# Standardizing sample factor scores: population or standard deviation

Please note: This question pertains to Q Methodology, a research method used to study people's subjectivity. Q embodies ontological and epistemological assumptions that sometimes differ markedly from mainstream ("R") / survey research. I'm trying to move some of the discussion on this methodology on to CrossValidated, so please don't downvote this because the question seems weird :). Also, I'd be great if someone could tag this with qmethod.

I need to standardize a bunch of factor scores for a sample of people. I obviously have all the factor scores to standardize (no sampling there), but the people from which these factors are extracted are obviously a sample.

So which do I use?

Population SD (uncorrected sample standard deviation) $$s_N = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - \overline{x})^2}$$

or

Sample SD (Corrected sample standard deviation) $$s_N = \sqrt{\frac{1}{n - 1} \sum_{i=1}^N (x_i - \overline{x})^2}$$

in the respective calculation of zscores:

$$z = {x- \mu \over \sigma}$$

Ps.: Some more context: this calculation is part of Q Methodology, a technique to study human subjectivity (still trying to get that tag qmethod for Cross Validated). In Q, people rank order a bunch of (30-90) items according to their agreement with the item. We then correlate the sorts of individual people (not the items!), and extract factors from that, so to speak, transposed matrix. (Yeah, it's wild, but it makes sense). We then calculate weighted, standardized factor scores on these items for each of the extracted factor, constructing an ideal-typically rank-ordering (for that factor). The weighted factor scores, are, of course a population, but the people are not (they're a sample, though Q people loathe to think of it that way – for Q, the items are the sample, and the people are the variables.

Pps.: yes, I know that standard, or zscores are conventionally defined by the population standard deviation. I'm asking nonetheless, because a) changing it to population sd would be huge validation hassle for our R package and b) it's a little more complicated in Q, because, see above, the items are the sample ...

Ppps: could someone with enough points be so kind as to tag this with qmethod – I'm trying to get this community on here. A workable short description from the qmethod webpage might be:

Q Methodology is a research method used to study people's subjectivity – that is, their viewpoint.
Q Methodology was originally developed by William Stephenson (1902-1989), an Englishman trained in physics (Ph.D., 1926), psychology (Ph.D., 1929) and psychometrics under the tutelage of Charles Spearman and Sir Cyril Burt.
It has been used both in clinical settings for assessing patients, as well as in research settings to examine how people think about a topic.


## 1 Answer

For standardizing (factor) scores, including in Q Methodology, the population standard deviation (see above) should be used. In this context, as elsewhere, the standard deviation is used to make scores comparable, and no statistical inference to a population is implied.

Leading Q methodologist Steven Brown and Q software maintainer Peter Schmolck confirmed via E-Mail that the population standard deviation would also be appropriate for Q Methodology.

They also pointed out that since the number of items was always the same, the divisor would affect all factors in the same way and that there would not be changes in the relative position of items on factor scores. In short: it doesn't even matter (much).