Normal Distribution Calculator
Calculate probabilities and areas under the normal (Gaussian) curve.
Area/Probability
Percentile
Quick Examples
Normal Distribution
The normal distribution (Gaussian) is a symmetric, bell-shaped probability distribution that is fundamental to statistics. It describes many real-world measurements — heights, test scores, measurement errors — and, through the central limit theorem, the behavior of averages even when the underlying data are not normal.
Z-Scores: Standardizing Any Value
A z-score tells you how many standard deviations a value sits from the mean, which lets you compare values from completely different normal distributions. This is the bridge between any normal distribution and the standard normal (μ = 0, σ = 1), whose cumulative probabilities live in z-tables.
A z-score of 1 means the value is one standard deviation above the mean; z = −2 means two standard deviations below it. Values beyond about |2| are relatively rare tail events under the model.
Tail Probability Formulas
Every normal probability is an area under the curve. Let Φ(z) be the standard normal cumulative probability (area to the left of z):
Convert each bound to a z-score, look up the cumulative areas, and subtract. Sketching which region you need — left tail, right tail, or middle — before calculating is the most reliable way to pick the right formula.
Probability Shortcuts from Symmetry
The standard normal curve is symmetric about zero, so you never need to look up both sides of the table:
- Mirror rule: P(Z ≤ −z) = P(Z ≥ z) — the left tail at −z equals the right tail at z.
- Subtraction rule: P(Z ≥ z) = 1 − P(Z ≤ z).
- Middle band: P(−z ≤ Z ≤ z) = P(Z ≤ z) − P(Z ≤ −z) = 2Φ(z) − 1.
For example, knowing Φ(1) = 0.8413 immediately gives P(Z ≥ 1) = 0.1587 and P(−1 ≤ Z ≤ 1) = 0.6826.
Reading a Z-Table
Printed z-tables tabulate Φ(z), the cumulative area to the left, usually to two decimal places of z. To find P(X ≤ x): standardize to z, split z into the row (ones and tenths) and column (hundredths), and read the intersection. For values between table entries, round z to the nearest hundredth — the rounding error is far smaller than the measurement error in most real data.
Properties
- Symmetric: About the mean
- Bell-shaped: Highest at the mean
- Mean = Median = Mode: All equal
- 68-95-99.7 Rule: Percent within 1, 2, 3 standard deviations
Empirical Rule
| Range | Coverage |
|---|---|
| μ ± 1σ | 68.27% |
| μ ± 2σ | 95.45% |
| μ ± 3σ | 99.73% |
Worked Examples
Example 1: Left-Tail Probability
Adult heights in a population follow a normal distribution with μ = 170 cm and σ = 10 cm. What share of people are shorter than 180 cm?
z = (180 − 170) / 10 = 1.0, and Φ(1.0) = 0.8413. About 84.1% of people are shorter than 180 cm.
Example 2: Probability Between Two Values
With the same heights, what share falls between 160 cm and 180 cm?
The bounds give z = −1 and z = +1, so P = Φ(1) − Φ(−1) = 0.8413 − 0.1587 = 0.6827 — the familiar 68.3% of the empirical rule.
Example 3: Inverse Problem — Finding a Percentile
How tall must someone be to be in the top 10% of heights?
The 90th percentile corresponds to z ≈ 1.282. Converting back: x = 170 + 1.282 × 10 = 182.82 cm. Someone taller than about 182.8 cm beats 90% of the population.
PDF vs CDF: Height Is Not Probability
The density f(x) is the height of the bell curve at x; the probability P(X ≤ x) is the area under the curve to the left of x. Probabilities come from areas, never from heights — a density value can even exceed 1 when σ is small, while the total area under the curve is always exactly 1. For the same reason, the probability of X equaling one exact value is zero: probability lives in ranges, not points.
Where the Normal Distribution Is Used
- Central limit theorem: in large samples, the sampling distribution of the mean approaches normality even when the raw data are not normal — this underpins most of inferential statistics.
- Hypothesis testing: t-tests, z-tests, and ANOVA all assume the data or the sampling distribution follows a normal distribution.
- Quality control: analyzing process variation, control charts, and tolerance limits in manufacturing (Six Sigma is built on this).
- Natural and social measurement: heights, weights, IQ scores, standardized test scores, and laboratory measurement errors.
- Medicine and finance: biological measurements like blood pressure, and as a first model for asset returns and risk assessment.
When a Normal Model Is a Poor Choice
The bell curve is not universal. It breaks down for strongly skewed data (incomes), values bounded near an edge (percentages close to 0 or 100), small counts, and heavy-tailed data (financial returns). In those cases normal probabilities can badly underestimate extreme outcomes — consider the lognormal, binomial, Poisson, or t-distributions instead.
Common Mistakes
- Asking for P(X = an exact value): for continuous distributions this is always zero. Ask about ranges — less than, greater than, or between — instead.
- Using σ = 0 or a negative σ: a standard deviation of zero does not define a valid normal curve.
- Swapping the bounds: for interval probabilities, make sure a ≤ b before subtracting cumulative areas.
- Forgetting to standardize: never look up x directly in a z-table. Convert to z = (x − μ) / σ first, every time.