# Determining the percentile rank of a score w.r.t. a known distribution

I have data from a SUS-survey (a standardized test for evaluating the usability of software) which gave me an average score of 73.25. As I understand, it is best practice to normalize the scores to produce a percentile ranking as indicated here.

I do not however understand how I would go about assigning a percentile rank to my score - if I understand correctly, I would require a reference distribution of SUS-Scores (or else there's no way of saying "my score is better or worse than X% of available scores", right?)

I have found that Bangor et al. have published the results from 10 years worth of SUS-evaluations, and here's the data they give:

• Count: 2324
• Mean: 70.14
• Median: 75
• Standard Deviation: 12.71

Is it possible to determine the percentile rank of the score of 73.25 with respect to this distribution, and how would I do that?

It depends on what you mean by distribution. Percentiles are expressed for a random variable and can be either empirically observed or theoretical. Empirical distributions would be published in some sort of long table in which you can look up for each score the corresponding percentile (rank). Theoretical distributions are those that follow a known functional form, such as the normal distribution.

Empirical distribution percentiles need to be published for distributions that do not follow a known functional form. For example if they are are skewed, multi-modal etc. it is more likely that it is very hard for statisticians to fit the observed distribution to a theoretical one without making crude approximations.

To attach a percentile rank to an observed value you need either of the two. In your example you only give first and second moment as well as the median of a distribution, but you do not say if this distribution is empirical or theoretical. If it is empirical, then there is not enough information to attach a percentile rank. Although you know the moments and median, it is not known how exactly the percentiles relate to these statistics. The only thing you know is that a score higher than 75 would have a percentile above 50, using the median which is the 50th percentile.

If the authors of that study tell you in addition that these statistics come from some sort of normal (super-)population you could perhaps use the mean and standard deviation as the parameter estimates for this normal distribution, and then use a normal distribution table or the R function pnorm to get percentiles. However I believe it is perhaps not very likely these data come from a normal distribution as the mean and median are not equal (they may of course be unequal due to sampling error but perhaps they deviate a bit too much here).

So the bottom line is that you need more information, either a better empirical distribution percentile table or knowledge on the true theoretical form that is reasonable to assume, in order to assign percentiles to your sample statistics.

• I understand - that makes perfect sense. I have also found this book: measuringu.com/book/… It seems the authors had access to the original dataset I'm referring to, as they give a conversion table from SUS-score to percentile that has entries for scores of 73 and 74 respectively - which will probably be sufficient for my purpose.
– DeX3
Feb 22, 2018 at 12:52