The classic bell curve with shaded sigma bands and a configurable percentile callout.
A free, animated normal distribution you can read here or embed on any website, from Scrollchart.
Normal Distribution
The normal distribution is a symmetric, bell-shaped probability density defined by two parameters: mean (mu) and standard deviation (sigma). The curve peaks at the mean and tapers symmetrically on both sides. Approximately 68% of values fall within 1 sigma, 95% within 2 sigma, and 99.7% within 3 sigma of the mean.\n\nThe vertical reference line locates a specific observed value. The floating callout translates that position into a percentile. A value at mu + 1.5 sigma sits at roughly the 93rd percentile, meaning it exceeds 93% of the distribution.\n\nNormal distributions describe data shaped by the accumulation of many small independent variations: adult heights in a population, repeated instrument-measurement errors in a lab, standardized test scores, blood pressure readings across a clinical sample.
Carl Friedrich Gauss derived this distribution in 1809 while modeling residual errors in astronomical observations. He showed that the least-squares estimator is optimal when errors follow this shape. The label normal was adopted later by Francis Galton and Karl Pearson, reflecting how often the shape recurs across natural data.\n\nThe Central Limit Theorem explains the distribution prevalence: the sampling mean of any sufficiently large collection of independent random variables converges to a normal distribution, regardless of the underlying distribution. This result underpins z-tests, t-tests, and the residual assumptions of linear regression.
Theoria Motus Corporum Coelestium , Gauss, C. F. (1809) - First formal derivation of the Gaussian (normal) error distribution
Good for
IQ score articles and percentile calculators
Exam score and standardized test explainers
Manufacturing tolerance and quality control posts
Source & accuracy
This normal distribution is an editorial illustration built to represent the concept accurately. Where it shows figures, they are typical or representative values chosen to make the relationship clear, not a single underlying dataset. The diagram and its explainer are reviewed and maintained centrally, and updated over time as understanding improves.
The ubiquitous bell curve and its sigma bands
The normal (Gaussian) distribution is perhaps the most important probability distribution in statistics. Defined by a mean and standard deviation, it produces the characteristic symmetric bell shape. Shaded bands often indicate standard deviation intervals: one sigma captures 68 percent of the data, two sigma captures 95 percent, and three sigma captures 99.7 percent.
These proportions are empirical rules that hold exactly for normal distributions and approximately for many real datasets. A horizontal percentile callout on the curve highlights a user-specified value, showing what fraction of the population falls below that threshold.
Central limit theorem and prevalence
The normal distribution is ubiquitous because of the central limit theorem: the sum (or average) of many independent random variables, regardless of their individual distributions, tends toward a normal distribution as the number of terms grows. This explains why heights, test scores, and measurement errors across diverse fields often follow normal curves.
Real-world data rarely follows a perfect normal distribution, but the normal is a robust approximation for many purposes. When data deviates significantly (skew, extreme outliers), transformations like log or square-root can normalize it, or specialized distributions (log-normal, Weibull) may fit better.
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Reference
What this is
A free, embeddable, animated normal distribution for any website.
Who uses it
Data journalists, Researchers & academics, Content marketers, News organizations.
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File size
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License
Free forever. Editorial explainer text included; updated centrally over time.
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Frequently asked questions
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