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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
$\mathrm{5\; +\; (1)(2)\; =\; 7}$ .
If 1 were also part of the data set, then 1 is
two standard deviations to the
left of 5 because
$\mathrm{5\; +\; (-2)(2)\; =\; 1}$ .
The equation value = mean + (#ofSTDEVs)(standard deviation) can be expressed for a sample and for a population:
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 $\sigma $ (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 $\sigma $ .
To calculate the standard deviation, we need to calculate the variance first. The variance is an average of the squares of the deviations (the $x-$ $\overline{x}$ values for a sample, or the $x-\mu $ values for a population). The symbol ${\sigma}^{2}$ represents the population variance; the population standard deviation $\sigma $ 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.
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