Statistics & Probability Calculators
Distributions, inference and correlation
Describing a sample, then inferring from it
Statistics splits cleanly into two jobs, and half the confusion in the subject comes from mixing them. Describing is the easy half: the statistics calculator returns mean, median, mode, range, variance and standard deviation for a data set you already have in full, and those numbers are facts about that data with no uncertainty attached. The outlier calculator belongs here too, flagging points beyond 1.5 interquartile ranges from the quartiles by Tukey’s rule.
Inferring is the hard half. Once your data is a sample standing in for a population you cannot see, every statement carries a margin. The confidence interval calculator gives the range a population mean plausibly occupies, using the t-distribution rather than the normal when the sample is small — which is the correction most spreadsheet formulas quietly skip.
Distributions, tests and survey design
The distribution tools answer probability questions directly. The normal distribution calculator gives the area under a bell curve below, above or between two values; the binomial calculator handles a fixed number of independent trials with a constant success rate; the z-score calculator converts a raw value into standard deviations from the mean and the percentile that implies.
For testing, the p-value calculator converts a z or t statistic into a probability, one-tailed or two-tailed. A p-value is the chance of seeing a result at least this extreme if nothing were going on — it is not the chance that your hypothesis is wrong, and treating it as such is the most consequential misreading in applied statistics.
Survey work runs the other way round. Sample size and margin of error are two views of one relationship: fix the precision you need and the sample size calculator tells you how many responses that takes; fix the sample you already have and the margin of error calculator tells you the ± figure you are entitled to quote. Correlation closes the loop, giving Pearson’s r, the least-squares line, and how much of the variance that line actually explains.
Statistics & Probability calculators — common questions
When should I use the t-distribution instead of the normal?
When you are estimating a population mean from a sample and do not know the population standard deviation — which is almost always. The t-distribution has heavier tails that widen the interval to account for that extra uncertainty, and the effect matters below about thirty observations. The confidence interval calculator selects it automatically based on your sample size.
Does a correlation mean one thing caused the other?
No. Pearson’s r measures how closely two variables move together in a straight line and says nothing about direction or cause. A strong correlation is consistent with causation in either direction, with both being driven by a third factor, or with coincidence. The r² figure the calculator reports is the honest summary: the share of variance explained, not the share of outcome caused.
What does a p-value of 0.05 actually mean?
That a result at least this extreme would occur 5% of the time if the null hypothesis were true. It is a statement about the data given the hypothesis, not about the hypothesis given the data. It also says nothing about effect size — a tiny, useless difference will produce a very small p-value given a large enough sample.
How many people do I need to survey?
That depends on the margin of error you can live with and the confidence level you want, not on how large the population is — beyond a few thousand, population size barely moves the answer. The sample size calculator takes both and returns the count; around 385 responses gives ±5% at 95% confidence for any large population.