Investment Studio > Expressions > Functions > Statistical > FISHER
Returns Fisher's z' transformation for argument Î ]-1, 1[, defined as
z'(argument) = 0.5 * ln((1 + argument) / (1 - argument))
This transformation is often applied to the sample correlations of two data sets (see correl).
If you sample two data sets repeatedly and plot the statistical frequency of the sample correlations obtained each time, the resulting distribution (the "sampling distribution") is skewed. On the other hand, running the correlations through Fisher's z' transform before plotting their statistical frequency yields a normal distribution with known standard error
sz' = 1 / sqrt(N - 3)
where N is the number of sample correlations.
This fact can be used to compute confidence intervals on correlation estimates: take Fisher's z' transform, compute its confidence interval using the above expression for sz', then compute the inverse transform (see fisherinv) of the interval's upper and lower limits.
Example
We want to know the correlation between SAT scores and first year grades at a large state university at the 5% significance level (95% confidence level). We sample 100 students, chosen at random, and find the sample correlation to be 0.59. Fisher's z' transform of this value is
=fisher(0.59)
= 0.677666067757962. To get the z' interval, we compute
=confidence(5%, 1, 100 - 3)
= 0.199004191550275. The z' limits are thus 0.677666067757962 ± 0.199004191550275. The lower correlation limit is then
=fisherinv(0.677666067757962 - 0.199004191550275)
» 0.45 and the upper correlation limit is
=fisherinv(0.677666067757962 + 0.199004191550275)
» 0.70.
Our final result is therefore that with 95% confidence, the correlation is somewhere in the range [0.45, 0.70].
See also confidence, correl, covar, fisherinv.