Certainly! The empirical rule, also known as the 68-95-99.7 rule, provides specifics about the distribution of data in a normal distribution. Here is the detailed breakdown:
For any variable whose distribution is approximately normal (bell-shaped):
- Property 1: Approximately 68% of all possible observations lie within one standard deviation to either side of the mean. This means if you plot the variable's distribution on a graph, around 68% of the data falls within the interval [tex]\((\mu - \sigma, \mu + \sigma)\)[/tex], where [tex]\(\mu\)[/tex] is the mean and [tex]\(\sigma\)[/tex] is the standard deviation.
- Property 2: Approximately 95% of all possible observations lie within two standard deviations to either side of the mean. In graphical terms, about 95% of the data will be within the interval [tex]\((\mu - 2\sigma, \mu + 2\sigma)\)[/tex].
- Property 3: Approximately 99.7% of all possible observations lie within three standard deviations to either side of the mean. This translates to nearly all the data (99.7%) being contained within the interval [tex]\((\mu - 3\sigma, \mu + 3\sigma)\)[/tex].
These properties help in understanding how data is spread around the mean and are especially useful in fields like statistics and quality control where normal distribution assumptions often apply.
