Empirical Rule Calculator (68-95-99.7 Rule)

How to Calculate Using the Empirical Rule

The Empirical Rule, also known as the 68-95-99.7 rule, is a statistical principle used to estimate the probability of data falling within certain ranges in a normal distribution. It's a powerful tool for understanding the spread of data around the mean.

Formula

The Empirical Rule states that for a normal distribution:

• Approximately 68% of the data falls within 1 standard deviation (σ) of the mean (μ)
• Approximately 95% of the data falls within 2 standard deviations (2σ) of the mean (μ)
• Approximately 99.7% of the data falls within 3 standard deviations (3σ) of the mean (μ)

Mathematically, this can be expressed as:

\[ P(\mu - \sigma < X < \mu + \sigma) \approx 0.6827 \]

\[ P(\mu - 2\sigma < X < \mu + 2\sigma) \approx 0.9545 \]

\[ P(\mu - 3\sigma < X < \mu + 3\sigma) \approx 0.9973 \]

Where X is a random variable, μ is the mean, and σ is the standard deviation.

Calculation Steps

1. Determine the mean (μ) and standard deviation (σ) of your dataset
2. Calculate the ranges:
   • For 1σ: (μ - σ) to (μ + σ)
   • For 2σ: (μ - 2σ) to (μ + 2σ)
   • For 3σ: (μ - 3σ) to (μ + 3σ)
3. The probabilities are approximately:
   • 68.27% for 1σ
   • 95.45% for 2σ
   • 99.73% for 3σ

Example

Let's calculate the ranges for a dataset with mean μ = 100 and standard deviation σ = 15:

1. Given: μ = 100, σ = 15
2. Calculate ranges:
   • For 1σ: (100 - 15) to (100 + 15) = 85 to 115
   • For 2σ: (100 - 2 × 15) to (100 + 2 × 15) = 70 to 130
   • For 3σ: (100 - 3 × 15) to (100 + 3 × 15) = 55 to 145
3. Interpret results:
   • Approximately 68.27% of the data falls between 85 and 115
   • Approximately 95.45% of the data falls between 70 and 130
   • Approximately 99.73% of the data falls between 55 and 145

Visual Representation

This diagram illustrates the Empirical Rule, showing the percentages of data falling within 1, 2, and 3 standard deviations of the mean in a normal distribution.