Appendix B — Formula sheet

B.1 General Part

Mean: \(\bar{X} = \frac{\Sigma_{i=1}^nx_i}{N}\)

Variance: \(S^2_x = \frac{\Sigma_{i=1}^n(x_i-\bar{x})^2}{n-1}\)

Standardized values (Z-values): \(Z = \frac{X-\mu}{\sigma}\)

Z-statistic in one sample Z-test: \(Z = \frac{\bar{x}-\mu}{\sigma_\bar{x}}\)

Standard error of the mean: \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\)

Cohen’s d: \(\frac{\bar{X}_1-\bar{X}_2}{s_{pooled}}\)

\(s^2_{pooled} = \frac{(n_1-1)*s_1^2 + (n_2-1)*s_2^2}{n_1+n_2-2}\)

\(s_{pooled} = \sqrt{s^2_{pooled}}\)

F-statistic in one-way ANOVA: \(F (df_b, df_w) = \frac{(SS_b/df_b)}{(SS_w/df_w)} = \frac{MS_b}{(MS_w}\)

Simple regression model: \(Y' = b_0+b_1X\)

Multiple regression model: \(Y' = b_0+b_1X_1 + b_2X_2\)

Explained variance: \(R^2 = \frac{s^2_{y'}}{s^2_y}\)

t-statistic in a one sample t-test: \(t = \frac{\bar{X}-\mu_{H0}}{se_x}\), where \(se_x = \frac{s_x}{\sqrt{n}}\), \(df = n - 1\)

t-statistic in an independent samples t-test: \(t = \frac{(\bar{X}_1-\bar{X}_2)-(\mu_1-\mu_2)_{H0}}{se_{x_1-x_2}}\)

\(se_{x_1-x_2} = \sqrt{s^2_{pooled}(\frac{1}{n_1}+\frac{1}{n_2})}\)

B.2 Business and economics

Logistic function: \(P(Y=1|X) = \frac{e^{(b_0 + b_1X)}}{1 + e^{(b_0 + b_1X)}}\)

From probability to odds: \(\text{odds} = \frac{P}{1 - P}\)

From odds to probability: \(P = \frac{\text{odds}}{1 + \text{odds}}\)

From odds to logit: \(\text{logit} = \ln(\text{odds})\)

From probability to logit: \(\text{logit} = \ln\left(\frac{P}{1 - P}\right)\)

From logit to odds: \(\text{odds} = e^{\text{logit}}\)

From logit to probability: \(P = \frac{e^{\text{logit}}}{1 + e^{\text{logit}}}\)

Wald test statistic: \(W = (\frac{b}{se_b})^2\)

B.3 Cognitive neuroscience

Number of Possible Pairwise Comparisons: \(k \times \frac{(k - 1)}{2}\)

Factorial ANOVA Linear Model: \(Y_{jkl} = \mu_Y + \alpha_k + \beta_l + \alpha\beta_{kl} + \epsilon_{jkl}\)

Eta-squared for Factor A: \(\eta_A^2 = \frac{SS_A}{SS_{total}}\)

Partial eta-squared for Factor A: \(\eta_{partial.A}^2 = \frac{SS_A}{SS_A + SS_w}\)

Adjusted Mean: \(\bar{Y}_{i(adj)} = \bar{Y}_i - b_w(\bar{X}_i - \bar{X})\)

t-Statistic in Paired Samples t-Test: \(t = \frac{\bar{d}}{\frac{s_{\bar{d}}}{\sqrt{n}}}, \quad \text{df} = n - 1\)

B.4 Social Sciences

Reliability: \(r_{xx'} = \frac{\text{var}(T)}{\text{var}(X)} = \frac{\text{var}(T)}{\text{var}(T) + \text{var}(E)}\)

Eigenvalue of Component 1 for 6 Items: \(\lambda_1 = a_{11}^2 + a_{21}^2 + a_{31}^2 + a_{41}^2 + a_{51}^2 + a_{61}^2\)

The proportion of Variance Accounted For by component 1 (when there are J items) is: \(\text{Proportion VAF} = \frac{\lambda_1}{\text{TotalVar}} = \frac{\lambda_1}{J}\)

Component loadings for component 1 and item j are represented as: \(a_{j1} = r_{X_jC_1}\)

Communality for 2 Components: \(h_{j2} = r_{XjC1}^2 + r_{XjC2}^2 = a_{j1}^2 + a_{j2}^2\)

Unicity for 2 Components: \(b_{j2} = 1 - h_{j2}\)