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The number line may help you understand standard deviation. If we were to put 5 and 7 on a number line, 7 is to the right of 5. We say, then, that 7 is one standard deviation to the right of 5 because
5 + (1)(2) = 7 .

If 1 were also part of the data set, then 1 is two standard deviations to the left of 5 because
5 + (-2)(2) = 1 .

A number line labeled from 0 to 7.

  • In general, a value = mean + (#ofSTDEV)(standard deviation)
  • where #ofSTDEVs = the number of standard deviations
  • 7 is one standard deviation more than the mean of 5 because: 7=5+ (1) (2)
  • 1 is two standard deviations less than the mean of 5 because: 1=5+ (−2) (2)

The equation value = mean + (#ofSTDEVs)(standard deviation) can be expressed for a sample and for a population:

  • sample: x = x + (#ofSTDEV)(s)
  • Population: x = μ + (#ofSTDEV)(σ)
The lower case letter s represents the sample standard deviation and the Greek letter σ (sigma, lower case) represents the population standard deviation.

The symbol x is the sample mean and the Greek symbol μ is the population mean.

Calculating the standard deviation

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 - 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 an average of the squares of the deviations (the x - 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. You can see that in the formulas below.

Formulas for the sample standard deviation

  • s = size 12{s={}} {} Σ ( x x ¯ ) 2 n 1 or s = size 12{s={}} {} Σ f · ( x x ¯ ) 2 n 1
  • For the sample standard deviation, the denominator is n-1 , that is the sample size MINUS 1.

    Formulas for the population standard deviation

  • σ = size 12{σ={}} {} Σ ( x μ ¯ ) 2 N or σ = size 12{σ={}} {} Σ f · ( x μ ¯ ) 2 N
  • For the population standard deviation, the denominator is N , the number of items in the population.

Practice Key Terms 2

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Source:  OpenStax, Collaborative statistics-parzen remix. OpenStax CNX. Jul 15, 2009 Download for free at http://legacy.cnx.org/content/col10732/1.2
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