Calculadora de Regra Empírica (68-95-99,7)
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
μ ± kσ (k = 1, 2, 3)
About this calculator
The Empirical Rule Calculator (68-95-99.7) computes the intervals around the mean that contain approximately 68%, 95%, and 99.7% of the data in a normal distribution. Simply enter the mean (μ) and standard deviation (σ). The output shows the lower and upper bounds for each k value (1, 2, and 3).
The operation is straightforward: for each k, the interval is [μ - kσ, μ + kσ]. For example, with μ = 100 and σ = 15, we get: for k=1, [85, 115]; for k=2, [70, 130]; for k=3, [55, 145]. This reflects the property that in a normal distribution, 68% of data lies within one standard deviation, 95% within two, and 99.7% within three.
Use this calculator in contexts such as test score analysis, quality control, population heights, or any dataset that is approximately normally distributed. It helps understand data spread and identify outliers (beyond 3σ).
Caution: the empirical rule is accurate only for approximately normal distributions. If data are highly skewed or heavy-tailed, actual percentages may differ. Always check normality before applying the rule.
Frequently asked questions
Does the empirical rule work for any distribution?
No, it is accurate only for approximately normal (bell-shaped) distributions. For other distributions, the percentages may differ.
What does a value beyond 3σ mean?
A value beyond 3σ is considered rare (less than 0.3% of data) and may be an outlier. In many contexts, it indicates an atypical value or error.
Can I use the empirical rule with small samples?
Yes, but accuracy depends on data normality. For very small samples (n < 30), the approximation may be less reliable.
How can I tell if my data is normally distributed?
You can use plots like histograms or Q-Q plots, or statistical tests like Shapiro-Wilk. The calculator assumes normality.
Does the calculator work with negative mean or standard deviation?
Yes, the mean can be negative and the standard deviation must be positive. Intervals are calculated normally.