To calculate the standard deviation, we need to calculate the variance first. If the sample has the same characteristics as the population, then s should be a good estimate of σ. The lower case letter s represents the sample standard deviation and the Greek letter σ (sigma, lower case) represents the population standard deviation. Therefore the symbol used to represent the standard deviation depends on whether it is calculated from a population or a sample. The calculations are similar, but not identical. The procedure to calculate the standard deviation depends on whether the numbers are the entire population or are data from a sample. For sample data, in symbols a deviation is x –. If the numbers belong to a population, in symbols a deviation is x – μ. The deviations are used to calculate the standard deviation. In a data set, there are as many deviations as there are items in the data set. If x is a number, then the difference “ x – mean” is called its deviation. If one were also part of the data set, then one is two standard deviations to the left of five because 5 + (–2)(2) = 1. We say, then, that seven is one standard deviation to the right of five because 5 + (1)(2) = 7. If we were to put five and seven on a number line, seven is to the right of five. The number line may help you understand standard deviation. (You will learn more about this in later chapters.) In general, the shape of the distribution of the data affects how much of the data is further away than two standard deviations. Considering data to be far from the mean if it is more than two standard deviations away is more of an approximate “rule of thumb” than a rigid rule. A data value that is two standard deviations from the average is just on the borderline for what many statisticians would consider to be far from the average.
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