# When to use Z scores

#### kowalsks

##### New Member
I am trying to look for significant differences between the test scores of two training programs (two groups undergoing different training, but taking the same exam). One group is larger than the other (n larger group = 200, n smaller group =80) .

I performed a One-Way Anova on the scores and there was a significant difference between the two groups.

However, I then converted the scores to z-scores (using the larger group as the population mean) and the smaller group had a z-score of -.382 (which would not be significant, correct?)

Under what circumstances do you convert to z-scores? Is it ok to use the raw means to do an ANOVA?

Thank you!

#### JohnM

##### TS Contributor
If the one-way ANOVA shows a significant difference, then it's significant.

When you converted raw scores to z-scores, you got an average z-score of -0.382, but that's the number of standard deviations away from the mean (of the other group) the individual scores are, on average - all this tells you is roughly how much separation there is between the two distribution curves, and is not enough info to draw inferences from.

You need to figure out how many "standard errors of the mean" they are away from each other, which is what an ANOVA does.

Also, assuming that one group exactly represents the population is risky, because in reality, that group is a sample, and is subject to the same sampling variation as the other group.

#### kowalsks

##### New Member
John, Thank you so much for replying! I'm glad to hear that ANOVA was the right thing to do. I was so confident about it at first, then I started second guessing myself after the z-scores weren't significant!

The reason I used the larger group as my baseline was because the smaller group is sort of a 'trial' program being compared with the established program (larger group). I have data for 3 years of exams. The reason I thought of doing a z-score comparison was so that I could account for any differences in exam difficulty (the second year seemed to have a harder exam than the other two years).

Basically, I'm trying to write a report on the progress of the trial group in relation to the established group. I think I am going to first discuss the ANOVA in detail (saying that the differences in means are significant). Then I thought I could plot the z-score data in a graph to show how the relationship has changed over the past 3 years? (i.e. the baseline group stays the same, the z-score of the trial group fluctuates).

So the z-scores would be used to show the relationship rather than to form any conclusions. Does this make sense?

I can't thank you enough for your help.

#### JohnM

##### TS Contributor
Yes, your approach should be OK.

Yes, converting to z-scores would be a good way to keep the baseline group "consistent."