Descriptive Measures
Mean (\(\bar{X}\)):
Discrete: \(\frac{\sum x_i f_i}{n}\) ; Continuous: \(\int x f(x) dx\)
Mode (\(M_o\)):
Discrete: \(max(f_i)\) ; Continuous (modal class): \(L + \frac{\Delta_1}{\Delta_1 + \Delta_2} \cdot h\)
Median (\(M_e\)):
Discrete: \(x\) such that \(F(x) \ge 0.5\) ; Continuous: \(L + \frac{h}{f_m}(\frac{n}{2} - F_{m-1})\)
Variance and Standard Deviation:
\(s^2 = \frac{\sum f_i x_i^2 - n\bar{X}^2}{n-1}\) ; \(s = \sqrt{s^2}\)
Coefficient of Variation: \(CV = \frac{s}{\bar{X}}\)
Discrete Distributions
Binomial \(B(n,p)\):
\(P(k) = \binom{n}{k}p^k q^{n-k}\) ; \(E=np, V=npq\)
Poisson \(Po(\lambda)\):
\(P(k) = \frac{e^{-\lambda}\lambda^k}{k!}\) ; \(E=\lambda, V=\lambda\)
Geometric \(G(p)\):
\(P(k) = q^{k-1}p\) ; \(E=\frac{1}{p}, V=\frac{q}{p^2}\)
Number of trials until first success.
Hypergeometric:
\(P(k) = \frac{\binom{D}{k}\binom{N-D}{n-k}}{\binom{N}{n}}\) ; \(E=n\frac{D}{N}\)
Sampling without replacement from a finite population.
Discrete Uniform \(U(n)\):
\(P(k) = \frac{1}{n}\) ; \(E=\frac{n+1}{2}, V=\frac{n^2-1}{12}\)
Equal probability for each value (e.g., fair die).
Continuous Distributions
General Density Condition:
\(\int_{-\infty}^{\infty} f(x) dx = 1\)
Area under any density curve must be 1.
Uniform \(U(a,b)\):
\(f(x) = \frac{1}{b-a}\) ; \(E = \frac{a+b}{2}, V = \frac{(b-a)^2}{12}\)
Exponential \(Exp(\lambda)\):
\(f(x) = \lambda e^{-\lambda x}\) ; \(E = \frac{1}{\lambda}, V = \frac{1}{\lambda^2}\)
Measures time until an event (e.g., waiting time).
Normal \(N(\mu, \sigma^2)\):
\(Z = \frac{X - \mu}{\sigma}\)
ANOVA (One-Way Analysis of Variance)
Sum of Squares (SS):
\(SST = \sum \sum x_{ij}^2 - \frac{G^2}{N}\) (Total)
\(SSB = \sum \frac{T_i^2}{n_i} - \frac{G^2}{N} = \sum n_i(\bar{x}_i - \bar{x}_{tot})^2\) (Between)
\(SSW = SST - SSB = \sum (n_i-1)s_i^2\) (Within)
\(SSB = \sum \frac{T_i^2}{n_i} - \frac{G^2}{N} = \sum n_i(\bar{x}_i - \bar{x}_{tot})^2\) (Between)
\(SSW = SST - SSB = \sum (n_i-1)s_i^2\) (Within)
Degrees of Freedom (df):
\(df_B = k-1\) ; \(df_W = N-k\) ; \(df_{tot} = N-1\)
Mean Squares (MS):
\(MSB = \frac{SSB}{df_B}\) ; \(MSW = \frac{SSW}{df_W}\)
Test Statistic:
\(F_{calc} = \frac{MSB}{MSW}\)
Hypothesis Testing
Single Sample (\(\sigma\) unknown):
\(t = \frac{\bar{X}-\mu_0}{s/\sqrt{n}}\) ; \(df = n-1\)
Paired Samples (Matched):
\(t = \frac{\bar{D}-\mu_D}{s_d/\sqrt{n}}\) ; \(df = n-1\)
D represents the paired differences (before/after).
Independent Samples (pooled \(s_p\)):
\(t = \frac{(\bar{X}_1-\bar{X}_2)-0}{s_p \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\) ; \(df = n_1+n_2-2\)
\(s_p^2 = \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}\)
\(s_p^2 = \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}\)
Proportion (Single Sample):
\(Z = \frac{\hat{p}-p_0}{\sqrt{\frac{p_0 q_0}{n}}}\)
Difference of Proportions (2 Samples):
\(Z = \frac{(\hat{p}_1-\hat{p}_2)-0}{\sqrt{\hat{p} \hat{q} (\frac{1}{n_1}+\frac{1}{n_2})}}\)
\(\hat{p} = \frac{x_1+x_2}{n_1+n_2}\) (pooled proportion)
\(\hat{p} = \frac{x_1+x_2}{n_1+n_2}\) (pooled proportion)
Chi-Square Test (\(\chi^2\))
Test Statistic Calculation: \(\chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}}\)
Expected Frequencies: \(E_{ij} = \frac{RowTotal \cdot ColTotal}{N}\)
Constructing expected values table for test of independence.
Regression and Correlation
Regression Line: \(\hat{y} = a + bx\)
Slope (b):
\(b = \frac{Cov(x,y)}{s_x^2} = r \frac{s_y}{s_x}\)
Intercept (a):
\(a = \bar{y} - b\bar{x}\)
Covariance:
\(Cov(x,y) = \frac{\sum x_i y_i - n\bar{x}\bar{y}}{n-1}\)
Pearson Correlation (r):
\(r = \frac{Cov(x,y)}{s_x \cdot s_y} = \frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum(x_i-\bar{x})^2 \sum(y_i-\bar{y})^2}}\)
Coefficient of Determination (R²):
\(R^2 = (r)^2\) (percentage of variance explained)