# Z-score from P-value

This online calculator calculates z score from p value

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This online calculator calculates z score from p value. Of course, there are some known values, like everybody (well, not everybody, but anyway) knows that z score for 0.05 significance level is roughtly 1.64. However, the calculator below can calculate z score for arbitrary p value. It is not that hard, if you can calculate inverse of CDF (cumulative distribution function) for standard normal distribution. And I can, thanks to excellent jStat library.

Now, let me give you couple of illustrations how it works for different hypothesis.

#### Left-tail event

For left-tail event, p-value is the probability of obtaining a result equal to or less than observed **x**. In this case, p-value is the value of cumulative distribution function of x, as shown on picture below.

To find out z-score, we just need to get inverse of CDF of p-value.

#### Right-tail event

For right-tail event, p-value is the probability of obtaining a result equal to or greater than observed **x**. In this case, p-value is the value of one minus cumulative distribution function of x, as shown on picture below.

To find out z-score, we just need to get inverse of CDF of one minus p-value.

#### Double-tail event

For double-tail event, p-value is the double probability of "smaller" of both. Knowing only p-value, we assume that we are talking about symmetrical x values. In this case, p-value can be found by doubling CDF of left-tail x, as shown on picture below.

To find out z-score, we just need to get inverse of CDF of p-value divided by 2. Note, that in this case the calculator below displays modulo of Z-score.

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