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The Empirical Rule, also known as the 68-95-99.7 Rule, is a statistical principle that applies to normally distributed data. It provides a simple way to understand the spread and clustering of data around the mean. According to this rule:

• Approximately 68% of data falls within one standard deviation ( \(\sigma\) ) of the mean ( \(\mu\) )
• About 95% is within two standard deviations
• Nearly all (99.7%) is within three standard deviations

This rule helps to visualize how data points distribute on a bell curve, describing the probability of a random variable falling within certain ranges. It's particularly useful for solving practical problems when comparing real-world data, like fuel economy estimates, to a theoretical normal model.

Standard deviation is a measure that quantifies the amount of variation or dispersion in a set of data values. A low standard deviation means that the data points are close to the mean, while a high standard deviation indicates a wider spread around the mean. In our problem, the standard deviation of 6.2 miles per gallon (mpg) tells us that the auto fuel economy readings differ from the mean in a range defined by this value.
To calculate standard deviation, consider how far each data point is from the mean, square these differences, average them, and then take the square root. This value helps to understand the confidence level around estimated averages, assessing the likelihood of variation.

In statistics, the mean is the average value of a dataset and serves as a measure of central tendency. It is found by adding all the data points and then dividing by the total number of points. For our exercise, the mean fuel economy is 24.8 mpg.
The mean gives an idea of the overall performance of automobile models in terms of fuel economy. It provides a baseline from which we can understand deviations, showing if certain cars perform significantly better or worse than others. Understanding the mean is crucial in many real-world applications, where it represents an "expected value" around which other results vary.

The 68-95-99.7 Rule is another way to apply the Empirical Rule to a normal distribution's bell curve. It illustrates the reach of standard deviations around the mean for predicting where most data points lie. In our problem:

• 68% of vehicles will achieve a gas mileage between 18.6 and 31.0 mpg (within 1 standard deviation of the mean).
• 95% should fall between 12.4 and 37.2 mpg (within 2 standard deviations).
• 99.7% are expected within 3 standard deviations.

These percentages are tools to forecast data behavior and to determine uncommon events, like gas mileage significantly above or below average (mean). This rule thus helps in making informed predictions and decisions based on statistical patterns.