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Standard deviation

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For other uses, see Standard deviation (disambiguation).

A plot of normal distribution (or bell-shaped curve) where each band has a width of 1 standard deviation – See also: 68–95–99.7 rule

Cumulative probability of a normal distribution with expected value 0 and standard deviation 1.

In statistics, the standard deviation (SD, also represented by the Greek letter sigma σ or the Latin letter s) is a measure that is used to quantify the amount of variation or dispersion of a set of data values.^[1] A low standard deviation indicates that the data points tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.
The standard deviation of a random variable, statistical population, data set, or probability distribution is the square root of its variance. It is algebraically simpler, though in practice less robust, than the average absolute deviation.^[2][3] A useful property of the standard deviation is that, unlike the variance, it is expressed in the same units as the data. There are also other measures of deviation from the norm, including average absolute deviation, which provide different mathematical properties from standard deviation.^[4]
In addition to expressing the variability of a population, the standard deviation is commonly used to measure confidence in statistical conclusions. For example, the margin of error in polling data is determined by calculating the expected standard deviation in the results if the same poll were to be conducted multiple times. This derivation of a standard deviation is often called the "standard error" of the estimate or "standard error of the mean" when referring to a mean. It is computed as the standard deviation of all the means that would be computed from that population if an infinite number of samples were drawn and a mean for each sample were computed. It is very important to note that the standard deviation of a population and the standard error of a statistic derived from that population (such as the mean) are quite different but related (related by the inverse of the square root of the number of observations). The reported margin of error of a poll is computed from the standard error of the mean (or alternatively from the product of the standard deviation of the population and the inverse of the square root of the sample size, which is the same thing) and is typically about twice the standard deviation—the half-width of a 95 percent confidence interval. In science, many researchers report the standard deviation of experimental data, and only effects that fall much farther than two standard deviations away from what would have been expected are considered statistically significant—normal random error or variation in the measurements is in this way distinguished from likely genuine effects or associations. The standard deviation is also important in finance, where the standard deviation on the rate of return on an investment is a measure of the volatility of the investment.
When only a sample of data from a population is available, the term standard deviation of the sample or sample standard deviation can refer to either the above-mentioned quantity as applied to those data or to a modified quantity that is an unbiased estimate of the population standard deviation (the standard deviation of the entire population).

Contents

 [hide] 

• 1 Basic examples
  • 1.1 Sample standard deviation of metabolic rate of Northern Fulmars
  • 1.2 Population standard deviation of grades of eight students
  • 1.3 Standard deviation of average height for adult men
• 2 Definition of population values
  • 2.1 Discrete random variable
  • 2.2 Continuous random variable
• 3 Estimation
  • 3.1 Uncorrected sample standard deviation
  • 3.2 Corrected sample standard deviation
  • 3.3 Unbiased sample standard deviation
  • 3.4 Confidence interval of a sampled standard deviation
• 4 Identities and mathematical properties
• 5 Interpretation and application
  • 5.1 Application examples
    • 5.1.1 Experiment, industrial and hypothesis testing
    • 5.1.2 Weather
    • 5.1.3 Finance
  • 5.2 Geometric interpretation
  • 5.3 Chebyshev's inequality
  • 5.4 Rules for normally distributed data
• 6 Relationship between standard deviation and mean
  • 6.1 Standard deviation of the mean
• 7 Rapid calculation methods
  • 7.1 Weighted calculation
• 8 History
• 9 See also
• 10 References
• 11 External links

Basic examples[edit]

Sample standard deviation of metabolic rate of Northern Fulmars[edit]

Logan ^[5] gives the following example. Furness and Bryant ^[6] measured the resting metabolic rate for 8 male and 6 female breeding Northern fulmars. The table shows the Furness data set.

Furness data set on metabolic rates of Northern fulmars

Sex	Metabolic rate	Sex	Metabolic rate
Male	525.8	Female	727.7
Male	605.7	Female	1086.5
Male	843.3	Female	1091.0
Male	1195.5	Female	1361.3
Male	1945.6	Female	1490.5
Male	2135.6	Female	1956.1
Male	2308.7
Male	2950.0

The graph shows the metabolic rate for males and females. By visual inspection, it appears that the variability of the metabolic rate is greater for males than for females.

The sample standard deviation of the metabolic rate for the female fulmars is calculated as follows. The formula for the sample standard deviation is

${\displaystyle s={\sqrt {\frac {\sum _{i=1}^{N}(x_{i}-{\overline {x}})^{2}}{N-1}}}.}$

where ${\displaystyle \scriptstyle \{x_{1},\,x_{2},\,\ldots ,\,x_{N}\}}$ are the observed values of the sample items, ${\displaystyle \scriptstyle {\overline {x}}}$ is the mean value of these observations, and N is the number of observations in the sample.
In the sample standard deviation formula, for this example, the numerator is the sum of the squared deviation of each individual animal's metabolic rate from the mean metabolic rate. The table below shows the calculation of this sum of squared deviations for the female fulmars. For females, the sum of squared deviations is 886047.09, as shown in the table.

Sum of squares calculation for female fulmars

Animal	Sex	Metabolic rate	Mean	Difference from mean	Squared difference from mean
1	Female	727.7	1285.5	-557.8	311140.84
2	Female	1086.5	1285.5	-199.0	39601.00
3	Female	1091.0	1285.5	-194.5	37830.25
4	Female	1361.3	1285.5	75.8	5745.64
5	Female	1490.5	1285.5	205.0	42025.00
6	Female	1956.1	1285.5	670.6	449704.36
	Mean =	1285.5		Sum of squared differences =	886047.09

The denominator in the sample standard deviation formula is N – 1, where N is the number of animals. In this example, there are N = 6 females, so the denominator is 6 – 1 = 5. The sample standard deviation for the female fulmars is therefore

${\displaystyle s={\sqrt {\frac {\sum _{i=1}^{N}(x_{i}-{\overline {x}})^{2}}{N-1}}}={\sqrt {\frac {886047.09}{5}}}=420.96.}$

For the male fulmars, a similar calculation gives a sample standard deviation of 894.37, approximately twice as large as the standard deviation for the females. The graph shows the metabolic rate data, the means (red dots), and the standard deviations (red lines) for females and males.