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Normal-test p-values by direction

For a standard-normal null statistic, the alternative determines which tail area is used.

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Use the standard-normal CDF

pU=1−Φ(z),pL=Φ(z),p2=2[1−Φ(∣z∣)]p_U=1-\Phi(z),\quad p_L=\Phi(z),\quad p_2=2[1-\Phi(|z|)]

Phi is the standard-normal cumulative distribution function; U and L denote upper and lower alternatives.

One statistic, distinct alternatives
zUpper pLower pTwo-sided p
20.02275010.97724990.0455003
−20.97724990.02275010.0455003
00.50.51

Check the direction before doubling

When z is positive, the upper tail is the smaller tail. When z is negative, the lower tail is smaller. The symmetric two-sided value doubles the smaller tail, not whichever one-sided tail happens to have been selected.

Keep the reference distribution explicit

These expressions use a continuous standard-normal statistic and the conventional symmetric two-sided rejection region. A t statistic needs a t distribution with its degrees of freedom. A discrete or asymmetric procedure can define two-sided evidence differently.

Preserve the unrounded probability

Near a planned threshold, compare with adequate numerical precision. A printed 0.05 may conceal either side of 0.05. Record the software method and the equality convention rather than using a rounded display as an exact input.

Further references