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.