DSST Principles of Statistics cheat sheet

Skewness measures asymmetry: for a symmetric distribution the third standardized central moment is 0; right-skewed distributions (tail to the right) have positive skewness, left-skewed have negative skewness.

Kurtosis is the fourth standardized central moment; excess kurtosis = kurtosis − 3, so the normal distribution has excess kurtosis 0, leptokurtic distributions exceed 0 (heavy tails), platykurtic fall below 0.

Sample skewness g1 = m3/m2^(3/2) and sample kurtosis g2 = m4/m2^2, where mk = (1/n)Σ(xi − x̄)^k; these are biased estimators, so software often reports adjusted versions (e.g., Fisher–Pearson or excess forms).

Standard error of the mean is σ/√n, estimated by s/√n; it shrinks with the square root of sample size, so quadrupling n halves the standard error.

The central limit theorem: for independent observations with finite variance, the sampling distribution of x̄ approaches normality as n grows, which justifies z- and t-based inference even for non-normal populations at moderate n.

Law of large numbers: the sample mean converges in probability to the population mean as n increases; this concerns accuracy of the estimate, not the shape of its distribution.

Type I error (α) is rejecting a true null hypothesis; Type II error (β) is failing to reject a false null. Power = 1 − β increases with larger effect size, larger n, smaller σ, and larger α.

A p-value is the probability, assuming H0 is true, of observing a test statistic at least as extreme as the one computed; it is not the probability that H0 is true.

Degrees of freedom reflect the number of independent pieces of information remaining after estimating parameters; for a one-sample t test with n observations, df = n − 1.

Binomial(n, p): P(X = k) = C(n,k) p^k (1−p)^(n−k); mean = np, variance = np(1−p). Conditions: fixed n, independent trials, constant success probability p.

Poisson(λ): P(X = k) = e^(−λ) λ^k / k!; mean = variance = λ. It models counts of rare events in a fixed interval; the binomial approaches Poisson when n is large and p is small with np moderate.

Geometric distribution: P(X = k) = (1−p)^(k−1) p for the trial of first success; mean = 1/p, variance = (1−p)/p². It is memoryless, like the exponential.

Hypergeometric models sampling without replacement from a finite population: mean = n(K/N), variance = n(K/N)(1−K/N)(N−n)/(N−1); the finite population correction factor is (N−n)/(N−1).

Uniform(a, b) continuous: density 1/(b−a) on [a, b]; mean = (a+b)/2, variance = (b−a)²/12.

Normal(μ, σ): symmetric, mean = median = mode = μ; about 68% of values lie within μ ± σ, 95% within μ ± 2σ (1.96σ), 99.7% within μ ± 3σ. Standardize with z = (x − μ)/σ.

Exponential(λ): density λe^(−λx) for x ≥ 0; mean = 1/λ, variance = 1/λ²; memoryless, so P(X > s+t | X > s) = P(X > t).

For any continuous distribution, probability equals area under the density curve, P(X = c) = 0 for a single point, and total area equals 1.

Sum of independent normals is normal: if X ~ N(μ1, σ1²) and Y ~ N(μ2, σ2²) are independent, then X + Y ~ N(μ1+μ2, σ1²+σ2²); variances add, standard deviations do not.

Simple linear regression fits ŷ = b0 + b1x, where b1 = r(sy/sx) and b0 = ȳ − b1x̄; the fitted line passes through the point (x̄, ȳ).

The slope b1 estimates the expected change in y per one-unit increase in x; the intercept b0 is the predicted y when x = 0 and may be meaningless outside the observed data range.

Coefficient of determination R² = r² in simple regression equals the proportion of total variation in y explained by the model: R² = SSR/SST = 1 − SSE/SST.

Partition of variance: SST = SSR + SSE, with df n − 1 = 1 + (n − 2) in simple regression; the residual standard error is s = √(SSE/(n − 2)).

Inference on the slope uses t = (b1 − β1)/SE(b1) with df = n − 2; the equivalent overall F test in simple regression has F = t².

Assumptions: linearity, independence of errors, constant error variance (homoscedasticity), normally distributed errors, and no severe multicollinearity in multiple regression.

Residuals e = y − ŷ have mean 0 and are uncorrelated with the fitted values and with x; residual plots reveal curvature, heteroscedasticity, and outliers.

Extrapolation beyond the observed range of x is unreliable because the linear relationship may not hold there.

Multiple regression ŷ = b0 + b1x1 + … + bkxk: adjusted R² penalizes added predictors, and each coefficient is interpreted holding the other predictors constant.

General form: estimate ± critical value × standard error. A 100(1−α)% interval uses z_(1−α/2) when σ is known, e.g., z = 1.645 for 90%, 1.96 for 95%, 2.576 for 99%.

Mean with unknown σ: x̄ ± t_(α/2, n−1) · s/√n; the t critical value exceeds the z value and shrinks toward z as n grows.

Proportion: p̂ ± z_(1−α/2) √(p̂(1−p̂)/n), valid when np̂ and n(1−p̂) are both at least about 10.

Margin of error halves when sample size is quadrupled; required n for a target margin E with a mean is n = (zσ/E)², rounded up.

Increasing confidence level widens the interval; increasing sample size narrows it. A 95% interval does not mean a 95% probability that μ lies inside this particular realized interval.

Interpretation: if the same sampling procedure were repeated many times, about 95% of the resulting intervals would contain the true parameter.

Relationship to tests: a two-sided test at significance α rejects H0 exactly when the parameter value under H0 falls outside the corresponding 100(1−α)% confidence interval.

Prediction interval for a new observation is wider than a confidence interval for the mean response, because it adds the residual variance: ŷ ± t·s√(1 + 1/n + (x0 − x̄)²/Sxx).

For a difference in means, the standard error is √(s1²/n1 + s2²/n2); when population variances are assumed equal, pool with sp² = [(n1−1)s1² + (n2−1)s2²]/(n1+n2−2) and df = n1+n2−2.