Arithmetic Mean

\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i

Weighted Mean

\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}

Median Position

Me = \text{value at } \frac{n+1}{2}

Population Variance

\sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2

Standard Deviation (Population)

\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2}

Combinations

C_n^k = \binom{n}{k} = \frac{n!}{k!(n-k)!}

Permutations

P_n = n!

Independent Samples t-Test

x̄₁, x̄₂ = group means, s₁², s₂² = group variances, n₁, n₂ = sample sizes.

t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}

Chi-Square Test Statistic

O_i = observed frequency, E_i = expected frequency, k = number of categories.

\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}

Confidence Interval (Known σ)

x̄ = sample mean, z = critical value, σ = population standard deviation, n = sample size.

\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}

Pearson Correlation Coefficient

Measure of linear relationship. r ∈ [−1, 1]; r = 1 is perfect positive, r = −1 perfect negative.

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

Cronbach's Alpha (Internal Consistency)

k = number of items, σ_i² = variance of item i, σ_T² = variance of total test score.

\alpha = \frac{k}{k-1} \left(1 - \frac{\sum_{i=1}^{k} \sigma_i^2}{\sigma_T^2}\right)

ANOVA F-Statistic

k = number of groups, N = total sample size, SS = sum of squares, MS = mean square.

F = \frac{MS_{between}}{MS_{within}} = \frac{SS_{between} / (k-1)}{SS_{within} / (N-k)}

Simple Linear Regression

β₀ = intercept, β₁ = slope, ε = error term. Fitted by least squares.

y = \beta_0 + \beta_1 x + \varepsilon

Cohen's d (Effect Size)

s_pooled = pooled standard deviation. 0.2 = small, 0.5 = medium, 0.8 = large effect.

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

Statistics

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