Normal Distribution Calculator
Calculate probabilities, z-scores, and percentiles for the normal (Gaussian) distribution. Find P(X ≤ x), probability density, or inverse normal values. See also our Z-Score Calculator and Standard Deviation Calculator.
How to Use the Normal Distribution Calculator
The normal distribution (also called the Gaussian distribution or bell curve) is the most important probability distribution in statistics. It describes how data clusters around a mean value with a characteristic bell-shaped curve. Approximately 68% of data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations — this is the empirical rule (68-95-99.7 rule).
This calculator supports three modes: (1) CDF mode calculates the cumulative probability P(X ≤ x) — the area under the curve to the left of x; (2) PDF mode calculates the probability density function value at a specific point; (3) Inverse mode finds the x-value corresponding to a given cumulative probability (percentile). Enter the mean and standard deviation of your distribution, then the x-value or probability depending on the mode.
The standard normal distribution has mean μ = 0 and standard deviation σ = 1. Any normal distribution can be converted to the standard normal using the z-score transformation: Z = (X - μ)/σ. This allows you to use standard normal tables or this calculator for any normal distribution regardless of its parameters. The z-score tells you how many standard deviations a value is from the mean.
Normal Distribution Formulas
Probability Density Function (PDF):
f(x) = (1/(σ√(2π))) × e^(-(x-μ)²/(2σ²))
Cumulative Distribution Function (CDF):
F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt
Z-Score Transformation:
Z = (X - μ) / σ
Empirical Rule:
P(μ-σ ≤ X ≤ μ+σ) ≈ 0.6827 (68.27%)
P(μ-2σ ≤ X ≤ μ+2σ) ≈ 0.9545 (95.45%)
P(μ-3σ ≤ X ≤ μ+3σ) ≈ 0.9973 (99.73%)
Properties:
Mean = Median = Mode = μ
Variance = σ², Skewness = 0, Kurtosis = 3
Example Calculation
IQ scores follow a normal distribution with μ = 100 and σ = 15. What percentage of people have an IQ above 130?
Given: μ = 100, σ = 15, x = 130
Z = (130 - 100)/15 = 30/15 = 2.0
P(X ≤ 130) = Φ(2.0) = 0.9772
P(X > 130) = 1 - 0.9772 = 0.0228 = 2.28%
About 2.28% of people have IQ above 130
Inverse: What IQ is the 95th percentile?
Z₀.₉₅ = 1.645
X = μ + Zσ = 100 + 1.645×15 = 124.67
Standard Normal Z-Table
| Z-Score | P(Z ≤ z) | P(Z > z) | Common Use |
|---|---|---|---|
| 0.0 | 0.5000 | 0.5000 | — |
| 0.5 | 0.6915 | 0.3085 | — |
| 1.0 | 0.8413 | 0.1587 | — |
| 1.28 | 0.8997 | 0.1003 | — |
| 1.5 | 0.9332 | 0.0668 | — |
| 1.645 | 0.9500 | 0.0500 | 90% CI (one-tail 5%) |
| 1.96 | 0.9750 | 0.0250 | 95% CI (two-tail 2.5%) |
| 2.0 | 0.9772 | 0.0228 | — |
| 2.326 | 0.9900 | 0.0100 | 99% CI (one-tail 1%) |
| 2.5 | 0.9938 | 0.0062 | — |
| 2.576 | 0.9950 | 0.0050 | 99% CI (two-tail 0.5%) |
| 3.0 | 0.9987 | 0.0013 | — |
Frequently Asked Questions
What is the normal distribution?
The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, symmetric about the mean. It is defined by two parameters: the mean (μ) which determines the center, and the standard deviation (σ) which determines the spread. Many natural phenomena follow a normal distribution due to the Central Limit Theorem — the sum of many independent random variables tends toward a normal distribution regardless of the original distributions.
What is the difference between PDF and CDF?
The PDF (probability density function) gives the relative likelihood of a continuous random variable taking a specific value — it is the height of the bell curve at that point. The CDF (cumulative distribution function) gives the probability that the variable is less than or equal to a value — it is the area under the curve to the left of that point. For continuous distributions, P(X = x) = 0; only intervals have non-zero probability, which is why we use the CDF.
How do I interpret a z-score?
A z-score tells you how many standard deviations a value is from the mean. Z = 0 means the value equals the mean. Z = 1 means one standard deviation above the mean (84th percentile). Z = -2 means two standard deviations below the mean (2.3rd percentile). Z-scores allow comparison across different normal distributions — a z-score of 2 is equally unusual whether the original data is heights, test scores, or temperatures.
When should I use the normal distribution?
Use the normal distribution when: (1) data is continuous and symmetric around the mean; (2) the Central Limit Theorem applies (sample means of large samples); (3) measurement errors; (4) natural biological measurements (height, weight, blood pressure); (5) quality control (process variation). Do NOT use it for: skewed data, count data (use Poisson), binary outcomes (use binomial), or data with hard boundaries (like income, which is right-skewed).
What is the 68-95-99.7 rule?
The empirical rule states that for a normal distribution: approximately 68.27% of data falls within ±1σ of the mean, 95.45% within ±2σ, and 99.73% within ±3σ. This means values beyond 3σ are extremely rare (0.27% or about 1 in 370). In quality control, the "six sigma" standard means defects beyond 6σ — only 3.4 per million opportunities. This rule provides quick probability estimates without calculation.
What is the Central Limit Theorem?
The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean approaches a normal distribution as sample size increases, regardless of the population distribution. For most distributions, n ≥ 30 is sufficient. The sampling distribution has mean = population mean and standard error = σ/√n. This is why the normal distribution is so important — it applies to sample means even when individual data is not normally distributed.
Applications of the Normal Distribution
The normal distribution appears throughout science and engineering: in physics (thermal noise, measurement uncertainty, Brownian motion), biology (height, weight, enzyme activity), psychology (IQ scores, reaction times), finance (stock returns under certain models, risk assessment), manufacturing (quality control, tolerance analysis), and education (standardized test scores). Understanding the normal distribution is fundamental to hypothesis testing, confidence intervals, and statistical inference.