20 Equations

20.1 Population Mean

$μ = \frac{\sum x_{i}}{n}$

20.2 Sample Mean

$\bar{x} = \frac{\sum x_{i}}{n}$

20.3 Standard error of the mean

$ _{m} = $

20.4 Pearson’s Correlation

$ρ = \frac{cov (X, Y)}{σ_{x} σ_{y}}$

and estimate:

$r = \frac{\sum_{i = 1}^{n} (x_{i} - \overset{―}{x}) (y_{i} - \overset{―}{y})}{\sqrt{\sum_{i = 1}^{n} (x_{i} - \overset{―}{x})^{2} \sum_{i = 1}^{n} (y_{i} - \overset{―}{y})^{2}}}$

20.5 Independent t-test

$t = \frac{\bar{x_{1}} - \bar{x_{2}}}{\sqrt{\frac{s_{p}^{2}}{n_{1}} + \frac{s_{p}^{2}}{n_{2}}}}$

$s_{p}^{2} = \frac{Σ (x_{i 1} - \bar{x_{1}})^{2} + Σ (x_{i 2} - \bar{x_{2}})^{2}}{n_{1} + n_{2} - 2}$

20.6 Repeared t-test

$t = \frac{Δ {\overset{―}{x}}_{i}}{s e_{d i f f}}$

$s e_{d} = \frac{\sum (d_{i} - {\overset{―}{x}}_{d i f f})^{2} \frac{1}{N}}{\sqrt{n}}$

20.7 Cohen’s d

$d = \frac{{\overset{―}{x}}_{1} - {\overset{―}{x}}_{2}}{s_{p o o l e d}}$

20.8 One Way ANOVA

$S S T = S S B + S S E$

$S S T = \sum_{i = 1}^{n} (x_{i} - \overset{―}{x})^{2}$

$d f_{T} = N - 1$

$S S E = \sum (x_{i j} - {\overset{―}{x}}_{j})^{2}$

$d f_{E} = N - k$

$S S B = \sum_{j = 1}^{n_{j}} n_{j} ({\overset{―}{x}}_{j} - {\overset{―}{x}}_{o v e r a l l})^{2}$

$d f_{K} = k - 1$

$M S X = \frac{S S_{x}}{d f_{x}}$

$F = \frac{M S B}{M S E}$

20.9 Eta^2

$η^{2} = \frac{S S B}{S S T}$

20.10 Tukey’s HSD

$T = q \times \sqrt{\frac{M S E}{n}}$

20.11 Repeated ANOVA

$S S T = \sum_{i = 1}^{n} (x_{i} - {\overset{―}{x}}_{g r a n d})^{2}$ with $N - 1$ degrees of freedom

$S S T = s_{o v e r a l l}^{2} (N - 1)$

$S S W = \sum_{i = 1, t = 1}^{n} (x_{i t} - {\overset{―}{x}}_{i})^{2}$

where $x_{i t}$ is the score for individual $i$ at time $t$ and ${\overset{―}{x}}_{i}$ is the mean for individual $i$ across all conditions. If you can quickly get the variances, you could also use the formula:

$S S W = \sum_{i = 1}^{n} s_{i}^{2} (n_{t} - 1)$

$d f_{w} = n_{i} (d f_{b})$

$d f_{b} = k - 1$

$S S M = \sum_{j = 1}^{n_{j}} n_{j} ({\overset{―}{x}}_{j} - {\overset{―}{x}}_{o v e r a l l})^{2}$

$S S E = S S W - S S B$

$d f_{e} = d f_{w} - d f_{b}$

$M S B$ , $M S E$ , and $F$ , same as One-way ANOVA

20.12 Chi-Square

$χ^{2} = \sum_{i = 1, j = 1}^{n} \frac{(O_{i j} - E_{i j})^{2}}{E_{i j}}$

where:

i is the row number
j is the column number
$O_{i j}$ is the observed frequency
$E_{i j}$ is the expected frequency and
$E_{i j} = \frac{n_{r o w} \times n_{c o l}}{N}$

$d f = (n_{r o w} - 1) \times (n_{c o l} - 1)$

Cramer’s V

$V = \sqrt{\frac{χ^{2} / n}{m i n (r - 1, c - 1)}}$

20.12.1 Pearson’s Residual

$r e s = \frac{O_{i} - E_{i}}{\sqrt{E_{i}}}$