List Statistics

How to use?

Type your numbers and separate them by line, comma or space. The results are worked out on their own. The mode shows you the value (or values) that repeat most often in the list.

Central tendency and dispersion

Three classic measures of central tendency: the arithmetic mean μ = Σxᵢ/n, the median (the middle value once the list is sorted, or the average of the two middle values when n is even), and the mode (the most frequent value — a list may be unimodal, bimodal, or have no mode). For dispersion, the population variance is σ² = Σ(xᵢ − μ)²/n and the sample variance s² = Σ(xᵢ − x̄)²/(n − 1) (Bessel's correction); the standard deviation is the square root of the variance and shares the unit of the original data. Example with {2, 4, 4, 4, 5, 5, 7, 9}: mean = 5, median = 4.5, mode = 4, sample std ≈ 2.14. Quartiles split the sorted data into four parts; IQR = Q3 − Q1, and points beyond Q1 − 1.5·IQR or Q3 + 1.5·IQR are flagged as outliers. For an approximately normal distribution, the empirical rule says 68% of data falls within 1σ, 95% within 2σ and 99.7% within 3σ of the mean.

Applications

Descriptive statistics underpin demographic indicators (IBGE/PNAD censuses and household surveys), psychometrics (IQ tests scaled to mean 100 and standard deviation 15), industrial quality control (Shewhart and Six Sigma control charts use μ ± 3σ as action limits), large-scale exams (ENEM applies Item Response Theory on top of basic descriptives), and product analytics in A/B testing, where comparing means with confidence intervals depends on the standard deviation of each group.

FAQ

When should I prefer the median over the mean? When the data has outliers or is skewed — the classic example: if Bill Gates walks into a bar, the average income soars while the median barely moves. Income, wealth and home prices typically report the median.

Population or sample standard deviation? Use the population formula (divide by n) when the data is the entire population; use the sample formula (divide by n − 1) when you're estimating a population from a sample — the −1 corrects for the bias of using x̄ in place of the unknown μ.

Can a list have more than one mode? Yes — when two or more values tie at the highest frequency, the distribution is multimodal. Some conventions report no mode in that case, others list all of them.

Why divide by n − 1 specifically? Because using x̄ instead of μ already "uses up" one degree of freedom; dividing by n − 1 yields an unbiased estimator of σ².