Normal Distribution Calculator

Calculate P(X ≤ x) and P(X > x) for any normal distribution from mean, standard deviation, and a value x, with z-score.

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Mean (μ) Std deviation (σ) Value (x)

95%

P(X ≤ x)

P(X > x)

1.65

Z-score

P(X ≤ x)	95%
P(X > x)	5%
Z-score	1.65
Mean (μ)	0
Std deviation (σ)	1
Value (x)	1.65

Probability & Statistics

Normal distribution calculator — probabilities for the bell curve

The normal distribution is the most important probability distribution in statistics. Its symmetric, bell-shaped curve describes natural variation in countless phenomena — heights, measurement errors, test scores, biological traits. This calculator computes the probability that a normally distributed variable falls above or below a given value, and returns the corresponding z-score.

About this calculator

Reviewed 2026-03-15

Methodology

The z-score is computed as (x − μ) / σ. The cumulative probability P(X < x) is calculated using the error function approximation Φ(z) = 0.5 × (1 + erf(z / √2)), implemented with a rational approximation that is accurate to better than 1.5 × 10⁻⁷ across the full real line. P(X > x) = 1 − P(X < x).

Limitations

The normal distribution is a mathematical model. Real-world data may deviate from normality, especially in the tails.
For very extreme z-scores (|z| > 8), floating-point precision limits the accuracy of the tail probability, but such values are astronomically unlikely in practice.
This calculator does not perform normality tests on your data. Always verify normality assumptions before drawing statistical conclusions.

Disclaimer

This calculator provides probabilities under a normal distribution model. It is intended for educational and analytical use. Statistical decisions should always be made with appropriate professional judgement.

Sources

See our calculation methodology and editorial standards for how this estimate is produced.

The normal distribution

A normal distribution is fully described by two parameters: the mean μ (the centre) and the standard deviation σ (the spread). About 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three — the empirical rule, also called the 68–95–99.7 rule.

The probability density function is f(x) = (1 / (σ√(2π))) × exp(−(x−μ)² / (2σ²)). The cumulative distribution function (CDF), which gives the probability that a value is below x, has no closed form and is computed numerically.

Z-score and standardisation

To find the probability for any normal distribution, we convert x to a z-score: z = (x − μ) / σ. The z-score measures how many standard deviations x is from the mean. We then look up this z in the standard normal distribution (μ = 0, σ = 1).

For example, for a distribution with μ = 100, σ = 15 (like many IQ scales), a value of x = 115 gives z = (115 − 100) / 15 = 1. The probability of scoring below 115 is P(Z < 1) ≈ 84.1%.

Probability below vs. probability above

The probability below x (P(X < x)) is the area under the normal curve to the left of x — the CDF value. The probability above x (P(X > x)) is the complementary area: 1 − CDF(z).

Together these two probabilities always sum to exactly 1 (100%). To find the probability within a range [a, b], compute P(X < b) − P(X < a). This calculator provides the two tail probabilities; for ranges you can combine two calculations.

Common applications

Quality control uses the normal distribution to set tolerance limits. A process with μ = 10 mm and σ = 0.1 mm: what fraction of parts fall outside 9.8–10.2 mm (±2σ)? P(outside) = 1 − P(−2 < Z < 2) ≈ 4.6%.

Hypothesis testing and p-values are based on the normal distribution. A z-score of 1.96 corresponds to a two-tailed p-value of 0.05 — the 95% confidence threshold. The standard normal CDF (z-score probability) is used constantly in confidence interval and significance calculations.

Frequently asked questions

What does the standard deviation tell me about the distribution?

A larger standard deviation means the distribution is wider and more spread out. A smaller standard deviation means values cluster tightly around the mean. Approximately 68% of data falls within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ.

Why is the standard normal distribution (μ=0, σ=1) special?

Any normal distribution can be converted to the standard normal via z = (x − μ) / σ. This allows one set of probability tables (or one CDF function) to handle all normal distributions. The standard normal is the reference distribution for z-scores, confidence intervals, and many hypothesis tests.

What is the 68–95–99.7 rule?

For any normal distribution, approximately 68% of values fall within one standard deviation of the mean (μ ± σ), about 95% within two standard deviations (μ ± 2σ), and about 99.7% within three standard deviations (μ ± 3σ). These percentages come directly from integrating the normal CDF.

What if my data is not normally distributed?

The normal distribution is a model, not a guarantee. Many real datasets are skewed, heavy-tailed, or multimodal. If your data departs significantly from normality, probabilities from this calculator may be inaccurate. Consider checking with a histogram, Q–Q plot, or normality test before applying normal-distribution-based conclusions.

Normal Distribution Calculator

Normal distribution calculator — probabilities for the bell curve

The normal distribution

Z-score and standardisation

Probability below vs. probability above

Common applications

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