Bijlage B — Formules Statistiek I
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B.1 Beschrijvende statistieken
B.1.1 Gemiddelde (Mean)
\[\bar{x} = \frac{\sum_{i = 1}^{n}x_{i}}{n}\]
B.1.2 Variantie (Variance)
\[s^{2} = \frac{\sum_{i = 1}^{n}{(x_{i} - \bar{x})}^{2}}{N - 1}\]
B.1.3 Standaardafwijking (Standard deviation)
\[s = \sqrt{s^{2}}\]
B.1.4 Interkwartielafstand (Interquartile range)
\[IQR = \ Q_{3} - Q_{1}\]
B.1.5 Verwachte waarde van een discrete stochastische variabele (Expected Value of a Discrete Random Variable)
\[E(X) = \ \sum_{i = 1}^{k}{x_{i}P(X = x_{i})}\]
B.1.6 Algemene Formule voor de Variantie (General Variance Formula)
\[\sigma^{2} = \sum_{j = 1}^{k}{{(x_{j} - \mu)}^{2}P(X = x_{j})}\]
B.2 Kansrekening
B.2.1 Kans op gebeurtenis A (Probability of event A)
\[ P(A) \]
B.2.2 Intersectie van gebeurtenissen (Intersection of events)
\[ P(A \cap B) = P(\text{A and B}) \]
B.2.3 Vereniging van gebeurtenissen (Union of events)
\[ P(A \cup B) = P(\text{A or B}) \]
B.2.4 Algemene somregel (General addition rule)
\[P(A\ or\ B) = P(A) + P(B) - P(A\ and\ B)\]
B.2.5 Voorwaardelijke kans (Conditional probability)
\[P\left( A \middle| B \right) = \ \frac{P(A\ and\ B)}{P(B)}\]
B.2.6 Algemene productregel (General multiplication rule)
\[P(A\ and\ B) = P\left( A \middle| B \right)*P(B)\]
B.3 Normaalverdeling
B.3.1 Z-score
\[z = \frac{x - \mu}{\sigma}\]
B.4 Proporties
B.4.1 Betrouwbaarheidsinterval 1 proportie (Confidence interval 1 proportion)
\[CI = \ \hat{p}\ \pm z^{*}SE\left( \hat{p} \right)\]
\[SE\left( \hat{p} \right) = \sqrt{\frac{\hat{p}(1 - \hat{p})}{N}}\]
B.4.2 Hypothesetoets 1 proportie (Hypothesis test 1 proportion)
\[Z = \ \frac{\hat{p} - p_{0}}{SE}\]
\[SE\left( p_{0} \right) = \sqrt{\frac{p_{0}(1 - p_{0})}{N}}\]
B.4.3 Betrouwbaarheidsinterval 2 onafhankelijke proporties (Confidence interval 2 independent proportions)
\[CI = {(\hat{p}}_{1} - {\hat{p}}_{2}) \pm z^{*}SE\left( {\hat{p}}_{1} - {\hat{p}}_{2} \right)\]
\[SE\left( {\hat{p}}_{1} - {\hat{p}}_{2} \right) = \sqrt{\frac{{\hat{p}}_{1}\left( 1 - {\hat{p}}_{1} \right)}{n_{1}} + \frac{{\hat{p}}_{2}(1 - {\hat{p}}_{2})}{n_{2}}}\]
B.4.4 Hypothesetoets 2 onafhankelijke proporties (Hypothesis test 2 independent proportions)
\[Z = \ \frac{\left( {\hat{p}}_{1} - {\hat{p}}_{2} \right) - \text{null value}}{SE}\]
\[{\hat{p}}_{pooled} = \frac{{\hat{p}}_{1}n_{1} + {\hat{p}}_{2}n_{2}}{n_{1} + n_{2}}\]
\[SE = \sqrt{\frac{{\hat{p}}_{pooled}(1 - {\hat{p}}_{pooled})}{n_{1}} + \frac{{\hat{p}}_{pooled}(1 - {\hat{p}}_{pooled})}{n_{2}}}\]
B.5 Chikwadraat en associatiematen
B.5.1 Chi2 (\(\chi^{2}\)) one-way
\[\chi^{2} = \sum_{i = 1}^{k}\frac{{(O_{i} - E_{i})}^{2}}{E_{i}}\]
\[df = k - 1\]
B.5.2 Chi2 (\(\chi^{2}\)) two-way
\[ E_{i,j} = \frac{kolomtotaal_j * rijtotaal_i}{tabeltotaal} \]
\[\chi^{2} = \sum_{i,\ j}^{}\frac{{(O_{i,j} - E_{i,j})}^{2}}{E_{i,j}}\]
\[ df = (R-1)(C-1) \]
B.5.3 Phi (\(\phi\))
\[\phi = \sqrt{\frac{\chi^{2}}{N}}\]
B.5.4 Cramérs V
\[V = \sqrt{\frac{\chi^{2}}{N*(Min.\ van\ r - 1,\ c - 1)}}\]
B.5.5 Lambda (\(\lambda)\)
\[\lambda = \frac{E1 - E2}{E1} = \frac{\text{original error} - \text{remaining error}}{\text{original error}}\]
B.5.6 Gamma (\(\gamma)\)
\[\gamma = \frac{N_{s} - N_{d}}{N_{s} + N_{d}}\]
B.6 Betrouwbaarheidsinterval gemiddelde en t-toets
B.6.1 Standaardfout van het gemiddelde (Standard error of the mean)
\[SE = \frac{s}{\sqrt{N}}\]
B.6.2 Betrouwbaarheidsinterval gemiddelde (Confidence interval mean)
\[CI = \bar{x} \pm t_{df}^{*}*SE\]
B.6.3 t (enkele steekproef / one sample)
\[t = \frac{\bar{x} - \mu_{0}}{s/\sqrt{N}}\]
\[ df = n - 1 \]
B.6.4 t (gepaarde steekproeven/paired samples)
\[t = \frac{{\bar{x}}_{diff} - \mu_{diff}}{s_{diff}/\sqrt{n_{diff}}}\]
\[ df = n - 1 \]
B.6.5 Welch’s t-test (onafhankelijke steekproeven/ independent samples)
\[t = \frac{{(\bar{X}}_{1} - {\bar{X}}_{2}) - (\mu_{1} - \mu_{2})}{SE}\]
\[SE = \ \sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}\]
B.6.6 Cohen’s D (one sample)
\[d = \frac{\bar{X} - \mu_{0}}{s}\]
B.6.7 Cohen’s D (gepaard/paired)
\[d = \frac{\bar{D} - \mu_{0}}{s}\]
B.7 ANOVA
B.7.1 Som en gemiddelde van kwadraten tussen groepen (Sum and Mean Square between Groups)
\[MSG = \frac{SSG}{{df}_{G}}\]
\[SSG = \sum_{i=1}^{k}{n_{i}{({\bar{x}}_{i} - {\bar{x}}_{grand})}^{2}}\]
\[\text{df}_{G} = k - 1\]
B.7.2 Som en gemiddelde kwadratische fout (Sum of squared error & mean squared error)
\[MSE = \frac{SSE}{{df}_{E}}\]
\[SSE = \sum_{}^{}{(x_{ik} - {\bar{x}}_{k})}^{2}\]
\[{df}_{E} = n\ –k\]
B.7.3 F (ANOVA)
\[F = \frac{MSG}{MSE}\]