# Statistical test methods

## Comparison of two mean values ​​(Student's t-test)

• Comparison of two mean values $xa¯$ and $xb¯$ from or $nb$ Data points with homogeneous variance. It should be checked whether the difference between the mean values ​​is due solely to a random error (both mean values ​​come from the population with the same mean value). If the test shows a significant difference, the two mean values ​​must not be compared with one another.
• First, after the error propagation, the standard deviation of the difference between two mean values ​​with the degrees of freedom $f=n1+n2−2$ calculated:$sx1¯−x2¯=s12n1+s22n2=sdv¯x⋅n1+n2n1⋅n2$
• The differences $xa¯−xb¯$ are random variables which, with the mostly available number, only follow a few measurements of a t-distribution. You will still be with $sdv¯x$ normalized and one obtains:$tˆ=xa¯−xb¯sdv¯x⋅na⋅nbna+nb$
• The tested hypothesis $m1=m2=m$ is with $α$ (possible error 1st type) to be discarded if calculated $tˆ$ larger than tabulated $t$-Is worth. Then there is a significant difference between the two mean values.
• The difference between the two mean values ​​is considered inconclusive, if $tber.fails. If applies $t>tα2f$, the mean values ​​are different within the given probability of error.
• The prerequisite for the calculation, however, is that the two standard deviations do not show any difference, otherwise they cannot be "pooled". An F-test must always be placed in front of it.
• Pooled variance:$var¯x=na−1⋅varxa¯+nb−1⋅varxb¯na+nb−2$