The most common measure of dispersion, or how spread out the data are from the mean. While the range estimates the spread of the data by subtracting the minimum value from the maximum value, the standard deviation roughly estimates the "average" distance of the individual observations from the mean. The greater the standard deviation, the greater the spread in the data.
Standard deviation can be used as a preliminary benchmark for estimating the overall variation of a process. For example, administrators track the discharge time for patients treated in the emergency departments of two hospitals. Although the average discharge times are about the same (35 minutes), the standard deviations are significantly different.
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Hospital 1 The standard deviation is about 6. On average, a patient's discharge time deviates from the mean (blue line) by about 6 minutes. |
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Hospital 2 The standard deviation is about 20. On average, a patient's discharge time deviates from the mean (blue line) by about 20 minutes. |
The standard deviation is calculated by taking the positive square root of the variance, another measure of data dispersion. Standard deviation is often more convenient and intuitive to work with, however, because it uses the same units as the data. For example, if a machine part is weighed in grams, the standard deviation of its weight is also calculated in grams, while its variance is calculated in grams2.
In a normal (bell-shaped) distribution, successive standard deviations from the mean provide useful benchmarks for estimating the percentage of data observations.
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About 95% of the observations fall within 2 standard deviations of the mean, shown by the blue shaded area.
About 68% of the observations fall within 1 standard deviation from the mean (-1 to +1), and about 99.7% of the observations would fall within 3 standard deviations of the mean (-3 to +3).
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The symbol s (sigma) is often used to represent the standard deviation of a population, while s is used to represent the standard deviation of a sample.
Variation that is not random or natural to a process is often referred to as noise.
