How do you calculate the Grand Variance?

[nukeman]

Homework Statement

Im working on this data set, and I can't 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?

Homework Equations

The Attempt at a Solution

 

[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?

 

[nukeman]

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?

5 people in each group...3 groups

 

[nukeman]

Crap, I corrected my post!

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

 

[haruspex]

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

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.

 

[nukeman]

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.

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

 

[nukeman]

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

Here is the data:

 

[haruspex]

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

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?

 

[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?

 

[nukeman]

Hey "I Like Serena"

I don't 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) ?

 

[I like Serena]

Yes.

 

[nukeman]

Yes.

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

 

[I like Serena]

So are you good now?

 

[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?

 

[I like Serena]

Correct.

 

[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.

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

 

[nukeman]

Wow, thanks! Appreciate it.