Statistics calculators on this page split into two jobs that get confused for each other constantly. Descriptive statistics — standard deviation, coefficient of variation, interquartile range, percentiles — summarise data you already have in full. Inferential statistics — confidence intervals, t-tests, z-tests, p-values, sample size — draw a conclusion about a larger population from a sample you can only partly see, and every one of them carries some stated degree of uncertainty about being wrong. Knowing which job a given calculator is doing is the first thing to get straight before reading anything it returns.
Calculators in this section
Population versus sample: why n − 1 shows up everywhere
The standard deviation calculator has two modes because there are genuinely two different calculations hiding behind one name. If your data is the entire population, you divide the sum of squared deviations by n. If your data is a sample meant to represent a larger population, you divide by n − 1 instead — Bessel's correction, which exists because a sample's own mean is calculated from the same data used to measure its spread, making the sample slightly less variable than the population it was drawn from unless the correction is applied. Standard error then takes that sample standard deviation and divides it by the square root of the sample size, which is why standard error shrinks as a sample grows even though the underlying standard deviation does not.
What a p-value actually claims
A p-value answers one specific question: assuming the null hypothesis is true, how likely is data at least this extreme? It is not the probability that the null hypothesis is true, and it is not the probability the result happened by chance — both are common misreadings of the same number. The conventional 0.05 threshold traces back to Ronald Fisher's work in the 1920s and is a widely adopted convention rather than a law of statistics; a result can clear that threshold on a large enough sample while representing an effect too small to matter in practice, which is why a p-value from the statistical significance calculator is worth reading alongside the actual size of the effect, not on its own.
Picking between a t-test, a z-test and chi-square
These three hypothesis tests answer related but distinct questions. A z-test assumes the population standard deviation is known or the sample is large enough that it barely matters. A t-test, built on work published in 1908 by William Sealy Gosset under the pen name "Student" while working at Guinness, handles the far more common situation of a small sample with an unknown population standard deviation, using a distribution with fatter tails to account for that extra uncertainty. Chi-square tests work on categorical counts rather than measured values — whether the distribution of observed categories differs from what would be expected — which is why it is the right tool for a 2×2 table of counts and the wrong one for comparing two averages.
Correlation, covariance and R² are the same idea at three scales
Covariance measures whether two variables move together, but its size depends on the units of both variables, which makes it useless for comparing one pair of variables against another. Correlation is covariance rescaled to sit between −1 and 1 regardless of units, which is what makes it comparable across completely different kinds of data. R², which turns up in regression output, is simply the correlation coefficient squared in a simple linear regression — it reframes the same relationship as the proportion of variance in one variable explained by the other, which is a more directly useful number when the question is how much of the picture a single input actually accounts for.