Answer :
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.
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.
