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layman here...

I have some sets of several hundred scores that are not distributed normally. I want to normalize these scores onto a percentile scale.

The mean-median-mode for these look like this:

3-2-0,
1-1-0,
9-3-0,
1-5-5,

I have calculated the variance/std dev and z scores for these, but now I'm not sure what to do with this info. What good are z scores when the distribution isn't normal? What's the best way to get a normalized percentile score for these?

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  • $\begingroup$ It would be, ... well, abnormal to use the term "abnormal" to describe data that is not normally distributed. "Non-normal" or simply 'not normal' would be more ... normal. $\endgroup$ – Glen_b Aug 23 '14 at 11:54
  • $\begingroup$ Are your data all integers? $\endgroup$ – Glen_b Aug 23 '14 at 22:19
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You can get percentiles from any distribution, normal or not. The percentile just says what percent of people did worse than the person. In fact, any reasonable transformation of the scores will leave the percentiles unchanged, because it will leave the order the same.

I am not sure I understand your mean, median, mode.

Were your scores all integers? Even if they were, the mean will be very unlikely to be an integer.

Why do you want to normalize the scores? What will you do with them once normalized?

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  • $\begingroup$ Thanks for your answer... How can I get percentiles? I understand if the data is a normal distribution, I can just use the normal distribution table or the appropriate equation to calculate it manually, but if the data is not distributed normally, what equation / function do I use to get the percentile? All of the scores were integers - the mean was not an integer but I provided it rounded to an integer for no particular reason :) I want to normalize the scores to make them more understandable / digestible to people who aren't familiar with the method used to calculate the score. $\endgroup$ – BrettH Aug 23 '14 at 12:03
  • $\begingroup$ How exactly to get percentiles depends on what software you have, but normality is irrelevant and you would not use a normal distribution table in any case. Rank your data. Then the highest score is 99% tile. If you had exactly 100 cases, all with different scores, then each person would be one percentile lower. Since you probably have some odd number of cases, you have to do the calculation or get a computer to do it. $\endgroup$ – Peter Flom Aug 23 '14 at 12:22
  • $\begingroup$ Yes, I am a programmer and this is part of an application I am making, so it will definitely be automated. I guess I'm just trying to find a formula (perhaps something similar to the standard table formula?) that I can apply to the z-scores... or do you think the z-scores aren't relevant for this case? Thanks again for your time! $\endgroup$ – BrettH Aug 23 '14 at 12:35
  • $\begingroup$ You don't need z scores. Formula something like (number higher)/total * 100. $\endgroup$ – Peter Flom Aug 23 '14 at 14:26

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