Statistics Definitions > Fisher Z-Transformation

## What is Fisher Z-Transformation?

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, r_{1} and r_{2} from independent samples. If r_{1} is larger than r_{2}, the z-value will be positive; If r_{1} is smaller than r_{2}, 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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