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The number line may help you understand standard deviation. If we were to put five and seven on a number line, seven is to the right of five. We say, then, that seven is one standard deviation to the right of five because 5 + (1)(2) = 7.
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.
The equation value = mean + (#ofSTDEVs)(standard deviation) can be expressed for a sample and for a population.
If x is a number, then the difference " x – mean" is called its deviation . In a data set, there are as many deviations as there are items in the data set. The deviations are used to calculate the standard deviation. If the numbers belong to a population, in symbols a deviation is x – μ . For sample data, in symbols a deviation is x – $\overline{x}$ .
The procedure to calculate the standard deviation depends on whether the numbers are the entire population or are data from a sample. The calculations are similar, but not identical. Therefore the symbol used to represent the standard deviation depends on whether it is calculated from a population or a sample. The lower case letter s represents the sample standard deviation and the Greek letter σ (sigma, lower case) represents the population standard deviation. If the sample has the same characteristics as the population, then s should be a good estimate of σ .
To calculate the standard deviation, we need to calculate the variance first. The variance is the average of the squares of the deviations (the x – $\overline{x}$ values for a sample, or the x – μ values for a population). The symbol σ ^{2} represents the population variance; the population standard deviation σ is the square root of the population variance. The symbol s ^{2} represents the sample variance; the sample standard deviation s is the square root of the sample variance. You can think of the standard deviation as a special average of the deviations.
If the numbers come from a census of the entire population and not a sample, when we calculate the average of the squared deviations to find the variance, we divide by N , the number of items in the population. If the data are from a sample rather than a population, when we calculate the average of the squared deviations, we divide by n – 1 , one less than the number of items in the sample.
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