Descriptive Statistics
Mean
xbar = sum(x_i) / nThe arithmetic average of a data set.
averagecenterdescriptive
Reference equations
Formula library
Descriptive Statistics
Sample Variance
s^2 = sum((x_i - xbar)^2) / (n - 1)Average squared distance from the sample mean with Bessel correction.
spreadvariancestandard deviation
Probability Foundations
Conditional Probability
P(A | B) = P(A and B) / P(B)Probability of A after restricting attention to cases where B occurred.
conditionalprobabilitygiven
Probability Foundations
Bayes' Theorem
P(A | B) = P(B | A)P(A) / P(B)Updates a prior probability after observing evidence.
bayespriorposterior
Distributions
Binomial PMF
P(X = k) = C(n,k)p^k(1-p)^(n-k)Exact probability of k successes in n independent Bernoulli trials.
binomialpmfcombinations
Distributions
Z-Score
z = (x - mu) / sigmaStandardizes a value by measuring standard deviations from the mean.
normalz scorestandardize
Sampling & CLT
Standard Error of xbar
SE_xbar = sigma / sqrt(n)Standard deviation of the sampling distribution of the sample mean.
samplingstandard errorclt
Inference
Confidence Interval for a Mean
xbar +/- critical value * standard errorEstimate plus or minus a margin of error.
confidence intervalmargin of errorinference
Inference
One-Sample Z Test
z = (xbar - mu_0) / (sigma / sqrt(n))Tests a sample mean against a null mean when population sigma is known.
hypothesisz testp value
Inference
Power
Power = 1 - betaProbability of correctly rejecting a false null hypothesis.
powertype ii errorbetahypothesis
Regression & Correlation
Least-Squares Regression Line
yhat = a + bxLine that minimizes the sum of squared residuals.
regressionslopeprediction
Regression & Correlation
Pearson Correlation
r = sum((x_i-xbar)(y_i-ybar)) / sqrt(sum((x_i-xbar)^2)sum((y_i-ybar)^2))Measures direction and strength of a linear association.
correlationassociationlinear