Work out percentile to z score instantly with clear inputs, formula shown and shareable results.
Converting a percentile to a z score inverts the normal cumulative distribution. The 90th percentile is z = 1.2816, which on a mean 100, SD 15 scale is a score of 119.2 and a T score of 62.8. The relationship is strongly non-linear at the extremes: the 99th percentile is z = 2.33 but the 99.9th is 3.09.
Inverse normal
z = probit(percentile / 100)
Score on a scale
score = mean + sd x z
Because the normal distribution has thin tails. Moving from the 99th to the 99.9th percentile takes only 0.76 of a standard deviation but excludes nine tenths of the remainder.
Yes. For a skewed distribution use empirical percentiles from the data rather than a normal transformation.