### Stata FAQ How should I analyze percentile rank data?

I have to do an analysis of variance on some test scores that were given to me as percentile scores. My question is, "How should I analyze percentile rank data?"

The problem, of course, is that percentile rank data are not normally distributed. Percentile ranks are ordinal and usually form a rectangular (uniform) distribution. The easiest solution is to transform the percentile rank scores into z-scores (standard normal scores) using an inverse normal function. The z-scores will be normally distributed with mean equal to zero and a standard deviation of one. The range of the z-scores will be between ±2.33. In Stata, the transformation would look like this:
generate zscore = invnorm(pctrank/100)
Specialists in testing often transform percentile ranks into NCE (normal curve equivalence) scores. NCEs are a type of standardized score with a mean of 50 and a standard deviation of 21.06. NCEs have a range of one to 99 and in many ways look a lot like percentile ranks. Here is how the NCE transformation would look in Stata:
generate nce = invnorm(pctrank/100)*21.06 + 50
Here is a table that gives a rank of percentile rank scores and their equivalent z-scores and NCE scores:
      pctrank     zscore        nce
1.        1  -2.326348   1.007114
2.        2  -2.053749   6.748048
3.        3  -1.880794   10.39049
4.        4  -1.750686   13.13055
5.        5  -1.644854   15.35938
6.       10  -1.281552   23.01052
7.       20  -.8416212   32.27546
8.       25  -.6744897   35.79525
9.       30  -.5244005   38.95612
10.       40  -.2533471   44.66451
11.       50          0         50
12.       60   .2533471   55.33549
13.       70   .5244005   61.04388
14.       75   .6744897   64.20476
15.       80   .8416212   67.72454
16.       90   1.281552   76.98948
17.       95   1.644854   84.64062
18.       96   1.750686   86.86945
19.       97   1.880794   89.60951
20.       98   2.053749   93.25195
21.       99   2.326348   98.99289 

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