# Fisher Z-Transformation

Statistics Definitions > Fisher Z-Transformation

## What is Fisher Z-Transformation?

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
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