1. What is the 68-95-99.7 Rule?

The 68-95-99.7 rule (Empirical Rule) describes the percentage of values that lie within certain standard deviations from the mean in a normal distribution. For normally distributed data:

• ≈68% of data falls within ±1σ of the mean
• ≈95% falls within ±2σ
• ≈99.7% falls within ±3σ

2. How the Calculator Works

The calculator determines what percentage of values fall below your specified value in a normal distribution:

Where:

• \( z \) — Standard score (how many σ from the mean)
• \( X \) — The value you're evaluating
• \( \mu \) — Mean of the distribution
• \( \sigma \) — Standard deviation

The calculator then uses the standard normal distribution to find the percentage of values below your specified value.

3. Importance of the Empirical Rule

Details: This rule is fundamental in statistics for understanding data distributions, identifying outliers, and making predictions about normally distributed data.

4. Using the Calculator

Tips: Enter the mean (average) of your distribution, the standard deviation (measure of spread), and the value you want to evaluate. The calculator will show what percentage of values fall below your specified value.

5. Frequently Asked Questions (FAQ)

Q1: When does the 68-95-99.7 rule apply?
A: Only to perfectly normal distributions. Many real-world distributions are approximately normal.

Q2: What if my data isn't normally distributed?
A: The percentages won't be accurate. Consider transforming your data or using non-parametric methods.

Q3: How precise is this calculator?
A: It provides a good approximation for most purposes. For exact values, use statistical software.

Q4: What does a z-score tell me?
A: How many standard deviations a value is from the mean. Positive = above mean, negative = below.

Q5: Can I use this for quality control?
A: Yes, it's commonly used in process control to identify when values fall outside expected ranges.