Statistics & Probability Calculators

What these statistics and probability calculators cover

Statistics answers two kinds of question. The first is descriptive: given a list of numbers, what is typical and how spread out are they? The second is inferential: given a sample, what can you honestly say about the larger group it came from, and how sure can you be? Probability underpins both, and adds questions of its own about chance and counting. The eleven calculators on this page cover the full path from pasting in raw data to reading off a p-value, and every one shows its working, so the method travels with the answer.

For describing data, start with the Average Calculator, the Mean, Median, Mode, Range Calculator, the Standard Deviation Calculator and the Statistics Calculator. For inference, the Z-Score Calculator, Confidence Interval Calculator, Sample Size Calculator and P-Value Calculator form a natural sequence. For chance and counting, use the Probability Calculator, the Permutation and Combination Calculator and the Random Number Generator.

Describing a data set: centre and spread

The mean, the median and the mode are three different claims about what is typical. The mean adds everything and divides by the count; the median is the middle value once the list is sorted; the mode is the value that appears most often. They disagree whenever data is skewed: a few very large salaries pull the mean up while the median barely moves, which is why house prices and incomes are usually reported as medians. The Mean, Median, Mode, Range Calculator works all four measures (range included) on your own figures, with the sorted list, a frequency table and a chart, so you can see why the numbers differ. The Average Calculator covers the same ground and adds a weighted average, which is what you need when some values count for more, such as coursework worth 40% and an exam worth 60%.

Centre is only half the description; two data sets can share a mean and behave nothing alike. The Standard Deviation Calculator measures spread as, roughly, the typical distance of a value from the mean, working through the variance step by step. It includes the toggle that trips up more students than any other: for a whole population you divide by n, but for a sample you divide by n - 1, a correction that compensates for the sample mean sitting closer to its own data than the true mean does. When you want the full picture in one pass, the Statistics Calculator returns the five-number summary, quartiles, interquartile range, variance and standard deviation, plus an honest outlier check using the IQR rather than a hunch.

The normal curve and z-scores

Many real measurements, heights, test scores, manufacturing tolerances, cluster around a mean in the bell-shaped normal distribution. A z-score places a single value on that curve by asking how many standard deviations it sits from the mean: z = (x - mean) / standard deviation. A z-score of 2 is two standard deviations above average, which is unusual; a z-score of 0 is exactly average. The Z-Score Calculator converts a value into its z-score, percentile and one- or two-tailed probability, and can find the chance of landing between two values, with the relevant region of the curve shown. Z-scores also solve a practical comparison problem: they put results from different scales onto one ruler, so you can say whether a 72 on one test is better than an 81 on another.

From sample to population: intervals, sample sizes and p-values

Surveys and experiments measure samples, but the question is always about the population, and three tools handle that leap. The Confidence Interval Calculator turns a sample mean or proportion into a range that is likely to contain the true population value, using the z or t distribution as the sample size demands, and showing the Wilson interval for proportions alongside the classic Wald result, which behaves badly for small samples or extreme percentages. Be careful with the interpretation: a 95% confidence level describes the reliability of the method, meaning that 95% of intervals built this way capture the truth; it is not a 95% probability that the truth sits in your particular interval.

Planning runs the same logic backwards. The Sample Size Calculator tells you how many people to survey for a target margin of error, or what margin of error an existing sample achieves, using Cochran's formula with the finite population correction for small groups. The honest headline is that precision is expensive: because the margin of error shrinks with the square root of the sample size, halving it needs roughly four times the respondents.

Hypothesis tests finish the story. The P-Value Calculator converts a z, t or chi-square test statistic into a p-value, shades the rejection region and explains it in words, with critical values at the 0.05 and 0.01 levels alongside. A p-value is the probability of seeing data at least this extreme if the null hypothesis were true. It is not the probability the null hypothesis is true, and a small p-value does not mean the effect is large or important; those are the two misreadings behind most bad statistics headlines.

Chance and counting

Probability puts a number between 0 and 1 on how likely something is. The Probability Calculator expresses a single event's probability as an exact fraction, decimal, percentage and odds, and combines two independent events for the standard cases: both happening, at least one happening, neither, and exactly one, with the working at every step. The core rules are worth having by heart: for independent events, P(A and B) = P(A) × P(B), and in general P(A or B) = P(A) + P(B) - P(A and B), where forgetting the subtraction double-counts the overlap. The other classic error is the gambler's fallacy; a coin that has landed heads five times running is no more likely to land tails next, because independent events have no memory.

Many probability problems reduce to counting arrangements. The Permutation and Combination Calculator computes both nPr and nCr from the same n and r: permutations count selections where order matters (podium places), combinations where it does not (lottery numbers). Since nPr = nCr × r!, permutations always outnumber combinations for the same inputs, and the tool shows that r! link, charts the relevant row of Pascal's triangle, and returns exact results up to n = 1,000 rather than rounded approximations. When you need randomness rather than a probability, the Random Number Generator draws numbers in any range, unique or with repeats, using genuine cryptographic randomness with a rejection-sampling step to keep every value equally likely; the Dice Roller does the same job in dice form.

Every calculator on this page is free, and most of them chain together naturally: describe your data with the Statistics Calculator, standardise a value with the Z-Score Calculator, then judge significance with the P-Value Calculator.

Frequently asked questions

Should I use the mean or the median?

Use the median when the data is skewed or contains outliers, incomes, house prices, waiting times, because a handful of extreme values distort the mean. Use the mean when data is roughly symmetrical or when later calculations, such as standard deviation, need it. The Mean, Median, Mode, Range Calculator shows both so you can compare them on your own data.

When do I divide by n and when by n - 1?

Divide by n when your data is the entire population you care about. Divide by n - 1 when it is a sample standing in for a larger group, which is the usual case. The Standard Deviation Calculator has a toggle for each and shows how the choice changes the answer.

What is the difference between a permutation and a combination?

Order. ABC and CBA are two different permutations but the same combination. If swapping the order creates a genuinely different outcome, count permutations; otherwise count combinations. The Permutation and Combination Calculator reports both from the same inputs.

What does a 95% confidence interval actually mean?

It means the procedure works 95% of the time: if you repeated the sampling many times and built an interval each time, about 95% of those intervals would contain the true population value. The Confidence Interval Calculator builds the interval and charts it, and the Sample Size Calculator tells you how many responses you need to narrow it.