Editing Student's t-distribution (section)

===Integral of Student's probability density function and {{mvar|p}}-value===
The function {{nobr|{{math|''A''(''t'' {{!}} ''ν'')}} }} is the integral of Student's probability density function, {{math|''f''(''t'')}} between &nbsp;{{mvar|-t}} and {{mvar|t}}, for {{nobr|{{math| ''t'' ≥ 0 }} .}} It thus gives the probability that a value of ''t'' less than that calculated from observed data would occur by chance. Therefore, the function {{nobr|{{math|''A''(''t'' {{!}} ''ν'')}} }} can be used when testing whether the difference between the means of two sets of data is statistically significant, by calculating the corresponding value of {{mvar|t}} and the probability of its occurrence if the two sets of data were drawn from the same population. This is used in a variety of situations, particularly in [[t test|{{mvar|t}}&nbsp;tests]]. For the statistic {{mvar|t}}, with {{mvar|ν}} degrees of freedom, {{nobr|{{math|''A''(''t'' {{!}} ''ν'')}} }} is the probability that {{mvar|t}} would be less than the observed value if the two means were the same (provided that the smaller mean is subtracted from the larger, so that {{nobr|{{math| ''t'' ≥ 0}} ).}} It can be easily calculated from the [[cumulative distribution function]] {{math|''F''{{sub|''ν''}}(''t'')}} of the {{mvar|t}}&nbsp;distribution:

:<math> A( t \mid \nu) = F_\nu(t) - F_\nu(-t) = 1 - I_{ \frac{\nu}{\nu +t^2} }\!\left(\frac{\nu}{2},\frac{1}{2}\right),</math>

where {{nobr| {{math| ''I{{sub|x}}''(''a'', ''b'') }}  }} is the regularized [[Beta function#Incomplete beta function|incomplete beta function]].

For statistical hypothesis testing this function is used to construct the [[p-value|''p''-value]].