25 Equations

25.1 Population Mean

$\mu = \frac{\sum{x_i}}{n}$

25.2 Sample Mean

$\bar{x} = \frac{\sum{x_i}}{n}$

25.3 Standard error of the mean

$ _{m} = $

25.4 Pearson’s Correlation

$\rho = \frac{\text{cov}(X,Y)}{\sigma_x \sigma_y}$

and estimate:

$r = \frac{\sum_{i=1}^{n} (x_i - \overline{x})(y_i - \overline{y})}{\sqrt{\sum_{i=1}^{n} (x_i - \overline{x})^2\sum_{i=1}^{n}(y_i - \overline{y})^2}}$

25.5 Independent t-test

$t = \frac{\bar{x_1} - \bar{x_2}} {\sqrt{\frac{s^2_p}{n_1}+\frac{s^2_p}{n_2}}}$

$s^2_p = \frac{\Sigma(x_{i1}-\bar{x_1})^2+\Sigma(x_{i2}-\bar{x_2})^2}{n_1+n_2-2}$

25.6 Repeared t-test

$t = \frac{\Delta{\overline{x}_i}}{se_{diff}}$

$se_d = \frac{\sum(d_i-\overline{x}_{diff})^2\frac{1}{N}}{\sqrt{n}}$

25.7 Cohen’s d

$d=\frac{\overline{x}_1-\overline{x}_2}{s_{pooled}}$

25.8 One Way ANOVA

$SST = SSB + SSE$

$SST = \sum_{i=1}^n(x_i-\overline{x})^2$

$df_T=N-1$

$SSE = \sum(x_{ij}-\overline{x}_j)^2$

$df_E=N-k$

$SSB = \sum_{j=1}^{n_j}{n_j}(\overline{x}_j-\overline{x}_{overall})^2$

$df_K=k-1$

$MSX=\frac{SS_x}{df_x}$

$F=\frac{MSB}{MSE}$

25.9 Eta^2

$\eta^2 = \frac{SSB}{SST}$

25.10 Tukey’s HSD

$T = q\times\sqrt{\frac{MSE}{n}}$

25.11 Repeated ANOVA

$SST=\sum_{i=1}^n(x_i-\overline{x}_{grand})^2$ with $N-1$ degrees of freedom

$SST=s_{overall}^2(N-1)$

$SSW=\sum_{i=1,t=1}^n(x_{it}-\overline{x}_{i})^2$

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

$SSW=\sum_{i=1}^ns_{i}^2({n_{t}-1)}$

$df_{w}=n_i(df_{b})$

$df_{b}=k-1$

$SSM = \sum_{j=1}^{n_j}{n_j}(\overline{x}_j-\overline{x}_{overall})^2$

$SSE=SSW-SSB$

$df_{e}=df_{w}-df_{b}$

$MSB$, $MSE$, and $F$, same as One-way ANOVA

25.12 Chi-Square

$\chi^2=\sum^n_{i=1, j=1}\frac{(O_{ij}-E_{ij})^2}{E_{ij}}$

where:

i is the row number
j is the column number
$O_{ij}$ is the observed frequency
$E_{ij}$ is the expected frequency and
$E_{ij}=\frac{n_{row}\times n_{col}}{N}$

$df=(n_{row}-1) \times (n_{col}-1)$

Cramer’s V

$V=\sqrt{\frac{\chi^2/n}{min(r-1, c-1)}}$

25.12.1 Pearson’s Residual

$res=\frac{O_i-E_i}{\sqrt{E_i}}$