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Standard deviation is one of the most important measures of dispersion in Business Mathematics and Statistics. It measures the extent to which individual observations in a data set differ from their arithmetic mean. While the mean indicates the central or average value of a series, standard deviation indicates how closely or widely the observations are distributed around that mean. A small standard deviation indicates that the observations are concentrated near the mean, whereas a large standard deviation indicates greater variability. Standard deviation is generally denoted by the Greek letter sigma, σ, for a population and by s for a sample. It is defined as the positive square root of the arithmetic mean of the squared deviations from the arithmetic mean. The basic formula for an individual or ungrouped series is: [ \sigma=\sqrt{\frac{\sum (X-\bar X)^2}{N}} ] where (X) represents each observation, (\bar X) represents the arithmetic mean, and (N) represents the total number of observations. For example, consider the following observations: [ 10,\ 12,\ 14,\ 16,\ 18 ] First, calculate the arithmetic mean: [ \bar X=\frac{10+12+14+16+18}{5} ] [ \bar X=\frac{70}{5}=14 ] Now calculate the deviation of each observation from the mean and square each deviation. | X | (X-\bar X) | ((X-\bar X)^2) | | --------- | ---------: | -------------: | | 10 | -4 | 16 | | 12 | -2 | 4 | | 14 | 0 | ________ _____ ________ ____ ____ ____ _______.
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