1. What is the Empirical Rule?

The Empirical Rule, also known as the 68-95-99.7 rule, describes the percentage of values that lie within a certain number of standard deviations from the mean in a normal distribution. This calculator computes the exact percentage for any number of standard deviations.

2. How Does the Calculator Work?

The calculator uses the Empirical Rule formula:

Where:

• \( P \) — Percentage of data within ±k standard deviations
• \( k \) — Number of standard deviations from the mean
• \( \text{erf} \) — Gauss error function

Explanation: The error function calculates the area under the normal distribution curve between -k and +k standard deviations from the mean.

3. Importance of the Empirical Rule

Details: The Empirical Rule is fundamental in statistics for understanding data distribution, identifying outliers, and making predictions in normally distributed datasets.

4. Using the Calculator

Tips: Enter the number of standard deviations (k) as a positive number. Common values are 1 (68.27%), 2 (95.45%), and 3 (99.73%).

5. Frequently Asked Questions (FAQ)

Q1: What does the Empirical Rule tell us?
A: It describes what percentage of values in a normal distribution fall within certain distances from the mean, measured in standard deviations.

Q2: Why is it called the 68-95-99.7 rule?
A: These are the approximate percentages for 1, 2, and 3 standard deviations respectively (68.27%, 95.45%, 99.73%).

Q3: Does this work for non-normal distributions?
A: No, the Empirical Rule specifically applies to normal (bell-shaped) distributions. Other distributions require different approaches.

Q4: What is the error function (erf)?
A: The error function is a special function that measures the area under the normal distribution curve between two points.

Q5: How accurate is this calculator?
A: The calculator uses a precise approximation of the error function, providing results accurate to several decimal places.