All calculators

Statistics: standard deviation and spread

• Standard deviation

  Sample or population σ from a list of numbers, with a dot plot and full working.

• Variance

  The squared spread, before the final square root. Sample and population.

• Relative SD (%RSD)

  Spread as a percentage of the mean. The laboratory precision measure.

• Coefficient of variation

  The same quantity as %RSD, reported as a ratio.

• Standard error

  How precisely the mean is known, and recovery of SD from a published SE.

Standard deviation and variance, by task

• Sample standard deviation

  Starts in sample mode: s with the n − 1 divisor.

• Population standard deviation

  Starts in population mode: σ, dividing by N.

• Sample variance

  s² with the n − 1 divisor, and the SD beside it.

• Population variance

  σ² dividing by N, with the shortcut formula worked.

• Mean and standard deviation

  The pair most reports quote, formatted as mean ± SD.

• Mean and variance

  The two parameters that define most distributions.

• Mean, variance and SD

  All three, each derived from the one before, in one worked table.

• Variance and standard deviation

  Both, in both sample and population form.

• SD and standard error

  Spread of the data against precision of the mean.

• Mean and standard error

  Mean ± SEM for error bars and lab reports.

• Z-scores for every value

  Mean and SD first, then each value standardised.

• Test score SD

  Class spread for teachers, with example score sets.

• SD from CSV data

  Paste exported rows and pick the column to read.

• SD for large datasets

  Tens of thousands of values, computed stably.

Spread, shape and position

• Standard deviation graph

  Histogram of your data with the fitted normal curve and the ±1, ±2 and ±3 SD bands shaded.

• Histogram maker

  Equal-width bins by Sturges’ rule, counts on every bar and a normal curve fitted to your data.

• Outliers & IQR

  Outliers by Tukey fences, z-scores and Grubbs’ test, with the quartiles and a box plot.

• Descriptive statistics

  Every summary at once: centre, spread, quartiles, skewness, kurtosis and a 95% CI.

• Five-number summary

  Min, Q1, median, Q3 and max, drawn as a box and whisker plot.

• Percentiles

  Any percentile from your data, plus the standard-deviation conversion table.

• Mean absolute deviation

  Spread without squaring, compared against the standard deviation.

From tables, groups or weights

• Grouped data SD

  From class midpoints and frequencies, when the raw values are gone.

• Frequency table SD

  Exact values with their counts, no midpoint approximation.

• Pooled SD

  One spread estimate across two or more groups, weighted by degrees of freedom.

• Weighted SD

  When values carry different weights: allocations, survey weights, precision.

Distributions

• Normal distribution

  Probability between any two values, with the area shaded on the curve.

• Binomial distribution

  Mean, variance and σ = √(npq), with the full distribution plotted.

• Empirical rule

  The 68–95–99.7 bands, with their exact coverages.

• Bell curve generator

  Draw and label a normal curve from any mean and standard deviation.

• Poisson distribution

  P(X = k) and cumulative probabilities for a rate λ, with σ = √λ and the mass function plotted.

Inference

• Z-score

  How many standard deviations a value sits from the mean, with percentile.

• Confidence interval

  An interval for a mean, using the t distribution.

• Margin of error

  The ± for a mean or a survey proportion, under t and z.

• CI for standard deviation

  An interval for the population SD itself, from the chi-square distribution.

• T-test

  One-sample, two-sample or paired, with Cohen’s d and an interval.

• Cohen’s d

  Effect size from two groups’ means, SDs and sizes, with Hedges’ g and a 95% CI.

• ANOVA

  Three or more group means at once, with the full sum-of-squares table.

• Chi-square

  Goodness of fit on counts, with each category’s contribution.

• Critical value

  Critical values for z, t, chi-square and F at any α, one- or two-tailed, with the rejection region.

• P-value

  Convert a z, t, chi-square or F statistic into a p value.

• Correlation coefficient

  Pearson’s r and r², with Spearman’s rho and a scatter plot.

• Linear regression

  The least-squares line, its slope, and how much it explains.

Applied: finance, quality and planning

• Portfolio SD

  Two-asset portfolio risk from weights, SDs and correlation, with the diversification benefit shown.

• Cpk & process capability

  Cp, Cpk, Cpu and Cpl from a mean, SD and spec limits, with expected ppm out of spec and sigma level.

• Sample size

  Minimum sample for a survey proportion or a mean, from the margin of error and confidence level.

• Batting average

  Hits ÷ at-bats, with how much of the average could be luck.

Statistics: means, medians and modes

• Geometric mean calculator

  The nth root of the product, or the average compound growth rate from percentages.

• Mean calculator (series and grouped data)

  The arithmetic mean of a list, a frequency table or class intervals, by the direct, assumed-mean or step-deviation method.

• Weighted mean calculator

  Σwx ÷ Σw with the full table: the weighted arithmetic mean used for index numbers and cost-of-living figures.

• Harmonic mean calculator

  The average for rates, such as speeds over equal distances, with a weighted version for unequal ones.

• Missing value and missing frequency calculator

  Find a missing observation, or one or two missing frequencies, when the mean is known.

• Mean, median, mode and range calculator

  All four from a list, with the working for each and every mode reported.

• Median and mode calculator (grouped data)

  Median and mode of a list, a discrete series or class intervals, checked with Mode = 3 Median − 2 Mean.

• Mode calculator

  The mode of a list, a frequency table or grouped data, including the grouping method.

• Arithmetic mean calculator

  Σx ÷ n for a list or frequency table, compared with the geometric and harmonic means of the same data.

• Weighted median calculator

  The value where the cumulative weight reaches half, with the exact-half tie handled.

Statistics: quartiles, deciles and dispersion

• Quartile calculator

  Q1, Q2 and Q3 for a list, a discrete (x, f) series or class intervals, with the textbook and Excel methods.

• Quartile deviation calculator

  Quartile deviation (Q3 − Q1)/2 and its coefficient for individual, discrete and grouped data.

• Mean deviation calculator

  Mean deviation about the mean, median or mode, with its coefficient, for all three series types.

• Decile calculator

  Any decile D1 to D9 from a list, a frequency table or class intervals.

• Decile deviation calculator

  Interdecile range D9 − D1, the decile deviation (D9 − D1)/2 and its coefficient.

• Coefficient of range calculator

  Range and the coefficient of range (L − S)/(L + S) for raw or grouped data.

• Percentile rank calculator

  The percentile rank of a value, by the half-ties or strictly-below rule, or the value at a given rank.

Statistics: shape, spread and outliers

• Dispersion calculator

  Range, IQR, quartile deviation, mean and median absolute deviation, variance, SD, CV and the index of dispersion in one table.

• Skewness calculator

  Fisher–Pearson g₁, adjusted G₁, both Pearson coefficients, the standard error of skewness and a z test.

• Outlier calculator

  Flag outliers by Tukey fences, z-scores, modified z-scores and Grubbs’ test, side by side.

• Upper and lower fence calculator

  Lower and upper fences Q₁ − k·IQR and Q₃ + k·IQR, from data or from the quartiles.

• Median absolute deviation calculator

  MAD, the 1.4826-scaled estimate of σ and modified z-scores for every value.

• Midrange calculator

  The midpoint of the smallest and largest values, compared with the mean and median.

• Covariance calculator

  Sample and population covariance of paired data with the deviations table and r.

• Index of qualitative variation calculator

  IQV for categorical data: (K/(K − 1))(1 − Σp²), from category counts.

• Average rating calculator

  Average star rating from the count at each score, with a Bayesian average and the distribution.

• Constant of proportionality calculator

  k = y/x from a pair or a table, with a check that the table is proportional.

• Standard deviation index calculator

  SDI = (lab mean − peer mean) ÷ peer SD for lab QC, with the CV ratio.

• Process capability index calculator

  Cp, Cpk, Pp and Ppk from measurements or summary figures, with ppm and sigma level.

Statistics: charts and frequency tables

• Box plot calculator

  Box-and-whisker plots for one or several data sets, with quartiles, fences and outliers.

• Dot plot calculator

  A stacked dot plot of your data, with counts, the mode, mean and median.

• Histogram calculator

  Bins by Sturges, square-root, Scott or Freedman–Diaconis, with the frequency table and histogram.

• Pie chart calculator

  Percentages and central angles in degrees for each category, drawn as a pie chart.

• Stem-and-leaf plot calculator

  Stem-and-leaf displays with any leaf unit, split stems, or two data sets back to back.

• Frequency distribution calculator

  Grouped or ungrouped frequency tables with relative, cumulative and cumulative relative frequency.

• Frequency polygon calculator

  Class midpoints and frequencies plotted as a closed frequency polygon, from data or a class table.

• Relative frequency calculator

  Relative and cumulative relative frequencies from raw values or category counts.

• Class width calculator

  Class width from the range and number of classes, with the class limits and boundaries.

• Venn diagram calculator

  Two- or three-set Venn diagrams from counts or element lists, with every region filled in.

• Scatter plot maker

  Plot x-y pairs as a scatter plot with an optional least-squares trend line, r, R² and flagged outliers.

• Least to greatest calculator

  Put fractions, decimals, negatives and percentages in order, least to greatest or the reverse.

Statistics: correlation, regression, probability and tests

• Pearson correlation coefficient calculator

  Karl Pearson's r by the actual-mean, direct and assumed-mean methods, with the t-test and a scatter plot.

• Spearman's rank correlation calculator

  Rank correlation from raw data or ranks, with the (m³ − m)/12 correction for tied ranks.

• Kendall's tau calculator

  Tau-a and tau-b from concordant and discordant pairs, with a z test.

• Multiple regression calculator

  Least-squares fit with up to four predictors: coefficients, t-tests, R² and the F-test.

• Logistic regression calculator

  Binary logistic regression by maximum likelihood, with odds ratios and predicted probabilities.

• Z-test calculator

  One- and two-sample z-tests for means and proportions, with p-value and decision.

• F-test calculator

  Test whether two variances are equal, from raw data or from s and n.

• Probability calculator

  Union, intersection, complement and conditional probabilities of two events, or P(at least one).

• Conditional probability calculator

  P(A|B) from probabilities or from a 2×2 table of counts.

• Bayes' theorem calculator

  Posterior probability for a medical test, with natural frequencies, or across several hypotheses.

Statistics: probability, odds and counting

• Combination calculator

  nCr with or without repetition, and a list of every combination for small sets.

• Permutation calculator

  nPr, nʳ with repetition, and the distinct arrangements of the letters in a word.

• Expected value calculator

  E[X], variance and standard deviation from a table of outcomes and probabilities, with a fair-game check.

• Joint probability calculator

  P(A ∩ B) for independent or dependent events, or from a two-way table of counts.

• Probability of 3 events calculator

  At least one, all three, exactly one, exactly two and none, for three independent events.

• Odds calculator

  Convert between probability, odds for and against, and fractional, decimal and American odds.

• Implied probability calculator

  Implied probabilities from betting odds, the bookmaker’s overround, and fair no-vig probabilities.

• Chebyshev's theorem calculator

  The minimum share of data within k standard deviations of the mean, for any distribution.

• P-hat calculator

  Sample proportion p̂ = x/n with its standard error, margin of error and confidence interval.

• Password combination calculator

  How many passwords a character set and length allow, their entropy, and the time to brute-force them.

• Lottery calculator

  Jackpot and prize-tier odds for any pick-k-from-n lottery with a bonus ball, and the expected value of a ticket.

• Quiz: Lottery calculator

  Random lottery-odds questions to answer, marked instantly with the working shown.

Statistics: coins, dice and games of chance

• Coin flipper

  Flip one coin or thousands, and compare the tally with a 50–50 split.

• Coin flip probability calculator

  The chance of exactly, at least or at most k heads in n flips.

• Coin toss streak calculator

  The exact chance of k heads (or either side) in a row, and the expected longest run.

• Dice average calculator

  The average, variance and SD of any NdS + M roll, with drop lowest or highest.

• Dice probability calculator

  The chance of rolling a sum exactly, at least, at most or in a range with N dice.

• Dice roller

  Roll any dice in NdS+M notation and keep a history of recent rolls.

• Quiz: Dice average calculator

  Random dice-average questions with instant marking and worked answers.

• Roulette payout calculator

  Payout, win probability, house edge and expected loss for every roulette bet.

• Random number generator

  Random whole numbers or decimals in a range, with no-repeat and sort options.

• Birthday paradox calculator

  The chance two people share a birthday, and the group size for any target.

• Monty Hall problem calculator

  Win chances for staying and switching with any number of doors, plus a simulation.

Statistics: diagnostic tests, risk and paradoxes

• Sensitivity and specificity calculator

  Sensitivity, specificity, PPV, NPV and likelihood ratios with Wilson or exact confidence intervals.

• Confusion matrix calculator

  Accuracy, precision, recall, F1, MCC, kappa and more from a 2×2 confusion matrix.

• Accuracy calculator

  Accuracy from TP/TN/FP/FN or from sensitivity and specificity, and percent accuracy of a measurement.

• Post-test probability calculator

  Pre-test probability and a likelihood ratio to post-test probability, drawn on a Fagan nomogram.

• Youden index calculator

  Youden's J from a 2×2 table, or the best cut-off from a table of thresholds.

• Relative risk calculator

  Relative risk with a 95% confidence interval, the risk difference and NNT or NNH.

• Risk calculator

  Absolute and relative risk, risk reduction, attributable risk, odds ratio and NNT from two groups.

• False positive paradox calculator

  Why an accurate test gives mostly false positives for a rare condition, with a frequency tree.

• Bertrand's box paradox

  Draw a gold coin: why the other coin is gold two times in three, with custom boxes and simulation.

• Bertrand's paradox simulator

  Random chords by three methods give 1/3, 1/2 and 1/4; simulated and drawn.

• Boy or girl paradox calculator

  The two-child problem: at least one boy, older boy, and the boy born on a Tuesday.

• Parrondo's paradox calculator

  Two losing games that win when mixed: exact drift and simulated runs.

• Two envelopes paradox calculator

  The 1.25× switching argument against the correct analysis, with simulated strategies.

Statistics: probability distributions

• Inverse normal distribution calculator

  The x and z that leave a given area to the left, right, middle or in both tails.

• Uniform distribution calculator

  Probabilities, mean and SD for a continuous range or whole numbers a to b.

• Exponential distribution calculator

  Waiting-time probabilities and percentiles from the rate λ or the mean.

• Geometric distribution calculator

  P(X = k), ≤ and ≥ for the trial of the first success, in both conventions.

• Negative binomial distribution calculator

  Probabilities for the number of failures or trials before the r-th success.

• Hypergeometric distribution calculator

  Successes in a sample drawn without replacement, with a full probability table.

• Beta distribution calculator

  CDF, density, percentiles and moments of Beta(α, β) on 0 to 1.

• Lognormal distribution calculator

  Lognormal probabilities from μ and σ of ln X, or from the mean and SD of X.

• Weibull distribution calculator

  Failure probabilities, reliability, B10 life and moments from shape k and scale λ.

• Rayleigh distribution calculator

  Probabilities, percentiles and moments of the Rayleigh distribution from σ.

• SMp(x) distribution calculator

  The six-parameter SMp(x) function: value, probabilities, percentiles and moments.

Statistics: sampling distributions and estimation

• Central limit theorem calculator

  Mean and standard error of x̄ from μ, σ and n, and the probability that the sample mean falls in a range.

• Normal probability calculator for sampling distributions

  Probabilities for a sample mean or sample proportion, with the finite population correction.

• Sampling distribution of the sample proportion calculator

  Mean and standard error of p̂, the np ≥ 10 check, and probabilities for p̂.

• Continuity correction calculator

  Turn P(X ≤ k), P(X = k) and the rest into the corrected normal bound, and compare with the exact answer.

• Normal approximation calculator

  Normal approximation to the binomial or Poisson, with and without the continuity correction, against the exact value.

• Standard deviation of the sample mean calculator

  σ/√n from a standard deviation or raw data, with the finite population correction.

• Point estimate calculator

  MLE, Laplace, Jeffrey and Wilson estimates of a proportion, the best one picked, or the estimate from a CI.

• Sampling error calculator

  Margin of sampling error for a survey proportion or a mean at any confidence level.

• Effect size calculator

  Cohen's d, Hedges' g and Glass's Δ from two groups, plus d ↔ r and η² from F.

• Exponential growth prediction calculator

  Predict x(t) = x₀(1 + r)^t or solve for the start value, rate or time, with the doubling time.

• Upper control limit calculator

  UCL and LCL for X̄-R, X̄-s, I-MR, p and c charts from data or summary values, with the chart.

• Benford's law calculator

  First-digit distribution against Benford, with the chi-square test and Nigrini’s MAD conformity.

Statistics: hypothesis tests

• Hypothesis testing calculator

  One-sample z or t test for a mean, or z test for a proportion, with the decision and a shaded rejection region.

• A/B test calculator

  Conversion rates, uplift, p-value and confidence interval for an A/B test, plus the sample size per variant.

• Bonferroni correction calculator

  The α/m threshold, and Bonferroni, Holm and Benjamini–Hochberg adjusted p-values for a list of tests.

• Fisher's exact test calculator

  Exact one- and two-sided p-values for a 2×2 table, with the odds ratio and every possible table.

• McNemar's test calculator

  Paired yes/no test: χ² with and without continuity correction, exact binomial p and the odds ratio b/c.

• Mann–Whitney U test calculator

  Rank-based comparison of two groups: U₁, U₂, exact or normal p-value and effect size r.

• Wilcoxon rank-sum test calculator

  The rank sum W with its expected value, standard error, z and p-value, ties handled.

• Power analysis calculator

  Sample size or power for t-tests, two proportions and correlations, with a power curve.

Statistics: regression and model fit

• Coefficient of determination calculator

  R² and adjusted R² from x, y data, from actual vs predicted values, or from sums of squares.

• Quadratic regression calculator

  Least-squares y = ax² + bx + c with R², a fitted-curve chart and predictions.

• Cubic regression calculator

  Fit y = ax³ + bx² + cx + d, with R², adjusted R² and a chart of the curve.

• Polynomial regression calculator

  Polynomial fit of degree 1 to 6, with a table comparing adjusted R² across degrees.

• Exponential regression calculator

  Fit y = a·e^(bx) and y = a·b^x by log-linear least squares, with growth rate and predictions.

• F statistic calculator

  F from ANOVA sums of squares, regression R² or two variances, with p-value and critical value.

• t statistic calculator

  One-sample, Welch, pooled and paired t from summary statistics, with df and p-value.

• Degrees of freedom calculator

  Degrees of freedom for t-tests, Welch, chi-square, ANOVA and regression, with formulas.

Statistics: error, uncertainty, diversity and information

• Error propagation calculator

  Carry uncertainties through sums, products, powers or any formula f(a, b, c).

• Uncertainty calculator

  Standard uncertainty s/√n and expanded uncertainty U = k·u from repeated readings.

• Absolute uncertainty calculator

  Convert between absolute and percentage uncertainty, or combine uncertainties of a sum.

• Relative error calculator

  Absolute, relative and percent error of a measurement against the true value.

• Relative standard error calculator

  RSE = SE ÷ estimate × 100 from data, or from a standard error and estimate.

• MSE calculator

  MSE, RMSE, MAE and MAPE from actual and predicted values.

• Residual calculator

  Residuals, a residual plot and SSE for a least-squares or given line.

• Raw score calculator

  Turn a z-score or a percentile back into a raw score with x = μ + zσ.

• Shannon entropy calculator

  Entropy from probabilities, counts or text, with the Shannon index H′ and evenness.

• Simpson's diversity index calculator

  D, 1 − D and 1/D from species counts, in the n(n − 1) or Σp² form.

• Quiz: Simpson's diversity index calculator

  Random quadrat questions: work out 1 − D and check it against the full working.

• Correlation coefficient calculator (Matthews)

  MCC from TP, TN, FP and FN, or from a multiclass confusion matrix.

Statistics: index numbers

• Laspeyres price index calculator

  Σp₁q₀ / Σp₀q₀ × 100: prices weighted by base-year quantities, with the full p₀q₀ and p₁q₀ table.

• Paasche price index calculator

  Σp₁q₁ / Σp₀q₁ × 100: prices weighted by current-year quantities, with the working.

• Fisher’s ideal index calculator

  √(Laspeyres × Paasche), with the time reversal and factor reversal tests checked on your numbers.

• Weighted aggregative index calculator

  Σp₁w / Σp₀w by six methods, from Laspeyres to Fisher, compared side by side on the same data.

• Marshall–Edgeworth price index calculator

  Σp₁(q₀ + q₁) / Σp₀(q₀ + q₁) × 100: the two years’ quantities added together as weights.

• Dorbish–Bowley price index calculator

  The arithmetic mean of the Laspeyres and Paasche indices, (L + P) / 2, with both shown.

• Kelly’s price index calculator

  Σp₁q / Σp₀q × 100 with a fixed basket q that does not change between the two periods.

• Simple aggregative method calculator

  Σp₁ / Σp₀ × 100, the unweighted index, with a quantity index option (Σq₁ / Σq₀).

• Simple average of price relatives calculator

  ΣR / N or antilog(Σ log R / N): an unweighted index from each item’s price relative.

• Weighted average of price relatives calculator

  ΣRW / ΣW with value weights W = p₀q₀ or your own weights, arithmetic or geometric.

• Price relatives calculator

  p₁/p₀ × 100 for each item, plus fixed-base, link and chain relatives for a price series.

Math: averages and sums

• Mean (average)

  The arithmetic mean on its own, with the sum and count that produced it.

• Median

  The middle value, sorted and shown: the average that ignores outliers.

• Mean, median, mode, range

  The full descriptive summary, with every tied mode reported.

• Sum of squares

  Σ(x − x̄)² by both formulas, with the variance and SD that follow from it.

• Root mean square

  RMS of any list, the sum and mean of squares, and how RMS relates to the SD.

• Weighted average

  Σwx ÷ Σw for grades, prices or survey weights, next to the unweighted mean.

• Grade average

  The mean of a set of scores, with a trimmed mean for dropping the extremes.

• Range

  Maximum minus minimum, with the midrange and how the range compares with the SD.

• Minimum and maximum

  The smallest and largest values, how often each occurs, and the runners-up.

Math: arithmetic and number theory

• Addition calculator

  Add any list of numbers exactly, with column addition and the carries shown.

• Subtraction calculator

  Subtract with column working that shows every borrow, including across zeros.

• Multiplication calculator

  Exact products of any size, with long multiplication and partial products.

• Division calculator

  Quotient, remainder, repeating decimal and a step-by-step long division.

• Percentage calculator

  Percent of a number, percentage change, increase or decrease, and reverse percentages.

• Ratio and proportion calculator

  Simplify ratios, solve a : b = c : x, and share an amount in a ratio.

• Prime number calculator

  Test any number for primality, find the next or previous prime, list primes, or the nth prime.

• Prime factorization calculator

  Prime factors with exponents, the division ladder, and every divisor.

• Divisibility calculator

  Test a number against 2 to 12 with each divisibility rule applied to its digits.

• Number theory calculator

  Euler’s totient, divisor functions, perfect numbers, digital root and Collatz steps.

• GCD and LCM calculator

  GCD and LCM by the Euclidean algorithm and by prime factors, with Bézout coefficients.

• Modular arithmetic calculator

  a mod n, modular powers, inverses and linear congruences, with the steps.

• Number system converter

  Convert between binary, octal, decimal, hex and any base from 2 to 36.

Math: algebra

• Quadratic equation calculator

  Roots of ax² + bx + c = 0 in exact surd or complex form, with the vertex and a graph.

• Linear equation calculator

  One linear equation step by step, or 2 × 2 and 3 × 3 systems by elimination or Cramer’s rule.

• Polynomial calculator

  Add, multiply and long-divide polynomials, evaluate them, and find every real and complex root.

• Simplify expression calculator

  Simplified, expanded and factored forms of an algebraic expression, with fractions combined.

• Function calculator

  Evaluate f(x), tabulate it, and find its domain, inverse and compositions, with a graph.

• Sequence and series calculator

  nth term, sum of n terms and sum to infinity for arithmetic, geometric and harmonic sequences.

• Binomial theorem calculator

  Expand (ax + b)ⁿ, find any term or the coefficient of xᵏ, with Pascal’s triangle.

Math: geometry

• Area calculator

  The area of thirteen shapes, with a labelled diagram and conversions to ft², acres and hectares.

• Perimeter calculator

  The distance round any common shape, including the ellipse by Ramanujan’s approximation.

• Volume calculator

  Volume and surface area of eleven solids, in cubic units, litres and gallons.

• Triangle calculator

  Area, angles, type, heights, medians and radii of a triangle from any three parts.

• Circle calculator

  Radius, diameter, circumference and area from any one, plus arcs, chords and segments.

• Rectangle and square calculator

  Length, width, area, perimeter and diagonal from any two known values.

• Polygon calculator

  Regular polygons from any one measurement, and any polygon’s area from coordinates.

Math: trigonometry

• Trigonometric functions calculator

  All six trig functions of an angle with exact values, plus the inverse functions.

• Sine calculator

  sin θ in degrees, radians or gradians, arcsin with every solution, and opposite over hypotenuse.

• Cosine calculator

  cos θ with exact values, arccos, adjacent over hypotenuse and the law of cosines.

• Tangent calculator

  tan θ, arctan, opposite over adjacent, and the angle of a slope or grade.

• Triangle solver

  Solve any triangle from three parts, with the ambiguous SSA case and a drawing to scale.

• Trigonometric equation solver

  Solve trig equations on an interval, with the general solution and a graph of both sides.

• Trig graph calculator

  Graph y = A sin(Bx − C) + D and friends: amplitude, period, phase shift and asymptotes.

Math: calculus

• Derivative calculator

  Derivatives up to the fourth, with each rule shown as a step, a tangent line and a graph.

• Integral calculator

  Antiderivatives with the method named, and definite or improper integrals, exact and numerical.

• Limit calculator

  Limits at a point or at ±∞, one- or two-sided, with L’Hôpital’s rule and a table of values.

• Differential equation calculator

  Solve y′ = f(x, y) numerically with RK4 and Euler, or linear equations exactly.

• Partial derivative calculator

  First and second partial derivatives, mixed partials and Clairaut’s check, at a point.

• Infinite series calculator

  Convergence tests with reasoning, partial sums, and the sum of a convergent series.

• Multivariable calculus calculator

  Gradient, directional derivative, Hessian, critical points and double integrals of f(x, y).

Math: coordinate geometry

• Distance and midpoint calculator

  Distance as an exact surd, midpoint, slope and the line through two points, in 2D or 3D.

• Straight line equation calculator

  A line’s equation from points, slope or intercepts, in slope-intercept, point-slope, general and intercept forms.

• Line calculator

  Where two lines meet, the angle between them, parallel-line distance and point-to-line distance.

• Point calculator

  Section formula, centroid and area of a triangle, collinearity, quadrants and reflections.

• Curve calculator

  Intercepts, turning points, tangent and normal, area and arc length for y = f(x), with a graph.

• Conic sections calculator

  Classify a second-degree equation and find its standard form, foci, directrices and eccentricity.

• 3D geometry calculator

  Direction cosines, lines and planes in space, angles between them and shortest distances.

Math: vectors and matrices

• Vector calculator

  Add, subtract and scale vectors; dot and cross products, angle, unit vector and projection.

• Matrix calculator

  Inverse, determinant, RREF, rank, trace, powers and eigenvalues, in exact fractions.

• Matrix operations calculator

  A + B, A − B, AB, BA and kA, or solve AX = B, with the row-by-column working.

• Determinant calculator

  Determinants up to 8×8 by cofactor expansion, Sarrus or row reduction, step by step.

• Cross product calculator

  a × b with the i, j, k determinant expanded, the area it spans and a perpendicularity check.

• Vector magnitude and direction

  Length, direction angle and bearing of a vector, 3D direction cosines, or components from an angle.

• Vector analysis calculator

  Triple products, volumes, coplanarity and independence tests, areas and Gram–Schmidt.

Math: discrete mathematics

• Set calculator

  Union, intersection, differences and complements of up to three sets, with a Venn diagram.

• Relation calculator

  Test a relation for the four properties and build its reflexive, symmetric and transitive closures.

• Graphing calculator

  Plot up to four functions of x and find their zeros and intersections.

• Discrete math calculator

  Solve linear recurrences, count with inclusion–exclusion and apply the pigeonhole principle.

• Permutation and combination calculator

  nPr, nCr, with-repetition counts, letter arrangements and circular permutations, exactly.

• Graph theory calculator

  Degrees, BFS and DFS, Dijkstra, Kruskal’s MST, Euler paths and a bipartite check for any edge list.

• Boolean algebra calculator

  Truth tables, canonical SOP and POS, and Quine–McCluskey minimisation of a Boolean expression.

Math: applied science and engineering

• Physics calculator

  SUVAT from any three values, F = ma, energy, power, momentum, projectile range and height, density, pressure and wave speed.

• Engineering calculator

  Ohm's law, resistor and capacitor networks, voltage dividers, stress and strain, torque, gear ratios and RC time constants.

• Financial mathematics calculator

  Present and future values, annuities and perpetuities, effective rates, and the NPV, IRR and payback of cash flows.

• Data analysis calculator

  Raw data to a frequency table: classes by width or Sturges, relative and cumulative frequency, histogram, ogives and grouped averages.

• Chemistry calculator

  Molar mass from a formula, moles and particles, molarity, dilution, the ideal gas law, pH and percent composition.

• Biostatistics calculator

  From a 2×2 table: sensitivity, specificity, predictive values, likelihood ratios, relative risk, odds ratio and NNT with CIs.

• Econometrics calculator

  Price, income and cross elasticity, the GDP deflator, CAGR, the Durbin–Watson statistic and simple OLS with standard errors.

Finance: saving, investing, tax and pay

• Compound interest calculator

  A = P(1 + r/n)^(nt) with regular deposits, or solve for the rate, time or principal.

• Investment calculator

  End balance from a lump sum and regular contributions, or the contribution a target needs.

• Retirement calculator

  Savings at retirement in future and today’s money, how long they last, and any shortfall.

• Inflation calculator

  Future cost and past value of money at an average inflation rate, or the rate between two prices.

• Finance calculator (TVM)

  Solve N, I/Y, PV, PMT or FV like a financial calculator, with a cash-flow timeline.

• Sharpe ratio calculator

  Sharpe ratio from a return series or from mean, SD and the risk-free rate, annualized.

• Stock beta calculator

  Beta, alpha and R² from paired stock and market prices or returns.

• Volatility calculator

  Historical volatility from prices: log returns, SD and the √252 rule.

Converters

Conversions the calculators above already do, each on the page that explains it.

• Variance ↔ standard deviation

  Square the SD for the variance, or take the root of a variance to get the SD back.

• Standard error → standard deviation

  Recover the SD behind a published standard error: s = SE × √n.

• Standard deviations → percentile

  How far above or below the mean a value sits, as a percentile of a normal distribution.

• Percentile ↔ value

  The value at any percentile of your data, or the percentile rank of a value.

• SD → coefficient of variation

  Express the spread as a ratio or percentage of the mean.

Learn: reference and guides

• Learn hub

  Everything below in a suggested order, from first principles to software.

• Formulas

  Every formula the calculators use, with a link to each one.

• FAQ

  Short answers to the questions people ask most about standard deviation.

• Quiz

  Fifteen questions with explanations, to check you have it.

• Worked examples

  Ten real-world problems worked in full.

• Practice problems

  Random problems on seven topics, marked instantly.

• Worksheets

  Four printable worksheets with answer keys, and a generator for new practice sets.

• Interpreting your result

  Is your SD high? Check it against the mean, the range and a benchmark.

• Use cases

  Where standard deviation is used, from classrooms to factory floors.

• What is standard deviation?

  What the number means, its units, and how to read high, low and zero.

• How to calculate it

  Six steps on real numbers, both formulas worked in full, and practice problems.

• SD vs variance

  The square root that separates them, and when each one is the right report.

• SD vs mean

  Centre and spread, mean ± SD vs ± SE, and what outliers do to each.

• SD and the normal curve

  How σ sets the width of the bell, and how to check data for normality.

• SD in probability

  The standard deviation of a random variable: dice, sums, averages, named distributions.

• The formula

  Both formulas taken apart term by term, and why each step is there.

• Symbols and notation

  σ, s, μ, x̄, Σ: what each means and how to type it.

• Sample vs population

  Which denominator to use, and how much it changes the answer.

• Standard deviation in Excel

  STDEV.S, STDEV.P, conditional SD, and the errors that catch people.

• Standard deviation on a TI-84

  Keystrokes, Sx versus σx, and frequency lists.

• Standard deviation on a Casio

  fx-991CW, fx-991EX, ES PLUS and MS keystrokes, and which result is sx or σx.

• Standard deviation in Python

  NumPy, pandas and the statistics module, and the ddof trap.

• Standard deviation in R

  sd(), var(), population SD, NA handling and by-group.

• SD in psychology

  IQ and T-score scales, APA reporting and Cohen’s d.