Normal Distribution Calculator

Last updated: 2026-05-18

Normal distribution calculator

The normal distribution is a continuous probability distribution that is symmetric around its mean, forming the familiar bell-shaped curve. It describes many natural phenomena: heights, measurement errors, IQ scores, exam results, and financial returns all approximate a bell curve when samples are large enough. From an observed value, the mean, and the standard deviation, this calculator returns the Z-score, the cumulative probability P(X < x), the complementary P(X > x), the percentile rank, and the probability density f(x).

The normal distribution

Its shape is fully determined by two parameters:

• μ (mean) — the center of the distribution
• σ (standard deviation) — how spread out the values are

The probability density function is:

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\!\left(-\frac{(x - \mu)^2}{2\sigma^2}\right)$$

The Z-score

The Z-score converts any value to units of standard deviations:

$$z = \frac{x - \mu}{\sigma}$$

Z-score	Interpretation
z = 0	Value equals the mean
z = 1	One σ above the mean
z = −2	Two σ below the mean
z > 3	Extreme — occurs < 0.3% of the time

Z-scores are used to compare values across distributions with different scales — for instance, comparing a score on a 100-point exam with a score on a 4.0 GPA scale.

Cumulative probability: P(X < x)

P(X < x) is the area under the bell curve to the left of x. It represents the probability that a randomly drawn value from this distribution is less than x.

$$P(X < x) = \Phi(z) = \frac{1}{2}\left[1 + \text{erf}\!\left(\frac{z}{\sqrt{2}}\right)\right]$$

Here Φ(z) is the cumulative distribution function (CDF) of the standard normal distribution N(0, 1), and erf is the error function — a special function defined as $\text{erf}(t) = \frac{2}{\sqrt{\pi}} \int_0^t e^{-s^2}\,ds$.

This calculator uses the rational approximation from Abramowitz & Stegun §7.1.26 (maximum error < 1.5 × 10⁻⁷), which is accurate to 6 significant figures.

The 68–95–99.7 rule

For any normal distribution, three round numbers define the practical range:

Range	Probability
μ ± 1σ	~68.3%
μ ± 2σ	~95.4%
μ ± 3σ	~99.7%

This means that only about 1 in 370 values fall more than 3 standard deviations from the mean. Quality control processes often use 3σ or 6σ as thresholds for "defect-free" production.

Worked example

A class of students scores on an exam have mean μ = 72 and standard deviation σ = 8. A student scored x = 85.

$$z = \frac{85 - 72}{8} = 1.625$$ $$P(X < 85) = \Phi(1.625) \approx 0.9479$$

The student scored in approximately the 95th percentile — better than about 94.8% of the class. The top 5.2% of the class scored higher.

PDF vs CDF

Quantity	Symbol	Meaning
Probability density	f(x)	Height of the curve at x — not a probability itself
Cumulative probability	P(X < x)	Area under the curve to the left of x — a true probability

f(x) can be greater than 1 for narrow (small σ) distributions because it is a density, not a probability. The probability of any exact single value in a continuous distribution is always zero; only intervals have non-zero probability.

Frequently Asked Questions (FAQ)

What is a Z-score and how is it calculated?

A Z-score measures how many standard deviations a value lies from the mean: z = (x − μ) / σ. A Z-score of 0 means the value equals the mean. Z = 1 means one standard deviation above the mean. Z = −2 means two standard deviations below. Z-scores allow you to compare values from distributions with different means and spreads by putting them on a common scale.

What does P(X < x) mean in a normal distribution?

P(X < x) is the cumulative probability — the probability that a randomly drawn value from this distribution falls below x. It equals the area under the bell curve to the left of x. For example, P(X < 75) = 0.69 means roughly 69% of values in this distribution are below 75. P(X > x) = 1 − P(X < x) covers the area to the right.

What is the 68-95-99.7 rule?

For any normal distribution, about 68% of values fall within 1 standard deviation of the mean (Z between −1 and +1), roughly 95% fall within 2 standard deviations, and 99.7% fall within 3 standard deviations. This is why values with |Z| > 3 are considered statistical outliers — they occur less than 0.3% of the time in a normal distribution.

How do I find the percentile from a Z-score?

The percentile equals P(X < x) × 100. A Z-score of 0 is the 50th percentile (exactly half the distribution lies below the mean). Z = 1.28 corresponds to approximately the 90th percentile; Z = 1.645 to the 95th; Z = 2.326 to the 99th. This calculator computes the percentile directly — no Z-table needed.