StatisticsCALCULATOR LAB
METHODS, NOT MYSTERY

How Our Statistics Calculators Work

Simple interfaces, explicit assumptions and traceable formulas. Calculations run locally in your browser; rounding is applied only to displayed results.

Numerical methods & limits

Dataset summaries use a shifted, compensated two-pass calculation, avoiding subtraction of two large squared totals. Sample variance divides by n − 1, while population variance divides by n. Sample quantities are undefined for one observation.

Normal probabilities use regularized incomplete gamma functions, evaluated with a convergent series or continued fraction. Small upper-tail probabilities are computed directly rather than by subtracting a near-one probability. Inverse-normal values use bounded bisection.

Binomial masses use logarithms and a Lanczos log-gamma approximation; selected masses are combined with log-sum-exp. Poisson cumulative and tail probabilities use incomplete gamma functions, with log-space summation for finite intervals. Charts are explanatory and may display only a window of the distribution.

Computations use browser double precision, not arbitrary precision. Extreme values can overflow or underflow. Positive probabilities below the representable range are shown as “below numeric precision,” not as a proven impossibility. Very close interval boundaries may lose precision; verify sensitive calculations independently.

Sources: NIST DLMF computation methods, gamma series, continued fractions, and gamma-function representations.

Z Score

z = (x − μ) / σ; below = Φ(z); above = Φ(−z); two-tailed = 2Φ(−|z|).

  • Standard deviation must be positive.
  • A z-score describes distance from a mean; normal probabilities and percentiles assume a normal model.
  • The two-tailed standard-normal p-value is not a complete hypothesis test.

Reviewed: October 2026

Normal Distribution

z = (x − μ) / σ; P(a ≤ X ≤ b) = Φ(zᵦ) − Φ(zₐ); percentile value = μ + σΦ⁻¹(p).

  • The variable is modeled by a normal distribution.
  • Standard deviation must be positive.
  • For a continuous variable, including or excluding a boundary does not change its probability.

Reviewed: October 2026

Binomial Distribution

P(X = k) = C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ; mean = np; variance = np(1 − p).

  • Trials are independent, with two outcomes each.
  • Success probability is constant across trials.
  • Counts are inclusive for at most, at least and between. Outside excludes both endpoints.
  • Up to 10,000 trials. Charts display a central window for large distributions.

Reviewed: October 2026

Poisson Distribution

P(X = k) = e⁻λ λᵏ / k!; mean = variance = λ; SD = √λ.

  • Events occur independently at a constant average rate.
  • The rate and observed count refer to the same interval.
  • Counts are inclusive for at most, at least and between. Outside excludes both endpoints.
  • Rate from 0 to 10,000; event counts up to 100,000. The chart may show only a central window.

Reviewed: October 2026

Standard Deviation

Population SD = √[Σ(xᵢ − x̄)² / n]; sample SD = √[Σ(xᵢ − x̄)² / (n − 1)].

  • Every entered observation has equal weight; repeat notation expands observations.
  • Sample standard deviation needs at least two values.
  • Standard deviation describes spread in the original units; it does not prove a normal distribution.

Reviewed: October 2026

Variance

Population variance = Σ(xᵢ − x̄)² / n; sample variance = Σ(xᵢ − x̄)² / (n − 1).

  • Observations have equal weight.
  • Sample variance requires at least two observations.
  • Variance is expressed in squared units, not the original measurement units.

Reviewed: October 2026

Descriptive Statistics

Mean = Σxᵢ / n; median = middle sorted value (average of the two middle values for even n); range = max − min.

  • Each observation has equal weight.
  • All tied most-frequent values are reported as modes; no mode is reported when every value occurs once.
  • Sample variance and SD require at least two observations.

Reviewed: October 2026

Probability Distribution

E(X) = ΣxP(x); variance = ΣP(x)[x − E(X)]²; F(t) = Σ P(x) for x ≤ t.

  • This tool is for discrete outcomes, not a continuous density function.
  • Probabilities must be nonnegative and total 1 (or 100%) within 1 × 10⁻¹⁰ in decimal form.
  • Inputs are never silently normalized. Duplicate outcomes are allowed and their masses are combined for the CDF.

Reviewed: October 2026

Coefficient of Variation

CV (%) = 100 × SD / mean; sample CV uses sample SD; population CV uses population SD.

  • Use ratio-scale measurements with a meaningful zero. Do not use Celsius or Fahrenheit temperatures.
  • This tool uses a positive-mean convention: zero and negative means are rejected, and dataset values must be nonnegative.
  • A mean close to zero makes CV sensitive to small changes. CV is not a confidence interval.
  • “Coefficient of variance” is a common search term for coefficient of variation (relative standard deviation).

Reviewed: October 2026

T Distribution

f(t) = Γ((ν + 1)/2) / [√(νπ) Γ(ν/2)] × (1 + t²/ν)⁻⁽ν⁺¹⁾/²; F(t) = P(T ≤ t); right tail = F(−t); two tails = 2F(−|t|).

  • This calculator uses the central Student's t distribution, symmetric around zero; it does not perform a complete t-test.
  • Degrees of freedom may be fractional. Supported df: 0.1 to 10,000. Enter df directly, not sample size.
  • Probabilities are areas under a continuous density curve; a single point has zero probability. Including an endpoint does not change an interval area.
  • Inverse inputs must be strictly between 0 and 1 (or 0% and 100%). Two-tailed α is the total area in both tails, not the confidence level.
  • Numerical evaluation uses regularized incomplete beta tails and a bounded inverse solver. Inputs outside the supported finite range fail explicitly.

Reviewed: October 2026

Statistical Power

t models: δ = d√n or d/√(1/n₁ + 1/n₂). Paired model uses dz. Proportions: h = 2asin√p₂ − 2asin√p₁. ANOVA: λ = Nf², df₁ = k−1, df₂ = N−k. Power is rejection probability under the specified alternative.

  • One-sample and paired t models assume normal observations or paired differences. Independent means use a pooled-variance t-test with equal population variances, not Welch’s test.
  • Balanced one-way ANOVA assumes independent normal groups, equal variances and fixed effects; no repeated-measures, factorial or clustered design.
  • Two-proportion power uses Cohen’s h normal approximation without continuity correction. Require at least 10 expected events and nonevents in each group. Rare-event and boundary rates need specialist consideration.
  • Signed d and dz describe the anticipated difference. Positive independent effects mean Group 2 minus Group 1. One-sided direction must be prespecified.
  • Supported α: 0.001–0.20; target power: 0.50–0.99. Up to 5,000 observations per group or complete pairs; ANOVA up to 500/group and 3–20 groups. Allocation ratio 0.2–5 for sample-size planning.
  • Results are analyzable sample sizes, not recruitment targets. Dropout, clustering, multiple testing and survival outcomes are not modeled. Power is not the probability that a hypothesis is true.
  • Use a scientifically meaningful anticipated effect. Small/medium/large conventions are rough illustrations, not justification. Observed post-hoc power is not a substitute for confidence intervals.

Reviewed: October 2026

Weighted Mean

x̄w = Σᵢ₌₁ⁿ wᵢxᵢ / Σᵢ₌₁ⁿ wᵢ

  • Each value must have a corresponding weight.
  • Weights must be nonnegative, and at least one weight must be positive.
  • Weights may be counts, proportions, percentages, or other consistent relative weights.
  • Weights do not have to add up to 1 or 100; the calculation normalizes them automatically.
  • Data values may be negative. A zero weight gives a row no influence on the result.
  • The calculator computes a weighted arithmetic mean, not the statistical uncertainty of that mean.

Reviewed: October 2026

Venn Diagram

|A ∪ B| = |A| + |B| − |A ∩ B|

  • A union includes elements in at least one specified set.
  • An intersection contains elements shared by the specified sets.
  • Inclusive pairwise intersections include the triple intersection.
  • Each exclusive Venn region is counted exactly once.
  • Counts must be nonnegative integers.
  • U is needed for the number outside the sets and complements relative to the complete universe.
  • The diagram is schematic, not area-proportional.

Reviewed: October 2026

Bell Curve Maker

f(x) = exp(−(x−μ)²/(2σ²)) / (σ√(2π)); z = (x−μ)/σ; P(a ≤ X ≤ b) = Φ(zᵦ) − Φ(zₐ).

  • Standard deviation must be strictly positive.
  • Fitted curves use sample SD with denominator n−1.
  • A normal fit does not establish that observations are normally distributed.
  • Histogram bar areas sum to 1; the y-axis shows probability density.
  • Central probability intervals are not confidence intervals for an estimated mean.
  • Parameters must have representable variance, density and graph bounds; rescale numerically unresolved inputs.

Reviewed: October 2026

Geometric Distribution

Trials: P(X = k) = (1 − p)^(k−1)p, k ≥ 1. Failures: P(Y = k) = (1 − p)^k p, k ≥ 0.

  • p is the probability of success on each trial.
  • 1−p is the probability of failure.
  • Trials must be independent.
  • p remains constant from trial to trial.
  • The process stops at the first success.
  • The distribution is discrete, with an infinite theoretical upper range.

Reviewed: October 2026

Probability Tree Generator

P(path) = p₁ × p₂ × … × pₙ; P(A | B) = P(A ∩ B) / P(B), for P(B) > 0.

  • Outgoing branches are mutually exclusive and exhaustive.
  • Later branch probabilities are conditional on the complete previous history.
  • Equal numerical probabilities alone do not establish independence.
  • Complete paths are mutually exclusive; selecting a union adds their probabilities.

Reviewed: October 2026

Hypotenuse

c = √(a² + b²)

  • Right triangles only; all lengths and area must be positive.

Reviewed: October 2026