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Normal Distribution

General

Gaussian Distribution

A symmetric, bell-shaped probability distribution defined entirely by its mean and standard deviation, where values cluster around the mean and taper off equally in both directions.

Definition

The normal distribution is a symmetric, bell-shaped probability distribution defined entirely by two parameters: its mean (μ), which sets the center of the curve, and its standard deviation (σ), which sets how spread out the values are. Values cluster most densely near the mean and taper off symmetrically in both directions, forming the familiar bell curve.

It is one of the most widely used distributions in statistics because so many real-world measurements — heights, blood pressure readings, standardized test scores, manufacturing tolerances — approximate this shape, and because the central limit theorem guarantees that averages of large samples tend toward normality regardless of the original data's distribution. The Normal Distribution Calculator lets you compute the probability density or cumulative probability for any value given a mean and standard deviation.

A key related concept is the z-score, which converts any normally distributed value into units of standard deviations from the mean. This standardization is what makes it possible to compare values across different normal distributions using a single common scale, calculated with the Z-Score Calculator.

Formula

The probability density function of the normal distribution is:

f(x) = [1 ÷ (σ√(2π))] × e^(−(x−μ)² ÷ (2σ²))

To find how many standard deviations a value x is from the mean, use the z-score:

z = (x − μ) ÷ σ

Worked Example

Suppose adult heights in a population are normally distributed with a mean (μ) of 170 cm and a standard deviation (σ) of 8 cm. For a person who is 186 cm tall:

z = (186 − 170) ÷ 8 = 16 ÷ 8 = 2

A z-score of 2 means this person's height is 2 standard deviations above the mean. Applying the 68-95-99.7 rule, about 95% of the population falls within ±2 standard deviations (154 cm to 186 cm), so this person is taller than roughly 97.5% of the population.

Key Things to Know

  • Defined by just two numbers: the entire shape and location of a normal distribution is captured by its mean and Standard Deviation — no other parameters are needed.
  • Symmetric around the mean: the mean, median, and mode are all identical and sit exactly at the center of the curve.
  • The 68-95-99.7 rule gives fast estimates: roughly 68% of data falls within 1 standard deviation, 95% within 2, and 99.7% within 3, without needing to calculate exact probabilities.
  • Z-scores standardize comparisons: converting values to z-scores lets you compare data measured on different scales or from different normal distributions directly.
  • Not every dataset is normal: skewed or multi-modal data does not follow this pattern, so it's worth checking the shape of a distribution before applying normal-distribution-based statistics.

Related Calculators

Normal Distribution Calculator →Calculate z-score and cumulative probability for any point on a normal distribution. Enter mean, standard deviation, and x-value to get P(X ≤ x) instantly, free.
Z-Score Calculator →Calculate z-score, percentile rank, and probability for any data point. Enter value, mean, and standard deviation to find where it falls in the distribution. Free online tool.

Related Terms

Standard DeviationStatistical Measure of Data Variability

Frequently Asked Questions

The 68-95-99.7 rule (also called the empirical rule) states that in a normal distribution, about 68% of values fall within 1 standard deviation of the mean, about 95% fall within 2 standard deviations, and about 99.7% fall within 3 standard deviations. It gives a quick way to estimate how unusual a given value is without doing a full calculation.
Many natural and social phenomena — heights, test scores, measurement errors — approximate a normal distribution, and the central limit theorem shows that the average of many independent samples tends toward normality even when the underlying data isn't. This makes it the foundation for confidence intervals, hypothesis testing, and quality control methods.
A standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Any normal distribution can be converted to the standard form by calculating a z-score for each value, which is what tools like the Z-Score Calculator do.
Yes — the normal distribution is defined over all real numbers, so it theoretically extends infinitely in both directions, including negative values. In practice, whether negative values make sense depends on what's being measured (temperature can be negative, height cannot).
Common checks include plotting a histogram to look for a symmetric bell shape, using a Q-Q plot to compare sample quantiles against theoretical normal quantiles, or running a formal statistical test such as Shapiro-Wilk. Skewness and kurtosis close to zero also support (but don't prove) approximate normality.
A normal distribution is perfectly symmetric, with the mean, median, and mode all equal at the center of the curve. A skewed distribution has a longer tail on one side, pulling the mean away from the median — income distributions, for example, are typically right-skewed because a small number of very high earners stretch the upper tail.