Empirical Rule (Leaving Cert Mathematics): Revision Notes

Empirical Rule

Overview

The Empirical Rule, also known as the 68-95-99.7 Rule, applies to bell-shaped, normal distributions. It provides a quick way to estimate the proportion of data that falls within one, two, or three standard deviations of the mean.

Key Proportions in the Empirical Rule

Within 1 Standard Deviation:

• Approximately 68% of the data falls within $\mu \pm \sigma$

Within 2 Standard Deviations:

• Approximately 95% of the data falls within $\mu \pm 2\sigma$

Within 3 Standard Deviations:

• Approximately 99.7% of the data falls within $\mu \pm 3\sigma$ Where:

• $\mu$: Mean of the data.

• $\sigma$: Standard deviation.

Applications of the Empirical Rule

1. Quick Estimation:

• Helps assess the spread and density of data in a normal distribution.

2. Outlier Detection:

• Data points beyond $\mu \pm 3\sigma$ are often considered outliers.

3. Prediction:

• Used to predict probabilities in scenarios with normal distributions.

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Worked Examples

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Summary

• The Empirical Rule applies to normal distributions and is summarised as:
  • 68% of the data falls within 1 standard deviation ($\mu \pm \sigma$).
  • 95% of the data falls within 2 standard deviations ($\mu \pm 2\sigma$).
  • 99.7% of the data falls within 3 standard deviations ($\mu \pm 3\sigma$).
• Useful for:
  • Estimating probabilities in a normal distribution.
  • Detecting outliers.
• Quick and practical for interpreting data sets with normal distributions.