## F-Test Example

In this example we test the equality of the variances of two data sets that belong to a normal distribution. We start this example by creating 3 waves of different statistics. The first pair (data1 and data2) have the same variance but different means. The second pair (data2 and data3) have the same mean but different variance. To create the data execute the commands:

Make/n=100 data1=100+gnoise(3)

Make/n=80 data2=80+gnoise(3)

Make/N=90 data3=80+gnoise(4)

### Comparing the variance of two waves using a two-tailed hypothesis

To run the test execute the command:

StatsFTest/T=1/Q data1,data2

The results of the test appear in the F-Test table:

n1 | 100 |

Mean1 | 99.8754 |

Stdv1 | 3.39174 |

degreesOfFreedom1 | 99 |

n2 | 80 |

Mean2 | 79.6029 |

Stdv2 | 3.10709 |

degreesOfFreedom2 | 79 |

F | 1.19162 |

lowCriticalValue | 0.659763 |

highCriticalValue | 1.53104 |

P | 0.418974 |

Accept | 1 |

The F statistic is within the critical range so the two-tailed hypothesis of equal variances is accepted.

### Testing in the case of unequal variances (two tails test)

To run the test execute the following command:

StatsFTest/T=1/Q data1,data3

The results of the test appear in the F-Test table:

n1 | 100 |

Mean1 | 99.8754 |

Stdv1 | 3.39174 |

degreesOfFreedom1 | 99 |

n2 | 80 |

Mean2 | 80.5489 |

Stdv2 | 4.43966 |

degreesOfFreedom2 | 79 |

F | 0.583641 |

lowCriticalValue | 0.659763 |

highCriticalValue | 1.53104 |

P | 0.0112429 |

Accept | 0 |

The rejection of H_{0} in this case is pretty sensitive to the choice of significance. It is apparent
from the P-value that it would have been accepted if alpha was set to 0.01.

### One-tail testing for the same data

First H_{0}: the variance of the first sample is greater than the variance of the second. To run the test execute the command:

StatsFTest/T=1/Q/TAIL=1 data1,data3

n1 | 100 |

Mean1 | 99.8754 |

Stdv1 | 3.39174 |

degreesOfFreedom1 | 99 |

n2 | 80 |

Mean2 | 80.5489 |

Stdv2 | 4.43966 |

degreesOfFreedom2 | 79 |

F | 0.583641 |

Critical | 0.70553 |

P | 0.00562143 |

Accept | 0 |

H_{0} is rejected here as one would expect. Similarly,

StatsFTest/T=1/Q/TAIL=2 data1,data3

n1 | 100 |

Mean1 | 99.8754 |

Stdv1 | 3.39174 |

degreesOfFreedom1 | 99 |

n2 | 80 |

Mean2 | 80.5489 |

Stdv2 | 4.43966 |

degreesOfFreedom2 | 79 |

F | 0.583641 |

Critical | 1.4289 |

P | 0.00562143 |

Accept | 1 |

Here the F is smaller than the critical value so the two-tailed hypothesis can't be rejected.