Statistics How To

Fisher Z-Transformation

Statistics Definitions > Fisher Z-Transformation

What is Fisher Z-Transformation?

fisher z

The normal distribution.

The Fisher Z-Transformation is a way to transform the sampling distribution of Pearson’s r (i.e. the correlation coefficient) so that it becomes normally distributed. The “z” in Fisher Z stands for a z-score.

The formula to transform r to a z-score is:
z’ = .5[ln(1+r) – ln(1-r)]

for example, if your correlation coefficient(r) is 0.4, the transformation is:
z’ = .5[ln(1+0.4) – ln(1-0.4)]
z’ = .5[ln(1.4) – ln(0.6)]
z’ = .5[0.33647223662 – -0.51082562376]
z’ = .5[0.84729786038]
z’ = 0.4236.
where ln is the natural log.

Instead of working the formula, you can also refer to the r to z’ table.

Fisher’s z’ is used to find confidence intervals for both r and differences between correlations. But it’s probably most commonly be used to test the significance of the difference between two correlation coefficients, r1 and r2 from independent samples. If r1 is larger than r2, the z-value will be positive; If r1 is smaller than r2, the z-value will be negative.

While the Fisher transformation is mainly associated with Pearson’s r for bivariate normal data, it can also be used for Spearman’s rank correlation coefficients in some cases.

R to z’ Table


The following table converts an r-value to Fisher’s Z and vice versa.

r z’
0.0000 0.0000
0.0100 0.0100
0.0200 0.0200
0.0300 0.0300
0.0400 0.0400
0.0500 0.0500
0.0600 0.0601
0.0700 0.0701
0.0800 0.0802
0.0900 0.0902
0.1000 0.1003
0.1100 0.1104
0.1200 0.1206
0.1300 0.1307
0.1400 0.1409
0.1500 0.1511
0.1600 0.1614
0.1700 0.1717
0.1800 0.1820
>0.1900 0.1923
0.2000 0.2027
0.2100 0.2132
0.2200 0.2237
0.2300 0.2342
0.2400 0.2448
0.2500 0.2554
0.2600 0.2661
0.2700 0.2769
0.2800 0.2877
0.2900 0.2986
0.3000 0.3095
0.3100 0.3205
0.3200 0.3316
0.3300 0.3428
0.3400 0.3541
0.3500 0.3654
0.3600 0.3769
0.3700 0.3884
0.3800 0.4001
0.3900 0.4118
0.4000 0.4236
0.4100 0.4356
0.4200 0.4477
0.4300 0.4599
0.4400 0.4722
0.4500 0.4847
0.4600 0.4973
0.4700 0.5101
0.4800 0.5230
0.4900 0.5361
0.5000 0.5493
0.5100 0.5627
0.5200 0.5763
0.5300 0.5901
0.5400 0.6042
0.5500 0.6184
0.5600 0.6328
0.5700 0.6475
0.5800 0.6625
0.5900 0.6777
0.6000 0.6931
0.6100 0.7089
0.6200 0.7250
0.6300 0.7414
0.6400 0.7582
0.6500 0.7753
0.6600 0.7928
0.6700 0.8107
0.6800 0.8291
0.6900 0.8480
0.7000 0.8673
0.7100 0.8872
0.7200 0.9076
0.7300 0.9287
0.7400 0.9505
0.7500 0.9730
0.7600 0.9962
0.7700 1.0203
0.7800 1.0454
0.7900 1.0714
0.8000 1.0986
0.8100 1.1270
0.8200 1.1568
0.8300 1.1881
0.8400 1.2212
0.8500 1.2562
0.8600 1.2933
0.8700 1.3331
0.8800 1.3758
0.8900 1.4219
0.9000 1.4722
0.9100 1.5275
0.9200 1.5890
0.9300 1.6584
0.9400 1.7380
0.9500 1.8318
0.9600 1.9459
0.9700 2.0923
0.9800 2.2976
0.9900 2.6467
------------------------------------------------------------------------------

Confused and have questions? Head over to Chegg and use code “CS5OFFBTS18” (exp. 11/30/2018) to get $5 off your first month of Chegg Study, so you can understand any concept by asking a subject expert and getting an in-depth explanation online 24/7.

Comments? Need to post a correction? Please post a comment on our Facebook page.

Check out our updated Privacy policy and Cookie Policy