Using the 6895997 Rule for Deviating Heights

Given the average height of an American adult male is 5'10" with a standard deviation of 3 inches, use the 68-95-99.7 rule to determine the percentage of men that deviate more than 9 inches from the average height.

The problem presented is a classic application of the empirical rule, commonly known as the 68-95-99.7 rule, which is a shortcut to understanding the distribution of data in a normal distribution. Specifically, this rule refers to the percentage of values that lie within a band around the mean in a normal distribution. In a normal distribution, roughly 68% of data points fall within one standard deviation of the mean, 95% fall within two standard deviations, and 99.7% fall within three standard deviations. These rough percentages help researchers and statisticians quickly understand the spread and potential outliers of the data set without requiring computational tools to determine the exact percentiles.

In this scenario, height is assumed to be normally distributed, a good example of several biological measurements which often fit a normal distribution pattern due to the Central Limit Theorem. Given the average height and standard deviation, the problem asks to determine the percentage of men who deviate more than a certain number of inches from the mean height. This is a practical way of using the empirical rule to estimate the proportion of data that lies outside the standard deviations specified, allowing us to identify the rarity or commonality of the height deviation in the population. Understanding how to use this rule effectively enables quick estimates for decision-making in statistical inferences.