Statistics Calculators

ANOVA Calculator Run a one-way ANOVA to compare the means of three or more groups, with the F statistic, p-value, ANOVA table and eta squared.
Binomial Distribution Calculator Find binomial probabilities — exactly k successes, at most, at least, fewer or more — for n trials with a fixed chance of success.
Chi-Square Calculator Run a chi-square goodness-of-fit test or a test of independence on a contingency table, with expected counts, p-value and Cramér's V.
Coefficient of Variation Calculator Find the coefficient of variation — the standard deviation as a percentage of the mean — from a data list or a known mean and SD.
Combination Calculator Count combinations — the ways to choose r items from n when order does not matter — with or without repetition, as exact numbers.
Confidence Interval Calculator Find a confidence interval for a mean, from data or summary figures with the t or z value, or for a proportion with the Wilson interval.
Correlation Calculator Find the Pearson correlation coefficient r between two sets of paired data, with r², a significance test and Spearman's rank correlation.
Covariance Calculator Find the sample or population covariance between two sets of paired data, with the direction of the relationship and the correlation.
Five Number Summary Calculator Find the five-number summary of a data set — minimum, Q1, median, Q3 and maximum — with the box plot, IQR and outlier fences.
Geometric Mean Calculator Find the geometric mean of positive numbers, or the true average growth rate of percentage returns, with the working shown.
Harmonic Mean Calculator Find the harmonic mean of positive numbers — the right average for speeds and rates over equal amounts — compared with the other means.
Interquartile Range Calculator Find the interquartile range (IQR) of a data set with Q1, the median and Q3, the 1.5 × IQR outlier fences and any outliers.
Linear Regression Calculator Fit a least-squares regression line ŷ = a + bx to paired data, with the slope, intercept, R², slope test and predictions.
Margin of Error Calculator Find the margin of error for a survey percentage or a sample mean at any confidence level, with an optional finite population correction.
Mean Median Mode Calculator Find the mean, median and mode of a list of numbers with each step shown, a frequency count and what the gap between mean and median shows.
Normal Distribution Calculator Find probabilities under a normal curve — below, above or between values — or the value at any percentile, for any mean and SD.
Odds Calculator Convert odds to probability and back, and between fractional, decimal and American odds, with the fair payout on a stake.
Odds Ratio Calculator Find the odds ratio from a 2 × 2 table of exposure and outcome, with its confidence interval, z test and p-value.
Outlier Calculator Find outliers in a data set with the 1.5 × IQR rule or a z-score threshold, and see how the mean and median change without them.
P-Value Calculator Find the p-value from a z score, a t statistic with degrees of freedom, or a chi-square statistic, one- or two-tailed, with the decision.
Percentile Calculator Find the value at any percentile of a data set, or the percentile rank of a value, with the position and interpolation shown.
Permutation Calculator Count permutations — the ways to arrange r items from n when order matters — with or without repetition, as exact numbers.
Poisson Distribution Calculator Find Poisson probabilities — exactly k events, at most, at least, fewer or more — from an average rate per interval.
Probability Calculator Find the probability of an event, of two independent events together or of either, and of something happening at least once in repeated tries.
Quartile Calculator Find the first, second and third quartiles of a data set by the textbook or spreadsheet method, with the halves, positions and IQR.
R-Squared Calculator Find R², the coefficient of determination, from paired data, from a correlation, or from sums of squares, with adjusted R².
Range Calculator Find the range of a list of numbers — the gap between the smallest and largest values — with the minimum, maximum and midrange.
Relative Risk Calculator Find the relative risk from a 2 × 2 table, with its confidence interval, the absolute risk difference and the number needed to treat or harm.
RMS Calculator Calculate the root mean square of a list of numbers, or the RMS value of a sine, square, triangle or sawtooth wave from its peak.
Sample Mean Calculator Find the sample mean x̄ of your data with its standard deviation, standard error and a confidence interval for the population mean.
Sample Size Calculator Find how many people to survey for a given margin of error and confidence level, for a percentage or a mean, with a finite population option.
Standard Deviation Calculator Find the sample or population standard deviation of a list of numbers, with the mean, variance and every step of the working.
Standard Error Calculator Find the standard error of a mean from data or from a standard deviation and sample size, or the standard error of a proportion.
T-Test Calculator Run a one-sample, two-sample (Welch) or paired t-test, with the t statistic, degrees of freedom, p-value and a confidence interval.
Variance Calculator Find the sample or population variance of a list of numbers, with a table of deviations and squared deviations and the standard deviation.
Weighted Average Calculator Find the weighted average of a list of values from their weights, such as grades by credit or percentage, with the working shown.
Z-Score Calculator Find the z-score of a value from the mean and standard deviation, or the value for a z-score, with the normal percentile and tail areas.

A Guide to Statistics Calculators

Statistics is the branch of mathematics where the right answer to the wrong question does the most damage. A p-value computed correctly on a sample that was not random, a correlation reported as though it showed cause, a mean quoted for a distribution that has no meaningful centre: in each case the arithmetic is sound and the conclusion is worthless.

The thirty-seven calculators here all show their working, and several of them refuse to proceed when an assumption they depend on is not met, which is more useful than a number. This guide explains what each group is for and which assumption each one is resting on.

Central tendency

Four calculators, because "average" is four different things. The Mean, Median and Mode Calculator gives all three at once, which is the quickest way to see a skewed distribution: when the mean sits well away from the median, the mean is being dragged by the tail and the median is the better summary.

The Weighted Average Calculator is for when observations do not count equally, which is more often than people assume. The Geometric Mean Calculator is the correct average for multiplicative quantities such as growth rates: three years of 10, 20 and minus 10 percent do not average to 6.67 percent of compound growth. The Harmonic Mean Calculator is the right one for rates over a fixed distance, which is why averaging two speeds arithmetically gives the wrong journey time.

Spread

Six calculators. The Range Calculator is the crudest measure and the most sensitive to a single bad value. Variance and Standard Deviation are the standard pair, and both ask whether your data is a sample or the whole population, because the divisor differs and the distinction matters most on the small datasets people actually paste in.

The Standard Error Calculator answers a different question that is constantly confused with standard deviation: the standard deviation describes the spread of the data, while the standard error describes the precision of your estimate of the mean. More data does not reduce the first and does reduce the second.

The Interquartile Range Calculator gives a spread measure that ignores the tails, and the Coefficient of Variation Calculator expresses spread relative to the mean, which is how you compare variability between quantities measured in different units.

Position and outliers

Five calculators for locating a value within a distribution. The Percentile Calculator and Quartile Calculator give position directly, the Five Number Summary Calculator gives the whole shape in one go, and the Z-Score Calculator expresses a value in standard deviations from the mean so that observations from different distributions can be compared.

The Outlier Calculator applies the standard rules for flagging extreme values. It flags them rather than removing them, which is the correct behaviour: an outlier may be an error or may be the most important observation in the set, and that is a judgement about the data, not about the arithmetic.

Relationships between variables

Four calculators, and the group where the most misreading happens. The Correlation Calculator gives the strength and direction of a linear relationship, and the Covariance Calculator the unstandardised version of the same thing.

The Linear Regression Calculator fits the line, and the R-Squared Calculator reports how much of the variation that line accounts for. A high r-squared means the line fits, nothing more. It does not mean one variable causes the other, it does not mean the relationship holds outside the range of the data, and it does not mean a line was the right shape to fit in the first place.

Probability and counting

Six calculators. The Probability Calculator handles the standard combinations of events, and the Odds Calculator converts between odds and probability, which are two ways of saying the same thing and are regularly mixed up: odds of 3 to 1 are a probability of 0.25, not 0.33.

Permutation and Combination calculators count arrangements, the difference being whether order matters. Choosing a committee is a combination; choosing a chair, secretary and treasurer is a permutation.

Two more are specific to research. The Odds Ratio Calculator and Relative Risk Calculator both compare two groups and are routinely reported as though interchangeable. They are not: an odds ratio exaggerates the apparent effect relative to a relative risk, and the gap widens as the outcome becomes more common. Both calculators show the other figure alongside for exactly that reason.

Distributions

Three calculators for the distributions that cover most practical cases. The Normal Distribution Calculator handles the continuous bell curve, the Binomial Distribution Calculator counts successes in a fixed number of trials, and the Poisson Distribution Calculator covers events in a fixed interval when the rate is known.

Choosing between them is the actual skill. The binomial needs a fixed number of independent trials with a constant probability. The Poisson needs a known average rate and independence between occurrences. Each calculator states its assumptions, because a distribution applied to data that violates them produces a precise and meaningless answer.

Inference and hypothesis testing

Seven calculators, and the group that most rewards reading the page as well as the result.

The P-Value Calculator and T-Test Calculator cover comparisons of means, the Chi-Square Calculator comparisons of counts and categories, and the ANOVA Calculator comparisons across more than two groups at once, which is what it exists for: running repeated two-group tests instead inflates the chance of a false positive.

For study design rather than analysis, the Sample Size Calculator tells you how much data you need before you collect it, and the Margin of Error Calculator and Confidence Interval Calculator express the precision of a result afterwards.

A confidence interval is the most useful of these and the most misdescribed. A 95 percent interval does not mean there is a 95 percent chance the true value lies inside this particular interval. It means that a procedure of this kind captures the true value 95 percent of the time over many repetitions. The distinction sounds academic and changes how much weight a single study deserves.

Two more

The Sample Mean Calculator handles the mean of a sample together with its standard error, and the RMS Calculator gives the root mean square, which is the meaningful average for a quantity that alternates in sign, such as an electrical waveform, where the ordinary mean is zero and tells you nothing.

Which calculator for which question

A rough route through the set. If you have one batch of numbers and want to describe it, start with the mean, median and mode together, then a five number summary for the shape. If you want to describe how spread out it is, use standard deviation for the whole picture or the interquartile range when the tails are untrustworthy.

If you have two variables and want to know whether they move together, use correlation, then regression if you want the line and r-squared to judge the fit. If you have two groups and want to know whether a difference is real, use a t-test for means and chi-square for counts, and ANOVA once there are three groups or more.

If you are planning rather than analysing, the sample size calculator comes first, before any data exists. That is the cheapest statistical intervention available, and the one most often skipped.

Before you report a result

Three checks are worth more than any calculator here.

Ask where the data came from. No amount of correct arithmetic repairs a sample that was not representative, and this is the failure that produces most wrong published conclusions.

Ask whether the test matches the design. A paired comparison analysed as unpaired throws away information; two groups compared with separate tests rather than ANOVA inflates the error rate.

Ask what the effect size is, not only whether it is significant. With a large enough sample almost any difference becomes statistically significant, including differences too small to matter. Significance says a difference is probably real. It says nothing about whether it is worth acting on, and that second question is usually the one you actually had.