Normal Distribution Calculator
Compute Z-scores, cumulative probabilities, and the area under the bell curve for any mean and standard deviation. Visualise your distribution instantly with precise statistical results.
Distribution Parameters
Set your population mean, standard deviation, and query value
Centre of the bell curve
Must be greater than 0
Find the probability that X is less than this value
Find the probability that X is greater than this value
Find the probability that X falls between two values
Convert any value to its standardised Z-score
Distribution Results
Probability, Z-score & bell curve
Set your mean (μ), standard deviation (σ), and a value or range, then click Calculate Probability to inspect your distribution.
Common Z-Score Reference Table
Key Z-score values and their cumulative probabilities for the standard normal distribution (μ = 0, σ = 1).
| Z-Score | P(X < z) — Left Tail | P(X > z) — Right Tail | Common Use |
|---|---|---|---|
| −3.000 | 0.0013 (0.13%) | 99.87% | Extreme outlier threshold |
| −2.576 | 0.0050 (0.50%) | 99.50% | 99% confidence interval (two-tail) |
| −2.000 | 0.0228 (2.28%) | 97.72% | 2σ below mean |
| −1.960 | 0.0250 (2.50%) | 97.50% | 95% CI lower bound (two-tail) |
| −1.645 | 0.0500 (5.00%) | 95.00% | 90% CI lower bound / 5th percentile |
| −1.000 | 0.1587 (15.87%) | 84.13% | 1σ below mean |
| 0.000 | 0.5000 (50.00%) | 50.00% | Mean — 50th percentile |
| +1.000 | 0.8413 (84.13%) | 15.87% | 1σ above mean |
| +1.645 | 0.9500 (95.00%) | 5.00% | 95th percentile / 90% CI upper |
| +1.960 | 0.9750 (97.50%) | 2.50% | 95% CI upper bound (two-tail) |
| +2.000 | 0.9772 (97.72%) | 2.28% | 2σ above mean |
| +2.576 | 0.9950 (99.50%) | 0.50% | 99% confidence interval (two-tail) |
| +3.000 | 0.9987 (99.87%) | 0.13% | Extreme outlier threshold |
Normal Distribution FAQ
Understand the core concepts behind Gaussian distributions, Z-scores, and probability calculations.
A normal distribution (also called a Gaussian distribution) is a continuous, symmetric, bell-shaped probability distribution. It is fully described by two parameters: the mean (μ), which sets the centre, and the standard deviation (σ), which controls the spread. Many natural measurements — height, blood pressure, test scores — closely follow this distribution. The total area under the curve always equals 1 (or 100% probability).
A Z-score tells you how many standard deviations a raw value (X) is from the mean (μ). The formula is: Z = (X − μ) / σ. A Z-score of 0 means the value equals the mean. A positive Z-score means the value is above the mean; a negative one means it is below. Z-scores allow you to compare values from different distributions on a common standardised scale.
The CDF at a value x gives P(X ≤ x) — the probability that a randomly drawn value from the distribution is less than or equal to x. Geometrically, this equals the area under the bell curve to the left of x. For example, a CDF of 0.975 means 97.5% of all values in the distribution fall below that point.
The empirical rule (also called the 68-95-99.7 rule) states that for any normal distribution: approximately 68.27% of data falls within ±1 standard deviation of the mean; 95.45% falls within ±2 standard deviations; and 99.73% falls within ±3 standard deviations. This rule is widely used in quality control, finance, and the natural sciences to quickly characterise how data is spread.
To find P(a < X < b), compute the CDF at the upper bound and subtract the CDF at the lower bound: P(a < X < b) = CDF(b) − CDF(a). Use the “Between Two Values” mode in this calculator and enter your lower (a) and upper (b) bounds. The result is the shaded area under the curve between those two points.
A left-tail probability P(X < x) is the area under the curve to the left of x — the chance a random value is below your threshold. A right-tail probability P(X > x) is the complementary area to the right — the chance a random value exceeds your threshold. They always sum to 1: P(X > x) = 1 − P(X < x). Right-tail values are commonly used in hypothesis testing as p-values.