Empirical Rule Calculator with Steps
Learn the empirical rule with detailed step-by-step solutions. Our interactive calculator shows you exactly how to apply the 68-95-99.7 rule with complete explanations for each step.
Step-by-Step Empirical Rule Calculator
Quick Results
Enter values to see results
Understanding Each Step
Step 1: Identify Given Values
First, identify the mean (μ) and standard deviation (σ) from your data or problem statement.
What to look for:
- Mean (μ): The average value of your dataset
- Standard Deviation (σ): A measure of data spread
- Ensure the data follows a normal distribution
Step 2: Calculate Standard Intervals
Calculate the boundaries for 1, 2, and 3 standard deviations from the mean.
Formulas used:
- 1σ: μ ± σ
- 2σ: μ ± 2σ
- 3σ: μ ± 3σ
Step 3: Apply Empirical Rule Percentages
Apply the 68-95-99.7 rule to determine probabilities within each interval.
The rule states:
- 68% of data falls within 1 standard deviation
- 95% of data falls within 2 standard deviations
- 99.7% of data falls within 3 standard deviations
Worked Example
Problem:
A standardized test has a mean score of 500 and a standard deviation of 100. Use the empirical rule to find the percentage of students scoring between 400 and 600.
Step 1: Identify Given Values
μ = 500, σ = 100
Step 2: Calculate the Interval
400 to 600 = 500 ± 100 = μ ± σ
This is exactly 1 standard deviation from the mean
Step 3: Apply Empirical Rule
Since the interval is μ ± σ, approximately 68% of students score between 400 and 600.
Answer:
68% of students score between 400 and 600 points.
Common Mistakes and Tips
Common Mistakes
Mistake 1: Wrong Formula Application
Using μ + σ instead of μ ± σ for interval calculation
Mistake 2: Percentage Confusion
Confusing 68% (within 1σ) with 34% (from mean to 1σ)
Mistake 3: Non-Normal Distribution
Applying the rule to non-normal distributions
Helpful Tips
Tip 1: Check Distribution
Always verify that your data follows a normal distribution
Tip 2: Draw a Diagram
Sketch the normal curve to visualize the problem
Tip 3: Double-Check Calculations
Verify your arithmetic, especially with negative numbers
Practice Problems
Problem 1
Heights of adult males are normally distributed with μ = 70 inches and σ = 3 inches. What percentage of males are between 67 and 73 inches tall?
Step 1: μ = 70, σ = 3
Step 2: 67 to 73 = 70 ± 3 = μ ± σ
Step 3: This is 1 standard deviation, so 68% of males are between 67 and 73 inches tall.
Problem 2
Test scores are normally distributed with μ = 85 and σ = 10. What percentage of students score between 65 and 105?
Step 1: μ = 85, σ = 10
Step 2: 65 to 105 = 85 ± 20 = μ ± 2σ
Step 3: This is 2 standard deviations, so 95% of students score between 65 and 105.
Problem 3
Manufacturing tolerances are normally distributed with μ = 2.5 mm and σ = 0.2 mm. What percentage of products fall between 1.9 mm and 3.1 mm?
Step 1: μ = 2.5, σ = 0.2
Step 2: 1.9 to 3.1 = 2.5 ± 0.6 = μ ± 3σ
Step 3: This is 3 standard deviations, so 99.7% of products fall between 1.9 mm and 3.1 mm.
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