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$$w \varpropto n$$

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