Probability, z-scores, percentiles, and inverse normal values — with an interactive bell curve.
The Normal Distribution Calculator is a statistical tool that helps you calculate probabilities, z-scores, percentiles, probability density values, and inverse normal values for a normal (Gaussian) distribution. Whether you're studying statistics, performing quality control, analyzing financial data, or conducting scientific research, this calculator makes it easy to solve normal distribution problems in seconds.
A normal distribution is one of the most important probability distributions in statistics. It is characterized by its familiar bell-shaped curve where data is symmetrically distributed around the mean. Many real-world measurements, including exam scores, heights, weights, manufacturing tolerances, and measurement errors, approximately follow a normal distribution.
Simply enter the distribution mean (μ), standard deviation (σ), choose the calculation mode, provide the required value, and the calculator instantly displays cumulative probability, upper-tail probability, z-score, percentile, probability density, and an interactive bell curve visualization.
A normal distribution, also known as the Gaussian distribution, is a continuous probability distribution where most values cluster around the mean while fewer values occur farther from the center. The distribution is perfectly symmetric, meaning the left and right halves are mirror images.
Two parameters completely define a normal distribution:
Approximately:
This is commonly known as the 68-95-99.7 Rule or the Empirical Rule.
Calculates the probability that a random variable is less than or equal to a specified value.
Calculates the probability that a value is greater than or equal to the specified point.
Finds the probability that a value falls between two limits.
Determines how many standard deviations a value is above or below the mean.
Formula:
Z = (X − μ) / σ
Calculates the percentile rank of a given value in the distribution.
Given a probability, this mode calculates the corresponding value (x) in the normal distribution.
Suppose exam scores follow a normal distribution with:
First calculate the z-score:
Z = (85 − 70) ÷ 10
Z = 1.5
The calculator returns approximately:
This means the student scored better than approximately 93% of all students.
The normal distribution — the "bell curve" — describes data that clusters symmetrically around a central average. Most values sit close to the mean (μ), and the standard deviation (σ) controls how spread out the rest of the data is: a small σ produces a tall, narrow curve, while a large σ produces a wide, flat one. This shape shows up everywhere from standardized test scores and heights to measurement error and manufacturing tolerances, which is why converting a raw value into a probability, percentile, or z-score is one of the most common calculations in statistics.
| Term | Meaning |
|---|---|
| Z-score | How many standard deviations a value sits from the mean |
| Cumulative Probability | P(X ≤ x) — the area under the curve to the left of a value |
| Upper Tail Probability | P(X ≥ x) — the area under the curve to the right of a value |
| Percentile | The cumulative probability expressed as a 0–100 rank |
| Inverse Normal | Working backward from a probability to find the corresponding x value |
This gives the height of the bell curve at any point x — it describes the relative likelihood of values near x, but it is not itself a probability. Probability comes from the area under this curve, not the height of a single point.
The CDF is the running total of area under the density curve up to x. It has no simple closed form, so this calculator evaluates it with a high-accuracy numerical approximation of the error function (erf).
Converting a raw value to a z-score rescales it in units of standard deviation, which is what lets any normal distribution be compared against the standard normal table (μ = 0, σ = 1).
Given a target probability p, the inverse normal function finds the z-score whose cumulative probability equals p, then converts it back to the original scale using μ and σ.
Click any row to load that z-score into the calculator above.
| Z-score | Cumulative Probability P(Z ≤ z) | Common use |
|---|---|---|
| 0.00 | 50.00% | Median |
| 0.6745 | 75.00% | Upper quartile |
| 1.00 | 84.13% | 1 SD above mean |
| 1.2816 | 90.00% | 90th percentile |
| 1.645 | 95.00% | One-tail 95% / 90% CI |
| 1.96 | 97.50% | Two-tail 95% CI bound |
| 2.00 | 97.72% | ~2 SD above mean |
| 2.3263 | 99.00% | 99th percentile |
| 2.576 | 99.50% | Two-tail 99% CI bound |
| 3.00 | 99.87% | 3 SD above mean |