BYTETOOLS

Normal Distribution Calculator

Compute normal distribution probabilities for any mean and standard deviation: P(X<x), P(X>x), P(a<X<b), the PDF, plus an inverse quantile mode.

0.975002
P(X < 1.96)
1.96
z-score(s)
97.5002%
As a percentage
0.058441
PDF f(x)

Bell curve — shaded area is the probability

CDF via the Abramowitz–Stegun erf approximation (max error ≈ 1.5 × 10⁻⁷); inverse via Acklam's rational approximation. Both exceed printed z-table precision.

What is the Normal Distribution Calculator?

The ByteTools Normal Distribution Calculator evaluates the Gaussian bell curve for any mean μ and standard deviation σ.

  • P(X < x), P(X > x) and P(a < X < b) for any μ and σ
  • Inverse mode: find the x value for a given probability
  • PDF value and z-score reported alongside each answer
  • Canvas-drawn bell curve with the probability region shaded
  • erf approximation accurate to about 7 decimal places
  • 100% private — runs entirely in your browser

How to use the Normal Distribution Calculator

  1. 1

    Enter the mean μ and standard deviation σ of your distribution.

  2. 2

    Choose a mode: P(X < x), P(X > x), P(a < X < b) or inverse (find x from p).

  3. 3

    Enter your x value(s) or the target probability.

  4. 4

    Read the probability, the z-score(s) and the shaded curve.

  5. 5

    Copy the results with one click.

About the Normal Distribution Calculator

The ByteTools Normal Distribution Calculator evaluates the Gaussian bell curve for any mean μ and standard deviation σ. It returns the probability density f(x), the cumulative probability P(X < x), the upper tail P(X > x) and the between-bounds probability P(a < X < b), along with the equivalent z-scores.

An inverse mode answers the reverse question: given a cumulative probability p, what x value cuts off that area? The CDF uses the Abramowitz–Stegun rational approximation of the error function (accurate to about 1.5 × 10⁻⁷), and the inverse uses the Acklam / Beasley-Springer-Moro rational approximation refined for practical accuracy — the same approach used in statistical software.

A canvas-drawn bell curve shades the probability region you asked for, so the answer is visual as well as numeric. Everything is computed 100% locally in your browser; no values are uploaded or stored.

Frequently asked questions

How do you find probability from a normal distribution?

Standardise your value to a z-score with z = (x − μ)/σ, then take the area under the standard normal curve. This tool computes that area with a high-accuracy error-function approximation, so P(X < x) for μ = 0, σ = 1, x = 1.96 comes out as 0.9750, matching the tables.

What is the difference between PDF and CDF?

The PDF (probability density function) is the height of the bell curve at a point — not a probability itself for continuous variables. The CDF (cumulative distribution function) is the accumulated area up to that point, which is the actual probability P(X ≤ x).

What is the empirical (68-95-99.7) rule?

For any normal distribution, about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. You can verify each figure with this calculator using the between mode, e.g. P(μ−2σ < X < μ+2σ) ≈ 0.9545.

How accurate is this calculator compared to statistical tables?

The CDF uses the Abramowitz–Stegun formula 7.1.26 with a maximum error near 1.5 × 10⁻⁷, and the inverse CDF uses Acklam's rational approximation, accurate to roughly 1.15 × 10⁻⁹ in relative terms. Both exceed the four-decimal precision of printed z-tables.

What does the inverse normal (quantile) mode do?

It answers questions like 'what score marks the top 5%?'. Enter the cumulative probability (0.95) and the tool returns the x with that much area to its left. For μ = 100, σ = 15 and p = 0.95, x ≈ 124.7.

Can I use this for values that are not standard normal?

Yes. Enter any mean and standard deviation — the tool standardises internally and converts back, so you work directly in your own units (test scores, heights, measurement errors) without doing the z-transformation yourself.

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