# p-Value Calculation

In hypothesis testing, the p-value is the probability of observing an effect larger than or equal to the measured metric delta, under the assumption that the null hypothesis is true. In other words, the smaller the p-value the higher the probability that there is a true difference between the test and control group.

The methodology used for p-value calculation depends on the number of degrees of freedom (*ν*). A two-sample z-test is appropriate for most experiments. Welch's t-test is used for smaller experiments with *ν < 100*. In both cases, the p-value depends on the metric mean and variance computed for the test and control groups.

### Two-Sample Z-Test

The z-statsitc of a two-sample z-test is:

The two-sided p-value is obtained from the standard normal cumulative distribution function:

### Welch's t-test

For smaller sample sizes, Welch's t-test is the prefered statistical test for lower false positive rates in cases of unequal sizes and variances. In Pulse, Welch's t-test is automatically applied when the degress of freedom *ν < 100*.

The t-statistic is computed in the same way as the two-sample z-statistic above. Additionally, we compute the degrees of freedom *ν* using:

The p-value is then obatined from the t-distribution with *ν* degrees of freedom.