Question
Compute the standard deviation.
Answer :
Word Count : 1145
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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 | 0 | | 16 | 2 | 4 | | 18 | 4 | 16 | | Total | 0 | 40 | Therefore, [ \sigma=\sqrt{\frac{40}{5}} ] [ \sigma=\sqrt{8} ] [ \sigma=2.828 ] Thus, the standard deviation is approximately 2.83. In business statistics, data are often presented in the form of discrete or frequency distributions. When frequencies are given, the standard deviation can be computed using: [ \sigma=\sqrt{\frac{\sum f(X-\bar X)^2}{N}} ] Here, (f) represents the frequency of each observation and (N=\sum f). Suppose the following data represent the number of units sold by a business on different days: | Units Sold (X) | Frequency (f) | | -------------- | ------------: | | 10 | 2 | | 20 | 3 | | 30 | 4 | | 40 | 3 | | 50 | 2 | The first step is to calculate (fX): | X | f | fX | | --------- | -----: | ------: | | 10 | 2 | 20 | | 20 | 3 | 60 | | 30 | 4 | 120 | | 40 | 3 |
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