Distributions

The distribution calculators compute probabilities and quantiles under the standard normal distribution (μ = 0, σ = 1) and any general normal distribution N(μ, σ²). Each calculator solves for any variable in its relationship and renders an interactive bell-curve visualization with the relevant region shaded.

Use these to convert between z-scores, percentiles, and probabilities; to find critical values from a target tail area; or to translate domain-specific scores (such as IQ on the Wechsler scale) into standardized z-units and percentile ranks.

When to use these calculators

Pick the Z-Score Calculator when you have a z-score and want the corresponding probability — left tail, right tail, or two-tailed. The Inverse Z-Score Calculator does the opposite: given a target probability or confidence level, it returns the critical z*.

For raw scores from a general normal distribution N(μ, σ²) rather than the standard normal, the Normal Distribution Probability Calculator standardizes inputs internally (z = (x − μ)/σ) and accepts any combination of one or two cutoffs.

The Z-Score to Percentile, P-Value, and IQ Score calculators are applied conveniences for the same underlying math: percentile rank from z, p-value from a test statistic, and IQ ↔ percentile ↔ z translations on the Wechsler scale (μ = 100, σ = 15).

Frequently Asked Questions

What's the difference between a z-score and a percentile?

A z-score expresses how many standard deviations a value sits above or below the mean. A percentile rank expresses what fraction of the distribution lies at or below the value. They convey the same positional information in different units — a z of 1.0 corresponds to roughly the 84th percentile under the standard normal.

When do I use a one-tailed vs two-tailed p-value?

Use a one-tailed p-value when the research hypothesis specifies direction (e.g., 'the new treatment increases scores'). Use a two-tailed p-value when the hypothesis is non-directional ('the new treatment differs from the control'). Two-tailed is the conservative default in most published research.

Why use the standard normal instead of the actual normal distribution?

Any normal distribution N(μ, σ²) can be standardized to N(0, 1) via the transformation z = (x − μ)/σ. This means a single tabulated distribution (the standard normal) serves every normal problem, and software only needs one routine to compute probabilities under it. The Normal Distribution calculator on this site does the standardization for you.

What does the IQ classification on the Wechsler scale mean?

The Wechsler scale (WAIS-IV, WISC-V) anchors IQ at μ = 100 and σ = 15. Classifications group ranges of IQ scores into descriptive labels — Extremely Low, Borderline, Low Average, Average, High Average, Superior, Very Superior. Boundaries derive from the test publisher's standardization, not from arbitrary statistical cutoffs.

Are these calculators sufficient for publication-grade statistics?

For everyday textbook problems and quick sanity checks, yes. For publication, peer-reviewed analyses, or regulatory submissions, verify against your institution's preferred software (R, SAS, SPSS, Stata). The calculators here use Abramowitz & Stegun for the standard normal CDF and Acklam's rational approximation for the inverse CDF — accuracy is on the order of 1e-7, which exceeds what most published reports require.