Our Methodology
Every calculator on this site is built the same way: from standard statistical formulas, checked by hand, and published only when the numbers are right. This page explains how we build and verify our tools, what this site is for, and where its limits lie.
How our calculators are built
Our tools start with the standard formulas for the normal distribution, not with shortcuts or guesses. The empirical rule itself is the starting point: for data that follows a normal distribution, about 68.27% of values fall within one standard deviation of the mean, 95.45% within two standard deviations, and 99.73% within three. Those are the exact bands our calculators use. The rounded figures 68%, 95% and 99.7% appear in explanations as shorthand, but they are never used in the calculations.
From there, each tool applies the formula it needs. A z-score is calculated as z = (x − μ) / σ, where x is the value, μ is the mean and σ is the standard deviation. Percentiles and probabilities come from the normal distribution, and ranges are calculated as the mean plus or minus one, two or three standard deviations. The formulas come from standard statistical references, including the NIST/SEMATECH e-Handbook of Statistical Methods.
Try the tools this method powers: the Empirical Rule Calculator, the Z-Score Calculator, the Normal Distribution Calculator and the Bell Curve Generator.
How we verify every tool before publishing
No tool is published until it has been tested with hand-verified values. For each calculator, we work through examples by hand first — for instance, with a mean of 100 and a standard deviation of 15, the range from 85 to 115 should contain 68.27% of values — and then check that the tool returns the same result. We test the edges too: very small and very large inputs, zero, and values far from the mean, so the tool behaves sensibly at the extremes.
When a tool changes, we test it again before the change goes live. And when a reader points out a possible error, we check it, and fix the tool if the reader is right. If you find a number that looks wrong, please tell us through the contact page.
Tools checked this way include the Standard Deviation Calculator, the Empirical Rule Percentile Calculator and the Empirical Rule Graph Generator.
What this site is for
This site is for stats, to help whoever needs it. It is a free set of tools for students working through a statistics class, teachers who need a quick check, and anyone exploring data — no sign-up, no subscription and no paywall. The site is built and run by Afan Shafiq, a BS AI student, who also writes the guides on the site. You can read more about the site and its author on the About page.
What our tools cannot do
Our calculators are an educational tool, not a substitute for professional statistical advice. The empirical rule is exact only for data that is, at least approximately, normally distributed. With skewed data, heavy outliers or very small samples, the mean and standard deviation can mislead, and the bands will not match the data. In those cases, other methods — for example Chebyshev’s theorem, which works for any distribution — may be more honest.
For graded work, use our tools to learn and to check your working, then check your answer with a teacher or your textbook. If a result matters — a grade, a report, a decision — confirm it from a second source.
Frequently asked questions
How accurate are the calculators?
They use the exact empirical rule bands (68.27%, 95.45% and 99.73%) and standard formulas, so results are precise for normally distributed data.
Why do you say 68.27% instead of 68%?
68% is a rounded shorthand. The exact value is 68.27%, and using the exact value keeps the calculations precise. The same applies to 95.45% and 99.73%.
Are the tools free?
Yes. Every tool on the site is free, with no sign-up, no subscription and no paywall.
Can I use the results for homework or exams?
Yes, to learn and to check your working — but for graded work, check your answer with a teacher or your textbook, as explained in “What our tools cannot do” above.
Who builds the tools?
Afan Shafiq, a BS AI student, builds and runs this site and writes its guides.
Check Tools
Put the method above to work. Explore all statistical tools on the site, including the Empirical Rule Calculator, the Bell Curve Generator, the Z-Score Calculator, the Normal Distribution Calculator and the Standard Deviation Calculator. Every tool is free, with no sign-up.
Check Articles
Learn the ideas behind the tools: What Is the Empirical Rule?, How to Use the Empirical Rule Calculator, Empirical Rule vs Chebyshev’s Theorem, Applications of the Empirical Rule, and Normal Distribution and the Empirical Rule.