Arithmetic mean
$$\bar{x}=\frac1n\sum_{i=1}^n x_i$$
Find the balance-point average of observed values.
High school · AP · UniversityStatistics formulas, grouped by topic.
21 statistics formulas across 11 topics, free and without an account. Statistics formulas summarize data and quantify how much a sample can be trusted. Descriptive measures such as the mean and standard deviation come first, then standardization, then the standard errors that drive every confidence interval and test statistic. Each inference formula carries conditions about independence, randomness, and sample size, and those conditions are what make the resulting number meaningful.
Every statistics formula on one page
Each entry links to its own page, where the symbols are defined, the conditions are stated, and the formula is substituted through a worked example. Levels covered here: Middle school, High school, AP, University.
Descriptive statistics
2 formulas in descriptive statistics.
Weighted mean
$$\bar{x}_w=\frac{\sum w_ix_i}{\sum w_i}$$
Average values that contribute unequal weights.
High school · APSpread
3 formulas in spread.
Population variance and standard deviation
$$\sigma^2=\frac1N\sum(x_i-\mu)^2,\qquad\sigma=\sqrt{\sigma^2}$$
Measure average squared distance from a population mean and return to original units.
High school · AP · UniversitySample variance and standard deviation
$$s^2=\frac1{n-1}\sum(x_i-\bar{x})^2,\qquad s=\sqrt{s^2}$$
Estimate population spread from a sample.
High school · AP · UniversityRange and interquartile range
$$\text{range}=x_{\max}-x_{\min},\qquad IQR=Q_3-Q_1$$
Measure total spread and the spread of the middle half of ordered data.
Middle school · High school · APStandardization
1 formula in standardization.
z-score
$$z=\frac{x-\mu}{\sigma}$$
Express a value as standard deviations above or below a mean.
High school · APAssociation
2 formulas in association.
Sample covariance
$$s_{xy}=\frac{\sum(x_i-\bar x)(y_i-\bar y)}{n-1}$$
Measure how two quantitative variables vary together.
AP · UniversityPearson correlation
$$r=\frac{\sum(x_i-\bar x)(y_i-\bar y)}{\sqrt{\sum(x_i-\bar x)^2\sum(y_i-\bar y)^2}}$$
Measure strength and direction of a linear sample relationship.
AP · UniversityRegression
3 formulas in regression.
Least-squares regression line
$$\hat y=b_0+b_1x,\quad b_1=r\frac{s_y}{s_x},\quad b_0=\bar y-b_1\bar x$$
Predict a response using the line minimizing squared residuals.
AP · UniversityResidual
$$e=y-\hat y$$
Measure vertical prediction error for one observation.
High school · APCoefficient of determination
$$R^2=1-\frac{SS_{\mathrm{res}}}{SS_{\mathrm{tot}}}$$
Measure the proportion of response variation explained by a fitted model.
High school · AP · UniversitySampling
2 formulas in sampling.
Standard error of a mean
$$SE_{\bar x}=\frac{\sigma}{\sqrt n}\quad\text{or}\quad\frac{s}{\sqrt n}$$
Describe typical sample-mean variation across repeated samples.
AP · UniversityStandard error of a proportion
$$SE_{\hat p}=\sqrt{\frac{p(1-p)}{n}}$$
Describe sampling variation of a sample proportion.
AP · UniversityConfidence intervals
2 formulas in confidence intervals.
One-mean confidence interval
$$\bar x\pm t^*\frac{s}{\sqrt n}$$
Estimate a population mean with a margin of error.
AP · UniversityOne-proportion confidence interval
$$\hat p\pm z^*\sqrt{\frac{\hat p(1-\hat p)}{n}}$$
Estimate a population proportion.
AP · UniversityHypothesis tests
2 formulas in hypothesis tests.
One-sample t statistic
$$t=\frac{\bar x-\mu_0}{s/\sqrt n}$$
Compare a sample mean with a null-hypothesis mean.
AP · UniversityChi-square statistic
$$\chi^2=\sum\frac{(O-E)^2}{E}$$
Compare observed categorical counts with counts expected under a null model.
AP · UniversityDistributions
1 formula in distributions.
Normal density
$$f(x)=\frac1{\sigma\sqrt{2\pi}}e^{-\frac12\left(\frac{x-\mu}{\sigma}\right)^2}$$
Describe the bell-shaped normal probability density.
AP · UniversityEstimation
1 formula in estimation.
Margin of error
$$ME=(\text{critical value})(\text{standard error})$$
Express the half-width of a confidence interval.
AP · UniversityInference
2 formulas in inference.
Two-sample t statistic
$$t=\frac{(\bar x_1-\bar x_2)-\Delta_0}{\sqrt{s_1^2/n_1+s_2^2/n_2}}$$
Compare two independent population means without assuming known population spreads.
AP · UniversityTwo-proportion z statistic
$$z=\frac{(\hat p_1-\hat p_2)-0}{\sqrt{\hat p(1-\hat p)(1/n_1+1/n_2)}}$$
Test whether two independent population proportions differ.
AP · UniversityKeep going with statistics
A formula sheet is a reference, not a method. Use the subject guide to see where these relationships come from, a calculator to check a substitution you are unsure about, and practice questions to find out whether you can actually retrieve them.
- Study statistics — the subject hub with lessons, worked methods, and common mistakes.
- Statistics calculator — enter your own values and compare the result with your working.
- Statistics Practice — retrieval practice across every topic listed above.
- Math formula library — search all 215 formulas by name, symbol, or condition.
Formulas in the other subjects
Mathematics does not stop at a subject boundary, and neither do its formulas. Each subject page below lists its own relationships, conditions, and worked examples.
Arithmetic
Review the core arithmetic relationships, their restrictions, and worked substitutions.
Algebra
Review the core algebra relationships, their restrictions, and worked substitutions.
Geometry
Review the core geometry relationships, their restrictions, and worked substitutions.
Trigonometry
Review the core trigonometry relationships, their restrictions, and worked substitutions.
Precalculus
Review the core precalculus relationships, their restrictions, and worked substitutions.
Calculus
Review the core calculus relationships, their restrictions, and worked substitutions.
Probability
Review the core probability relationships, their restrictions, and worked substitutions.
Linear Algebra
Review the core linear algebra relationships, their restrictions, and worked substitutions.
Differential Equations
Review the core differential equations relationships, their restrictions, and worked substitutions.
Discrete Math
Review the core discrete math relationships, their restrictions, and worked substitutions.