🔔 Normal Distribution Calculator

Calculate probabilities and z-scores for a normal distribution. Find P(X < x), P(X > x), and P(x₁ < X < x₂) with a visual bell curve.

Mean (μ)

Std Dev (σ)

x value

x₂ value (for range probability, optional)

What is the Normal Distribution?

The normal distribution (also called the Gaussian distribution or bell curve) is the most important probability distribution in statistics. It is a continuous, symmetric, bell-shaped distribution fully characterised by two parameters: the mean (μ), which sets the centre, and the standard deviation (σ), which controls the width. The normal distribution describes the probability of a continuous random variable and arises naturally whenever a quantity is the sum of many small, independent random effects — from measurement errors to heights of a population.

The Central Limit Theorem is the reason the normal distribution is so ubiquitous: it states that the sum or mean of a large number of independent, identically distributed random variables approaches a normal distribution regardless of the original distribution's shape — as long as the variance is finite. This theorem underpins parametric statistical inference: hypothesis tests, confidence intervals, regression, and ANOVA all rely on the assumption that sample means follow a normal distribution, justified by the Central Limit Theorem for sufficiently large samples.

The empirical rule (68-95-99.7 rule) describes the proportion of data within 1, 2, and 3 standard deviations of the mean respectively: ~68% within ±1σ, ~95% within ±2σ, and ~99.7% within ±3σ. Z-scores standardise any normal distribution to the standard normal (μ=0, σ=1), allowing probability lookups from a single standard table. The CDF (cumulative distribution function) gives the probability that a value falls below a given threshold — essential for hypothesis testing, process capability analysis, and risk quantification.

Normal Distribution Formulas

Probability Density Function (PDF)

f(x) = (1/σ√(2π)) × e^(−(x−μ)²/2σ²)

Z-Score

z = (x − μ) / σ

CDF (using error function)

P(X < x) = ½ × [1 + erf((x−μ) / (σ√2))]

How to Use This Calculator

1

Set Distribution Parameters

Enter the mean (μ) and standard deviation (σ) of your normal distribution.
2

Enter x Values

Enter x for single-tail probabilities, or both x₁ and x₂ for a range probability.
3

Read Probabilities

The calculator shows P(X < x), P(X > x), range probability, and the z-score.
4

View the Bell Curve

The SVG diagram highlights the shaded area corresponding to your probability.

How the Normal Distribution Calculator Works

Formula, assumptions, and calculation steps for this statistics tool.

Methodology

Statistics calculators organize sample data, apply the selected descriptive or inferential formula, and report the statistic with interpretation.

Calculation Steps

Enter raw values or summary statistics.
Clean separators and count the sample size.
Apply the relevant statistic, probability, or confidence formula.
Display the result with context such as degrees of freedom, percentile, or strength.

Assumptions and Limits

Samples should be representative of the population being studied.
Normality or independence assumptions apply only where the selected method requires them.
Rounded results may differ slightly from spreadsheet software.

Frequently Asked Questions

What is a normal distribution?

A normal distribution is a symmetric, bell-shaped probability distribution defined by its mean (μ) and standard deviation (σ). Many natural phenomena follow a normal distribution, such as heights, test scores, and measurement errors.

What is a z-score?

A z-score measures how many standard deviations a value is from the mean. z = (x − μ) / σ. A z-score of 1.96 corresponds to the 97.5th percentile of a standard normal distribution.

What is the 68-95-99.7 rule?

For any normal distribution: 68% of data falls within ±1σ, 95% within ±2σ, and 99.7% within ±3σ of the mean. This is also called the empirical rule.

What is the CDF?

The Cumulative Distribution Function (CDF) gives the probability that a random variable X takes a value less than or equal to x. It is computed using the error function (erf) approximation in this calculator.

What is the standard normal distribution?

The standard normal distribution is a special case with μ = 0 and σ = 1. Any normal distribution can be converted to standard normal using the z-score transformation, making z-tables universally applicable.

Real-World Applications

🎓

Standardised Testing & IQ Scores

IQ is defined with μ=100 and σ=15. The normal distribution tells us exactly what percentile a score of 130 falls at (97.7th) — enabling standardised comparison of individual performance.

🏭

Manufacturing Process Control

A production process making bolts with μ=10mm and σ=0.1mm — the normal distribution quantifies the probability of any bolt falling outside specified tolerances (±0.3mm).

📈

Financial Risk (Value at Risk)

Assuming normally distributed daily returns, calculate Value at Risk (VaR) at 95% or 99% confidence — the loss not expected to be exceeded on 95% or 99% of days.

🧪

Clinical Trial Analysis

Z-tests and t-tests for comparing treatment group means rely on the normal distribution (or its approximation via the CLT) to compute p-values and confidence intervals.

🌡️

Measurement Uncertainty

Repeated measurements of the same physical quantity follow a normal distribution centred on the true value — statistical process control uses this to distinguish signal from noise.

🎯

Grading on a Curve

Convert raw exam scores to a normalised distribution with a desired mean and standard deviation — assigning grades based on standard deviation bands from the mean.

Common Mistakes

Assuming all data is normally distributed

Many real-world distributions are not normal — income is log-normal, stock returns have fat tails, and waiting times follow exponential or Weibull distributions. Always check normality with a Q-Q plot or Shapiro-Wilk test before applying normal-based methods.

Confusing standard deviation with standard error

Standard deviation measures the spread of individual data points; standard error = σ/√n measures the uncertainty of the sample mean. These are different quantities — using one where the other is needed produces wrong results.

Applying the 68-95-99.7 rule to non-normal distributions

The empirical rule (68-95-99.7%) applies only to normal distributions. For skewed or heavy-tailed distributions, far more or fewer observations may fall within ±2σ — applying the rule blindly gives wrong probability estimates.

Treating the tails as impossibly rare

For a normal distribution, events beyond ±4σ have probability ~0.006% — seemingly negligible. But in finance, the tails are typically fatter than normal, meaning "six-sigma events" occur far more often than the normal model predicts.

Confusing Z-score with percentile

A Z-score of 1.96 is not the 1.96th percentile — it is the 97.5th percentile (for a one-tailed test) or the critical value for a 95% two-tailed confidence interval. Always use the CDF to convert Z-scores to percentiles.

Z-Score to Percentile Quick Reference

Z-Score	Percentile	Two-tailed CI
±1.00	84.1%	68.3% within ±1σ
±1.645	95.0%	90% within ±1.645σ
±1.960	97.5%	95% CI critical value
±2.000	97.7%	95.4% within ±2σ
±2.576	99.5%	99% CI critical value
±3.000	99.865%	99.73% within ±3σ

References

Gauss, C.F. Theoria Motus Corporum Coelestium. Perthes and Besser, 1809.
Casella, G. and Berger, R.L. Statistical Inference. Cengage, 2001.
Montgomery, D.C. Introduction to Statistical Quality Control. Wiley, 2019.
Abramowitz, M. and Stegun, I.A. Handbook of Mathematical Functions. Dover, 1965.
NIST/SEMATECH. e-Handbook of Statistical Methods. nist.gov, 2024.