User:Martin Arrowsmith/maths

Z-transformation: $Z_{s}={\frac {\sum _{i=1}^{k}Z_{n}}{\sqrt {k}}}$

Weighted Z-method: $Z_{W}={\frac {\sum _{i=1}^{k}w_{i}Z_{i}}{\sqrt {\sum _{i=1}^{k}w_{i}^{2}}}}$

where $w={\frac {1}{(SE_{\overline {x}})^{2}}}$

for $SE_{\overline {x}}={\frac {s}{\sqrt {n}}}$

where $s={\sqrt {{\frac {1}{N-1}}\sum _{i=1}^{N}(x_{i}-{\overline {x}})^{2}}}$

therefore $w={\frac {n}{s^{2}}}$

where $s^{2}={\frac {1}{N-1}}\sum _{i=1}^{N}(x_{i}-{\overline {x}})^{2}$

If $s$ of all trials is similar, then $SE_{\overline {x}}\varpropto {\frac {1}{\sqrt {n}}}$

and$w\varpropto n$

${\frac {n}{s^{2}}}$