Statistics calculators

Z-scores, confidence intervals, t-tests, p-values, and regression

5 Number Summary CalculatorCompute the five-number summary (min, Q1, median, Q3, max) and IQR of a dataset for box-plot construction.A/B Test Sample Size CalculatorCalculate the required sample size per variant for an A/B test from baseline conversion rate, minimum detectable effect, significance level, and powerAverage Rating CalculatorCompute the average star rating (1-5) from individual counts, plus a Bayesian-smoothed estimate that resists small-sample skew.Bayes' Theorem CalculatorCompute the posterior probability from a prior, sensitivity, and false-positive rateBinomial Distribution CalculatorCalculate exact and cumulative binomial probabilities with mean, variance, and standard deviationChi-Square CalculatorRun a chi-square goodness-of-fit or test of independence with the chi-squared statistic, degrees of freedom, and p-value. Free.Coefficient of Variation CalculatorCompute coefficient of variation (CV = SD/mean) as both a ratio and a percentage, from a data list or from mean and SD.Cohen's d CalculatorCompute Cohen's d (standardized effect size for two-sample mean difference) and Hedges' g with classification.Cohen's d Effect SizeCalculate Cohen's d effect size to measure the practical significance of the difference between two group means. Free, instant.Cohen's Kappa CalculatorCalculate Cohen's kappa for inter-rater agreement from a two-by-two agreement table, with observed agreement and a strength labelCoin and Dice Probability CalculatorCalculate binomial coin-flip probabilities and dice-sum probabilities for any number of fair coins or standard dice. Free, instant.Confidence Interval CalculatorCalculate confidence intervals for a population mean at any confidence level using the z or t distribution. Free, instant results.Confidence Interval for a Proportion CalculatorCompute Wald and Wilson score confidence intervals for a sample proportion at any confidence level. Free, instant statistics tool.Critical Value CalculatorFind critical values for the z and t distributions at any α level (one- or two-tailed) for hypothesis tests and confidence intervals.Degrees of Freedom CalculatorCompute degrees of freedom for one-sample t, two-sample t, F-test, ANOVA, chi-square goodness-of-fit, and contingency tables.Dice Average CalculatorCompute the expected sum, variance, SD, min, and max of rolling n dice with s sides plus an optional modifier - useful for D&D and tabletop games.Expected Value CalculatorCompute the expected value E[X] = Σ x·P(x) of a discrete probability distribution, plus variance and standard deviation.F-Statistic CalculatorCompute the F statistic = s₁² / s₂² for comparing two sample variances, with df₁, df₂ and a two-sided p-value.Geometric Mean CalculatorCalculate the geometric mean of a list of positive numbers along with the arithmetic mean and the count of valuesHarmonic Mean CalculatorCalculate the harmonic mean of a list of positive numbers, the right average for rates and ratios such as average speedHypothesis Testing CalculatorOne-proportion z-test: compare a sample proportion to a hypothesized population proportion with z, p-value, and decision.Linear Regression CalculatorFit y = mx + b by ordinary least squares from paired data. Returns slope, intercept, correlation r, R², and standard error of slope.Mann-Whitney U Test CalculatorNonparametric test for two independent samples. Compute the U statistic, normal-approximation z, and two-sided p-value without assuming normality.Margin of Error CalculatorCalculate the margin of error for a survey proportion at any confidence level from sample size and observed proportion. Free, instant.McNemar's Test CalculatorCompare paired proportions on a 2x2 table of matched pairs. Returns the chi-square statistic, exact-style continuity correction, and p-value.Normal Distribution CalculatorFind the probability between bounds, tail areas, and inverse values for any normal distributionNormal Probability Sampling CalculatorFind the probability that a sample mean falls in a given range using the sampling distribution of the mean (CLT) for a normal population.Number Needed to Treat CalculatorCalculate the number needed to treat, absolute risk reduction, and relative risk reduction from control and treatment event ratesOdds Ratio & Relative Risk CalculatorCalculate the odds ratio and relative risk from a two-by-two contingency table of exposed and unexposed casesOne-Way ANOVA CalculatorCompare means across three or more groups with a one-way ANOVA table including sum of squares, F statistic, and p-value. Free.P-Value CalculatorFind the p-value from a z, t, chi-square, or F statistic for one- or two-tailed testsPearson Correlation CalculatorCalculate the Pearson r correlation coefficient for two variables, test its significance, and interpret the strength of the relationship. Free.Percentile Rank CalculatorFind the percentile rank of any value in a data set, or find the value at any given percentile using standard interpolation. Free.Permutation with Repetition CalculatorCount arrangements of r items chosen from n distinct items where repetition is allowed: n^r. Useful for license plates, PINs, and passwords.Poisson Distribution CalculatorCalculate exact, cumulative, and complementary Poisson distribution probabilities for any event rate and count. Free, instant.Power Analysis CalculatorCompute the sample size needed to detect a given effect size (Cohen's d) at any α and power level (one- or two-tailed).Prevalence Rate CalculatorCalculate the prevalence of a condition as a percentage and as a rate per one hundred thousand from existing cases and populationRange CalculatorCompute the range (max − min) of a list of numbers, plus minimum, maximum, and count.Sample Size CalculatorDetermine the required sample size for a survey or experiment at a given margin of error and confidence level. Free, instant.Scatter Plot CalculatorSummarize paired data: means, ranges, covariance, Pearson correlation, and an ASCII scatter to eyeball the relationship.Standard Error of the Mean CalculatorCalculate the standard error of the mean and the 95 percent margin of error from a sample standard deviation and sizet-Statistic CalculatorCompute the one-sample t-statistic, degrees of freedom, standard error, and p-value from sample mean, hypothesized mean, sample SD, and n.T-Test CalculatorRun one-sample, two-sample, or paired t-tests with t, df, p-value, and a decisiont-Test vs Z-Test Comparison CalculatorDecision tool: pick between z-test, t-test, and nonparametric alternatives based on σ knowledge, sample size n, and normality of the data.Z-Score CalculatorCalculate the standard score and find the percentile rank in a normal distributionZ-Test CalculatorOne-sample z-test for a population mean when σ is known. Returns z, p-value, and reject/fail-to-reject decision.

About Statistics Calculators

Statistics calculators take the formulas behind A/B tests, surveys, and research papers and put them in front of you without the SPSS license fee. Whether you're sizing a study, interpreting a regression output, or just trying to figure out what a z-score actually means, this category covers the inferential and descriptive tools that come up over and over in business, social science, and quality engineering. Every calculator is built on the standard textbook definitions used by Casella & Berger, OpenIntro, and the NIST Engineering Statistics Handbook, so the numbers you get here line up with what your stats professor or peer reviewer expects.

The normal-distribution tools are the busiest. The z-score calculator converts a raw value into standard deviations from the mean and returns the cumulative probability to four decimal places using the same Hastings approximation that Excel's NORM.S.DIST uses. The confidence interval calculator switches automatically between the z-distribution (when σ is known or n ≥ 30) and the t-distribution (with the right degrees of freedom) when you supply a sample standard deviation, which is a step a lot of online calculators get wrong. The margin of error tool handles both proportion and mean intervals, with the finite population correction available when you're sampling more than 5% of a known population.

For experimental design, the sample size calculator works backwards from your target margin of error, expected proportion or standard deviation, and confidence level, the inverse of what most people need but rarely find in a single tool. Cohen's d effect size lets you turn a p-value-significant result into a practically-meaningful one using the conventional small (0.2), medium (0.5), and large (0.8) thresholds Jacob Cohen published in 1988. Pearson's r and the simple linear regression calculator pair naturally: get the correlation, then fit the line and check the R² value before predicting.

Most visitors land here for one of three reasons: a research methods class, a marketing or product analytics question, or a Six Sigma / quality project. The calculators show the formula and the worked numbers underneath each result so you can paste them into a thesis appendix or a deck without losing the audit trail. Critical values come from the same lookup tables (Student's t, z, and chi-squared) that statistical software uses internally, with no rounding to two decimals like a printed table forces. The percentile rank tool is also a quiet favorite for parents interpreting standardized test scores, where understanding that the 84th percentile means roughly +1σ above the mean is more useful than the raw number itself.

When to Use a Statistics Calculator

  • Planning a survey, A/B test, or experiment and need a sample size for a given margin of error
  • Interpreting a z-score, percentile, or confidence interval from a research paper or report
  • Calculating effect size (Cohen's d) to gauge whether a statistically significant result is practically meaningful
  • Fitting a regression line to two-variable data and predicting Y from X
  • Computing Pearson's r to measure the linear association between two variables
  • Translating standardized test scores between percentile, z-score, and raw score formats

Frequently Asked Questions

When should I use a z-score versus a t-score?

Use a z-score when you know the population standard deviation σ, or when your sample size is large (n ≥ 30) and the central limit theorem makes the sampling distribution approximately normal. Use a t-score when σ is unknown and you're estimating it from a small sample (n < 30). The t-distribution is wider, which produces appropriately wider confidence intervals for the extra uncertainty.

What sample size do I need for a 95% confidence level?

For a survey with a 5% margin of error and a 50% expected proportion (the most conservative case), you need roughly 385 respondents. Tighten the margin to 3% and you need about 1,068. Our sample size calculator handles all the standard combinations of confidence level, margin, and expected proportion.

How do I interpret a Cohen's d effect size?

Cohen's 1988 conventions are: 0.2 is small, 0.5 is medium, and 0.8 is large. A d of 1.0 means the two group means differ by one full pooled standard deviation. Effect size is independent of sample size, which is why it's the right metric for judging practical importance, unlike a p-value.

What does R² tell me in a regression?

R² (coefficient of determination) is the fraction of variance in Y explained by X. R² = 0.80 means 80% of the variability in your dependent variable is accounted for by the regression line. Above 0.7 is generally strong for social-science data and above 0.9 for physical-science data.

Are these calculators suitable for academic and published research?

Yes. The formulas and critical values match the standards used by R, SPSS, SAS, and Stata. For peer-reviewed work you should still cite the underlying methods and verify edge cases, but the numbers will reconcile to what statistical software produces.

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