Skip to main content
Calculator Answers

Normal Distribution Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A normal distribution calculator is a statistical tool that computes probabilities and percentiles for a normally distributed dataset. Using the population mean and standard deviation, the calculator determines the probability of a random variable falling within a specific range, above a threshold, or below a threshold by calculating the corresponding z-scores. It can also perform the inverse operation, finding the raw data value that matches a target percentile. Students, statisticians, and data analysts use this utility to evaluate standard distributions and test hypothesis models.

Pick raw-score or z-score mode and a probability direction (left, right, or between two values). Enter the inputs and the calculator returns the probability under the standard normal curve as a decimal and a percent.

See also: Z Score Calculator, P Value Calculator, and Standard Deviation Calculator – Variance & SD.

Quick Answer

Calculate normal distribution probabilities and percentiles. Enter your mean, standard deviation, and range bounds to find areas under the bell curve.

Probability

e.g. 110

e.g. 100

Must be greater than 0. · e.g. 15

Uses Hart's rational approximation of the standard normal distribution. Each tail is computed directly rather than by subtracting the other from 1, so probabilities stay accurate far out in the tails instead of rounding to zero.

Normal probability

P(X ≤ x) · z = 0.667

0.747507

74.7507% area under the curve

Z-score (z₁)0.667
Φ(z₁) area to left0.747507
1 − Φ(z₁) area to right0.252493
Probability0.747507
As percent74.7507%
Was this helpful?

Examples

Left tail at x = 110, μ = 100, σ = 15

z = 0.667, P ≈ 0.747508 (74.75%)

Right tail at z = 1.96

P(Z ≥ 1.96) ≈ 0.024998 (2.50%)

Between z = −1 and z = +1

P ≈ 0.682689 (68.27%)

How it works

Convert any raw score to a z-score using z = (x − μ) ÷ σ. The standard normal CDF Φ(z) is the cumulative area under the standard normal curve to the left of z. The calculator uses a closed-form polynomial approximation, then derives the requested probability:

Left tail · Φ(z)

Right tail · 1 − Φ(z)

Between · Φ(z₂) − Φ(z₁)

68.27% within ±1σ · 95.45% within ±2σ · 99.73% within ±3σ. Common critical values: ±1.645 (90%) · ±1.96 (95%) · ±2.576 (99%).

Related probability and stats calculators

Related Calculators

More tools from Education

Z Score CalculatorCalculate a z score from a raw score, mean, and standard deviation using z = (x − μ) / σ, with the formula, step by step work, and a plain English reading.P Value CalculatorConvert a z-score to a p-value for a one-tailed or two-tailed test, with the area on each side of the curve.Standard Deviation Calculator – Variance & SDUse this sd calculator to compute sample and population standard deviation, variance, mean, and count from a list of numbers.

Frequently asked questions

The normal distribution is a symmetric, bell-shaped distribution defined by two parameters: mean μ and standard deviation σ. It is the most-used distribution in statistics because many natural processes are approximately normal, and because the Central Limit Theorem makes sums and averages of many independent random variables tend toward a normal distribution. The standard normal has μ = 0 and σ = 1.

The calculator first converts the raw score x to a z-score using z = (x − μ) ÷ σ. Then it computes the area under the standard normal curve using Hart's rational approximation. Each tail is calculated directly rather than by subtracting the other from 1, because subtracting a number very close to 1 destroys the accuracy of a small tail. That is why the left-tail and right-tail results agree exactly for values of equal magnitude, and why probabilities far out in the tail stay accurate instead of rounding to zero.

Use raw score mode when you have the actual values from your data, plus the mean and standard deviation of the underlying distribution. Use z-score mode when you already have the z-score (a standardized value) and just want the corresponding probability. The math is the same; the modes just save you the conversion step in one direction or the other.

Between mode computes P(a ≤ X ≤ b), the area under the curve strictly between two values. It is the difference between two CDFs: Φ(z₂) − Φ(z₁). For example, the probability that a value falls within 1 standard deviation of the mean (between z = −1 and z = +1) is about 0.6827, the well-known 68 percent rule for the normal distribution.

Φ(0) = 0.5 (the mean). About 68.27 percent of the area lies within z = ±1, 95.45 percent within z = ±2, and 99.73 percent within z = ±3. Common critical values: z = ±1.645 (90 percent two-sided), ±1.96 (95 percent), ±2.576 (99 percent). The calculator returns these directly when those z-scores are entered.

It is an approximation, but a close one. Checked against SciPy's normal distribution functions, the underlying calculation is correct to at least 8 significant figures everywhere tested, and to roughly 13 significant figures for z-scores within ±4 — the range that covers every conventional significance threshold. The same implementation is shared with the p-value calculator on this site, so the two always agree. Ordinary results are shown to six decimal places. A probability too small to survive that rounding switches to scientific notation instead — P(Z ≥ 8.3) reads 5.2056e-17 rather than 0.000000 — so a real value is never displayed as zero. A displayed zero therefore means a genuine zero: the probability has fallen below the smallest number the browser can represent, which happens past roughly z = ±38.6.