# Z-scores, standardized tests and population means

I have been using z-scores by subtracting the mean (and then dividing by the SD) of the sample, whereas I recently read they actually need to be the population mean and SD. Assuming you are transforming raw scores to z-scores in order to compare within-sample variables (various test scores of the same sample of subjects), does it really invalidate the comparison if you have been using sample rather than population statistics?

Also, is it ever realistic that one would know the population statistics? I always assumed this is more of a theoretical concept than an actual statistic. For instance, in order for the population mean of the IQ test to be set to 100, surely this didn't imply testing every single human on the planet and adjusting the raw-to-standardized tables to reflect the population average! Many thanks.

If you're interested in generalizing the within-subjects differences across various tests that you find in your dataset to the larger population of people just like your subjects (the population from which you sampled your subjects), then you need to work with sample statistics. Technically, standardizing scores with sample means and $SD$s produces $t$-statistics, not $z$-scores, as I've mentioned in my recent response to Triangular distribution, and as @Henry said just now :)

I wouldn't say the difference invalidates your comparisons altogether, but since $t$-statistics have different distributions depending on your degrees of freedom ($\nu$), inferences you base on the probability of differences among $t$-stats would also depend on $\nu$, whereas this isn't true of $z$-scores. Here's Wikipedia's of Student's $t$ (only the PDF for $\nu=\infty$ is the same as that of $z$):
by Skbkekas.

• That's helpful, thanks. What about my second question, about the meaning of population means (100 for the IQ score) and the impossibility of it having really reflect the actual "entire population" mean? Mar 15 '14 at 12:36
• Despite being a personality psychologist, I don't know enough about IQ to say how close to the mark the scale mean is to the population mean. Regardless, it's very simple to know the entire population mean if you define your population restrictively enough. As I said, if you're really only interested in your subjects, not in others like them (and don't plan to generalize beyond them), then your subjects comprise your population of interest by definition. Mar 15 '14 at 14:27
• But surely in the case of the IQ test (or other standardised tests), the population of interest is not just the sample that was initially used to "calibrate" the mean to 100, and the purpose is, in fact, to generalise and compare across lots of people rather than define your population restrictively as you say Mar 15 '14 at 18:20
• Agreed, actually working with the entire population of interest is probably quite rare, and probably doesn't apply in the case of IQ testing. Mar 15 '14 at 22:05

If you are using the sample mean and standard deviation to standardise, then strictly speaking you have a $t$-statistic.

The difference between this and a $z$-statistic using the population mean and standard deviation matters more when you have a smaller sample.