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Mathematics for Technology I (Math 1131)

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Durham College, Mathematics
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Variance and Standard Deviation

The standard deviation (SD) is a measure of the spread of the data (how far from the normal it is). A number such as the mean or the standard deviation may be found either for an entire population (symbolized as σ or σx) or for a sample (symbolized as s) drawn from that population.

Basically, when your standard deviation, it means that the values in a statistical data set are close to the mean of the data set, on average, and a large standard deviation means that the values in the data set are farther away from the mean, on average.

The variance (symbolized s² or σ²) is defined as the average of the squared differences from the mean. In other words, once you find the mean, you subtract the mean from each sample, and square that number. You then take the average of these squared numbers by adding them up and dividing by the number of observations. The formula is shown below for a population:

s2=i=1nxi2n   or   σ2=i=1nxi2N   
  • Keep in mind that and n is used for sample populations while σ² and N for the entire population.

Once we have the variance, it is a simple matter to get the standard deviation (s or σ – notice the lack of the power).  As mentioned earlier, SD is the most common measure of dispersion.

s=i=1nxi2n   or   σ=i=1nxi2N

Let’s look at an example where variance and standard deviation are calculated:

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