Empirical Rule Calculator
Visualize the 68-95-99.7 rule for any normal distribution.
Distribution Results
68% Range
-
95% Range
-
99.7% Range
-
This tool is for informational purposes and assumes a normal distribution.
About the Empirical Rule Calculator
The Empirical Rule, also known as the 68-95-99.7 rule, is a fundamental concept in statistics for understanding normally distributed data. This calculator allows you to input a mean and standard deviation and instantly see the data ranges where 68%, 95%, and 99.7% of your data points are expected to fall, complete with a visual bell curve representation.
Formula Explained
The calculator applies the three core tenets of the Empirical Rule:
- 68% of data falls within 1 standard deviation of the mean: μ ± 1σ
- 95% of data falls within 2 standard deviations of the mean: μ ± 2σ
- 99.7% of data falls within 3 standard deviations of the mean: μ ± 3σ
How to Interpret the Results
The Empirical Rule provides a quick way to check for normalcy and identify potential outliers:
Check for Normality
If your actual data closely matches the 68-95-99.7% predictions, it's a strong indication that your dataset is normally distributed.
Identify Outliers
Any data point that falls outside of 3 standard deviations (the 99.7% range) is extremely rare and can be considered a potential outlier worth investigating.
Frequently Asked Questions
What is the Empirical Rule? →
The Empirical Rule, also known as the 68-95-99.7 rule, is a statistical rule which states that for a normal distribution, almost all data falls within three standard deviations of the mean. Specifically, about 68% of the data falls within one standard deviation, 95% within two, and 99.7% within three.
When can you use the Empirical Rule? →
The Empirical Rule can only be used when the data is confirmed to follow a normal distribution, meaning it is symmetric and bell-shaped. It is a quick way to get an overview of the data's spread without performing more complex calculations. It is not applicable for datasets that are skewed or not bell-shaped.
How do you calculate the 68% range? →
The 68% range is found by taking one standard deviation and adding it to and subtracting it from the mean. The formula is: [Mean - 1 × Standard Deviation] to [Mean + 1 × Standard Deviation]. This range contains the middle 68% of all data points in the distribution.
What is the difference between the Empirical Rule and Chebyshev's Inequality? →
The Empirical Rule only applies to normal (bell-shaped) distributions and provides precise percentages (68%, 95%, 99.7%). Chebyshev's Inequality is more general and can be applied to any distribution, regardless of its shape. However, it provides more conservative, less precise estimates (e.g., it guarantees that at least 75% of data is within two standard deviations, compared to the Empirical Rule's 95%).
What are some real-world examples of the Empirical Rule? →
The Empirical Rule applies to many real-world phenomena that follow a normal distribution. Examples include heights of a population, blood pressure readings, IQ scores, and measurement errors in scientific experiments. For instance, if the average IQ is 100 with a standard deviation of 15, the rule tells us that 95% of the population has an IQ between 70 and 130.