Statistics Formulas

Z-Test Formula

The Z-test formula compares a sample mean to a population mean. Learn when to use it, how to calculate the Z-statistic, and interpret p-values.

Mann-Whitney U Test Formula

Reference for the Mann-Whitney U test, a non-parametric method that compares two independent groups. Includes formula, worked example, and interpretation.

Error Function (erf)

The error function erf(x) gives the probability that a normally distributed variable falls within a given range. Includes examples and applications.

F-Distribution Formula

The F-distribution is the ratio of two chi-squared distributions. Used in ANOVA, comparing variances, and testing overall regression model significance.

Log-Normal Distribution Formula

The log-normal distribution models quantities whose logarithm is normally distributed. Common in finance, biology, and environmental science.

Spearman Rank Correlation Formula

Spearman rank correlation measures monotonic relationships using data ranks not raw values. A non-parametric alternative to Pearson correlation with examples.

Weibull Distribution Formula

Reference for the Weibull distribution for time-to-failure modeling. Covers shape and scale parameters with applications in reliability and survival analysis.

Coefficient of Variation Formula

Calculate the coefficient of variation: CV = (standard deviation / mean) x 100. Compare variability between datasets with different units or scales.

Geometric Distribution

Learn the geometric distribution formula for probability of first success on trial k, with mean, variance, and worked examples.

Linear Regression Slope Formula

Calculate the slope and intercept of a linear regression line using the least squares method. Predict outcomes from data relationships.

Moment Generating Function

Learn the moment generating function (MGF) formula, how to derive moments of probability distributions, with worked examples.

Weighted Average Formula

Learn the weighted average formula for calculating means where some values count more than others, with practical examples.

ANOVA F-Test Formula

The ANOVA F-test compares means of three or more groups using F = MS_between / MS_within to determine if group differences are significant.

ANOVA Formula (F-Test)

The one-way ANOVA F-test compares means across multiple groups. Learn how to calculate the F-statistic with step-by-step examples.

Bayes' Theorem

Bayes' theorem: P(A|B) = P(B|A)*P(A)/P(B). Foundation of Bayesian statistics with worked examples in medical testing, spam filtering, and decision-making.

Chi-Square Test Formula

The chi-square test statistic χ² = Σ(O−E)²/E measures how observed frequencies differ from expected frequencies in categorical data.

Chi-Squared Distribution

The chi-squared test measures how well observed data fits expected values. Learn the formula and chi-squared test with worked examples.

Combinations and Permutations

Reference for C(n,r) = n!/(r!(n-r)!) and P(n,r) = n!/(n-r)!. Explains when order matters with worked examples for cards, teams, and lottery problems.

Covariance Formula

Reference for covariance Cov(X,Y) = E[(X-μx)(Y-μy)]. Explains positive, negative, and zero covariance with examples for portfolio analysis and data science.

Effect Size (Cohen's d)

Cohen's d measures the practical significance of a difference between two group means. Learn to calculate and interpret effect sizes.

Expected Value

Reference for expected value E(X) = Σ[x·P(x)] and E(X) = ∫x f(x)dx. Covers variance, linearity property, and applications in gambling and risk analysis.

Hypergeometric Distribution

The hypergeometric distribution models drawing successes from a finite population without replacement. Learn the formula with examples.

Markov Chain Formula

Markov chains model systems that transition between states with fixed probabilities. Learn transition matrices and steady-state formulas.

Mode Formula

Reference for finding the mode — the most frequent value in a data set. Covers unimodal, bimodal, and multimodal distributions and grouped data mode formula.

P-Value Formula and Interpretation

Reference for calculating and interpreting p-values in hypothesis testing. Covers null rejection, one-tailed vs two-tailed, and z-test vs t-test.

Percentile Rank Formula

Calculate percentile rank to find what percentage of values fall below a given score. Used in test scores and data analysis.

Regression Formula (Least Squares)

Learn the least squares regression formula for finding the line of best fit. Includes slope and intercept derivation with examples.

Sample Size Formula

Formula for calculating the minimum sample size needed for surveys and research studies at a given confidence level and margin of error.

Standard Error Formula

Calculate the standard error of the mean with SE = s / sqrt(n). Understand sampling variability and confidence intervals with examples.

T-Distribution Formula

The Student's t-distribution formula for hypothesis testing with small samples when population standard deviation is unknown.

Binomial Distribution Formula

Reference for the binomial distribution formula. Calculate the probability of exactly k successes in n independent trials with worked examples.

Chi-Squared Test Formula

Reference for χ² = Σ(O-E)²/E for goodness-of-fit and independence tests. Covers degrees of freedom, critical values, p-values, and contingency tables.

Confidence Interval Formula

Calculate confidence intervals for any sample using CI = x̄ ± z*(σ/√n). Returns 90%, 95%, and 99% intervals for surveys, experiments, and statistical inference.

Interquartile Range (IQR)

Reference for IQR = Q3 - Q1 with calculation steps, box plot interpretation, and outlier detection using the 1.5 x IQR rule. Includes worked examples.

Poisson Distribution Formula

Calculate the probability of a given number of events in a fixed interval. Models rare events like calls, defects, or arrivals.

Basic Probability Formula

Calculate probability using P(A) = favorable outcomes / total outcomes. Covers basic, conditional, complement, and joint probability rules with worked examples.

Bayes' Theorem

Apply Bayes' Theorem with P(A|B) = P(B|A) × P(A) / P(B). Update probabilities based on new evidence and conditional information.

Correlation Coefficient Formula

Calculate the Pearson correlation coefficient r to measure the strength and direction of a linear relationship between two variables.

Linear Regression Formula

Calculate the line of best fit with y = mx + b. Predict outcomes using linear regression and understand the slope and intercept.

Mean (Average) Formula

Reference for arithmetic mean formula Mean = Σx/n. Compares arithmetic, geometric, and harmonic mean with guidance on when each best represents a data set.

Median Formula

Find the median of a data set. The median is the middle value when data is sorted. Learn how to handle both odd and even data sets.

Normal Distribution Formula

Understand the normal distribution (bell curve) with f(x) = (1/σ√(2π)) × e^(-(x-μ)²/(2σ²)). The most important distribution in statistics.

Standard Deviation Formula

Calculate standard deviation using σ = sqrt(Σ(x-μ)^2/N). Measure how spread out a dataset is from its mean. Covers population vs sample standard deviation.

Variance Formula

Reference for population variance σ² = Σ(x-μ)²/N and sample variance s² = Σ(x-x̄)²/(n-1). Explains Bessel correction with step-by-step examples for statistics.

Z-Score Formula

Reference for z-score z = (x − μ) / σ measuring standard deviations from the mean. Covers z-table lookup, p-value interpretation, and test score comparison.