2.7 Measures of the spread of the data  (Page 2/25)

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.

• In general, a value = mean + (#ofSTDEV)(standard deviation)
• where #ofSTDEVs = the number of standard deviations
• #ofSTDEV does not need to be an integer
• One is two standard deviations less than the mean of five because: 1 = 5 + (–2)(2).

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

• sample: $x\text{ = }\overline{x}\text{ + }(\#ofSTDEV)(s)$
• Population: $x=\mu +(\#ofSTDEV)(\sigma )$

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 – $\overline{x}$ .

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.

Formulas for the sample standard deviation

• $s=\sqrt{\frac{\Sigma {(x-\overline{x})}^2}{n-1}}$ or $s=\sqrt{\frac{\Sigma f{(x-\overline{x})}^2}{n-1}}$
• For the sample standard deviation, the denominator is n - 1 , that is the sample size MINUS 1.

Formulas for the population standard deviation

• $\sigma = \sqrt{\frac{\Sigma {(x-\mu )}^2}{N}}$ or $\sigma = \sqrt{\frac{\Sigma f{(x–\mu )}^2}{N}}$
• For the population standard deviation, the denominator is N , the number of items in the population.