Normal Distribution Calculator

Enter the mean, the standard deviation, and a value x to get the probability density (PDF), the cumulative probability P(X ≤ x), the z-score, and the upper-tail probability P(X > x).

Quick Facts

Method
Exact Gaussian PDF and CDF
CDF uses P(X ≤ x) = ½·[1 + erf((x − μ) / (σ√2))]; erf is evaluated with a high-accuracy numeric approximation (error < 1e-7).

Your Results

Calculated
P(X ≤ x) — CDF
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Area to the left of x
P(X > x) — upper tail
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Area to the right of x
Z-score
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(x − μ) / σ
PDF f(x)
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Curve height at x

Ready

Enter μ, σ, and x, then press Calculate.

The normal distribution formula

The normal (Gaussian) distribution is defined by two parameters: the mean μ, which locates the center of the bell curve, and the standard deviation σ, which sets its width. Its probability density function is:

f(x) = 1 / (σ√(2π)) · e−(x−μ)² / (2σ²)

Because the variable is continuous, the density f(x) is not itself a probability — the probability of landing on any single exact value is zero. Probabilities come from the area under the curve. The cumulative distribution function (CDF) gives the area to the left of a point, P(X ≤ x), and is written in terms of the error function:

P(X ≤ x) = ½ · [1 + erf((x − μ) / (σ√2))]

There is no elementary closed form for this integral, so this calculator evaluates erf with the Abramowitz & Stegun rational approximation (formula 7.1.26), whose maximum absolute error is under 1.5×10−7 — accurate to roughly six decimal places for everyday statistics.

The z-score and the standard normal curve

Any normal distribution can be converted to the standard normal distribution (μ = 0, σ = 1) with the z-score:

z = (x − μ) / σ

The z-score is the number of standard deviations x sits above (positive) or below (negative) the mean. This is exactly the lookup value in a printed z-table: P(X ≤ x) for your distribution equals Φ(z), the standard normal CDF. This calculator reports the z-score alongside the probabilities so you can cross-check against a table.

Key reference points (the 68–95–99.7 rule)

  • ±1σ: about 68.27% of values fall within one standard deviation of the mean (z from −1 to +1).
  • ±2σ: about 95.45% fall within two standard deviations.
  • ±3σ: about 99.73% fall within three standard deviations.
  • z = 1.645 marks the 95th percentile (one-tailed); z = 1.960 marks the boundary that leaves 2.5% in each tail, the basis of the common 95% confidence interval.

A worked example

IQ scores are scaled to μ = 100, σ = 15. For x = 115, the z-score is (115 − 100) / 15 = 1.00, so x is exactly one standard deviation above the mean. The CDF gives P(X ≤ 115) ≈ 0.8413, meaning about 84.13% of people score at or below 115 and about 15.87% score higher — consistent with the 68% rule, since half of the 100% − 68.27% = 31.73% outside ±1σ sits in the upper tail.

Frequently Asked Questions

What is the difference between the PDF and the CDF?
The PDF gives the height of the bell curve at a single point x. It is a density, not a probability, and for a continuous variable the probability of any exact value is zero. The CDF gives P(X ≤ x) — the area under the curve to the left of x — which is a genuine probability between 0 and 1. To get the probability that X lands in an interval [a, b], subtract: CDF(b) − CDF(a).
How is the z-score used with this calculator?
The z-score standardizes your value with z = (x − μ) / σ, giving the number of standard deviations x is from the mean. Every normal distribution maps onto the same standard normal curve after this transformation, so P(X ≤ x) equals Φ(z), the value a z-table looks up. A z of 0 means x is the mean (CDF = 50%); a z of +1.96 leaves about 2.5% in the upper tail.
How accurate are the probabilities?
The CDF has no closed form, so it is computed from the error function using the Abramowitz & Stegun 7.1.26 approximation, accurate to within about 1.5×10−7. That is more than enough for coursework, quality control, and everyday statistical work. Results are exact given your inputs — the real limit on accuracy is whether the data are actually normally distributed and whether your μ and σ are well estimated.

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