Empirical Rule Calculator

About this tool

What Is the empirical rule? Empirical Rule Calculator. The empirical rule calculator applies the 68-95-99.7 guideline to a given mean and standard deviation, reporting the share of data inside 1, 2, and 3 standard deviations as concrete ranges. For a bell-shaped distribution, about 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three. This holds for any normal or approximately normal dataset. The rule is a fast sanity check: a value more than 3 deviations from the mean is remarkable, while roughly one in twenty values falls beyond two deviations. P(μ±1σ) ≈ 68%, P(μ±2σ) ≈ 95%, P(μ±3σ) ≈ 99.7%. How to Use Enter the mean μ and standard deviation σ of your data. Read the ranges for 1, 2, and 3 deviations with their percentages. If you also enter a specific value x, the tool reports its z-score and which ranges it falls inside. Worked Example With mean 100 and standard deviation 15, the tool reports that 68% of values sit between 85 and 115, 95% between 70 and 130, and 99.7% between 55 and 145. Entering x = 115 gives z = 1. Common Use Cases exam score distributions and bell curves quality-process capability checks sampling distribution reasoning explaining variability to students Important Notes The rule presumes normality; for skewed data it is only approximate. The z-score uses the same mean and σ you entered. All computation is local