Blazon Research Resources · Statistical testing
How to calculate P values from test statistics
Why the distribution matters
A numerical test statistic does not determine a P value by itself. The relevant sampling distribution and, where required, its degrees of freedom are part of the calculation. A t statistic, for example, is interpreted using the Student t distribution, whereas a z statistic is interpreted using the standard normal distribution.
Calculate P from a t statistic
For a two-sided t test:
P = 2 × Pr(Tdf ≥ |t|)
The degrees of freedom must accompany the t statistic because the tails of the t distribution depend on df.
One-sided versus two-sided
For t and z statistics, Statistical Converter asks for the alternative hypothesis:
- two-sided — differences in either direction;
- greater — upper-tail alternative;
- less — lower-tail alternative.
Calculate P from a z statistic
A z statistic uses the standard normal distribution. For a two-sided test:
P = 2 × [1 − Φ(|z|)]
where Φ is the standard normal cumulative distribution function.
Calculate P from chi-square
For a conventional chi-square test statistic:
P = Pr(Χ²df ≥ χ²observed)
This is an upper-tail probability. The tool therefore does not ask for a one-sided/two-sided selector for χ².
Calculate P from F
An F statistic requires both numerator and denominator degrees of freedom:
P = Pr(Fdf1,df2 ≥ Fobserved)
Again, the conventional inferential P value is the upper-tail probability.
Why tiny P values should not display as zero
Very large test statistics can produce probabilities so small that ordinary floating-point arithmetic approaches its numerical limit. Statistical Converter uses upper-tail/survival calculations where appropriate to reduce cancellation error. It shows scientific notation for representable small probabilities and reports numerical underflow explicitly rather than displaying P = 0.
P values are not effect sizes
A P value addresses compatibility with a null model under a specified test. It does not quantify the magnitude or practical importance of an effect. Effect sizes should be considered separately. Lakens’ practical primer emphasizes reporting standardized effects alongside inferential statistics when appropriate. Lakens 2013
Read: How to convert test statistics to effect sizes →
Frequently asked questions
Can I calculate P from a t statistic and degrees of freedom?
Yes. The t statistic and df define the relevant Student t distribution, but you must also know whether the test is two-sided, greater or less.
Can a chi-square P value be two-sided?
The conventional P value from a chi-square test statistic is an upper-tail probability. Statistical Converter therefore does not provide a two-sided selector for χ².
Why does an F statistic need two degrees of freedom?
The F distribution is indexed by numerator and denominator degrees of freedom, both of which affect the tail probability.
Is a small P value the same as a large effect?
No. P values depend on both effect magnitude and sampling uncertainty, including sample size. Report or calculate an appropriate effect size separately. Lakens 2013
Why doesn’t the calculator round very small P values to zero?
A non-zero probability should not be reported as zero merely because it is tiny. Scientific notation or an explicit numerical-limit message is more informative.
References
- R Core Team. R documentation for probability distributions and related functions.
- Lakens D. Calculating and reporting effect sizes to facilitate cumulative science: a practical primer for t-tests and ANOVAs. Front Psychol. 2013;4:863. Lakens 2013