This online statistical tool calculates left-tailed and right-tailed P-values from various test scores (z-score, chi-square, Student’s t-value). Choose the type of the statistics distribution and enter the input data in the appropriate fields of this P-value Calculator to get the corresponding P-value.
How to calculate P-value
The P-value (probability value) is a quantitative parameter used in statistical hypothesis testing to determine whether a null hypothesis (or claimed hypothesis) is true, or in other words, whether the obtained test results are significant.
Simply speaking, the P-value is the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct. A very small P-value means that such an extreme observed outcome would be very unlikely under the null hypothesis.
For instance, if the appropriate (left-tail or right-tail) P-value found is higher than conventional criteria for statistical significance (0.001-0.05), we usually do not reject the null hypothesis and assume that all the differences between observed result and expected value are due to chance.
• Normal Distribution.
For a given test statistic \(z\) in the case of Standard Normal Distribution, the right-tail P-value is defined as:
$${ P }_{ right } ( z ) = \frac{ 1 } {\sqrt{2\pi}}\int _{ z }^{ \infty }{{ e }^{ -\frac { { t }^{ 2 } }{ 2 } }dt },$$
and the left-tail P-value is defined as:
$${ P }_{ left }\left( z \right) = 1 – { P }_{ right }\left( z \right).$$
• Chi-Square Distribution.
For a given test statistic \({ \chi } ^{2}\) and \(k\) degrees of freedom in the case of Chi-Square Distribution, the right-tail P-value is defined as:
$${ P }_{ right }\left( { \chi }^{ 2 },k \right) = { \left[ { 2 }^{ k/2 }\Gamma \left( \frac { k }{ 2 } \right) \right] }^{ -1 }\int _{ { \chi }^{ 2 } }^{ \infty }{ { \left( t \right) }^{ \frac { k }{ 2 } -1 }{ e }^{ -\frac { t }{ 2 } }dt },$$
where \(Γ\) is the gamma function, which is the generalization of the factorial function to real and complex arguments:
$$\Gamma \left( x \right) =\int _{ 0 }^{ \infty }{ { t }^{ x-1 }{ e }^{ -t }dt }. $$
The left-tail P-value is defined as:
$${ P }_{ left }\left( { \chi }^{ 2 },k \right) = 1 – { P }_{ right }\left( { \chi }^{ 2 },k \right).$$
• Student’s t-Distribution.
For a given test statistic \({ t }_{ e }\) and \(k\) degrees of freedom in the case of Student’s t-Distribution, the right-tail P-value is defined as:
$${ P }_{ right }\left( { t }_{ e },k \right) = \frac{\Gamma(\frac{k+1}{2})} {\sqrt{k\pi}\,\Gamma(\frac{k}{2})}
\int _{ { t }_{ e } }^{ \infty } \left(1+\frac{t^2}{k} \right)^{\!-\frac{k+1}{2}}dt,$$
where \(Γ\) is the gamma function, and the left-tail P-value is defined as:
$${ P }_{ left }\left( { t }_{ e },k \right) = 1 – { P }_{ right }\left( { t }_{ e },k \right).$$
Related calculators
Check out our other statistics calculators such as Chi Square Calculator or Multinomial Distribution Calculator.