Sure! The empirical rule, also known as the 68-95-99.7 rule, applies to any variable whose distribution is approximately normal. Here's a detailed explanation of this rule with the specified properties:
1. Property 1: Approximately 68% of all possible observations lie within one standard deviation to either side of the mean.
- This means that if you consider all the data points in a normal distribution, about 68% of them will fall within the range of the mean (µ) minus one standard deviation (σ) to the mean plus one standard deviation (µ ± 1σ).
2. Property 2: Approximately 95% of all possible observations lie within two standard deviations to either side of the mean.
- Similarly, about 95% of the data points will fall within the range of the mean minus two standard deviations to the mean plus two standard deviations (µ ± 2σ).
3. Property 3: Approximately 99.7% of all possible observations lie within three standard deviations to either side of the mean.
- Finally, about 99.7% of the data points will be within the range of the mean minus three standard deviations to the mean plus three standard deviations (µ ± 3σ).
So, summarizing the empirical rule for a normal distribution:
1. 68% of observations are within one standard deviation (µ ± 1σ) of the mean.
2. 95% of observations are within two standard deviations (µ ± 2σ) of the mean.
3. 99.7% of observations are within three standard deviations (µ ± 3σ) of the mean.
These properties are key to understanding the distribution of data in a normal distribution and can be very useful for statistical analysis and probability estimates.
