the sum of the squared differences of each score with the grand mean,
$N$ is the total number of scores (15 in your case).​

If you want, you can also calculate it without using the actual scores.
But then you'll need a couple of additional formulas, so you can calculate SST differently.
Do you have formulas for that?

10. Jan 19, 2013

### nukeman

Hey "I Like Serena"

I dont quite understand the SST part of that equation.

Do I take the squared differences of ALL 15 data points, then add them up?

then divide by n-1 ? (14) ?

11. Jan 19, 2013

### I like Serena

Yes.

12. Jan 19, 2013

### nukeman

ahhg I did that!! Don't tell me I just messed up on my calculator and thats why im trying all these different things!! lol :)

13. Jan 19, 2013

### I like Serena

So are you good now?

14. Jan 19, 2013

### nukeman

Oh yes, I got it! Thanks!!

While you are here, your formula make more sense than the one I was given. What is the formula for the Grand SD?

I just square root the Grand Variance correct?

15. Jan 19, 2013

### I like Serena

Correct.

16. Jan 19, 2013

### I like Serena

If you're interested, here are a couple of other identities ($n=5$).
They come from (one-way) ANOVA theory which is what you are doing.

$$SST = SSM + SSE$$
$$SSM=n((\bar X_1 - \text{Grand Mean})^2 + (\bar X_2 - \text{Grand Mean})^2 + (\bar X_3 - \text{Grand Mean})^2)$$
$$SSE=(n-1)(s_1^2 + s_2^2 + s_3^2)$$

17. Jan 19, 2013

### nukeman

Wow, thanks! Appreciate it.