For example, the standard deviation of a random variable which follows a Cauchy distribution is undefined because its expected value is undefined. Not all random variables have a standard deviation, since these expected values need not exist. The standard deviation of a ( univariate) probability distribution is the same as that of a random variable having that distribution. In the case where X takes random values from a finite data set, with each value having the same probability, the standard deviation is That is, the standard deviation РџС“ ( sigma) is the square root of the average value of ( XР’В РІв‚¬вЂ™ РћС) 2.

Then the standard deviation of X is the quantity Here the operator E denotes the average or expected value of X. Let X be a random variable with mean value РћС: See the section Estimation below for an explanation.Р”Вefinition Probability distribution or random variable If the 8 values are obtained by random sampling from some parent population, then computing the sample standard deviation would use a denominator of 7 instead of 8. Therefore, the above has a population standard deviation of 2. Next divide the sum of these values by the number of values and take the square root to give the standard deviation: To calculate the population standard deviation, first compute the difference of each data point from the mean, and square the result: There are eight data points in total, with a mean (or average) value of 5:

6 Relationship between standard deviation and meanР”В¬onsider a population consisting of the following values:.

5.4 Rules for normally distributed data.

4 Identities and mathematical properties.

3.1 With standard deviation of the sample.

2.1 Probability distribution or random variable.

When only a sample of data from a population is available, the population standard deviation can be estimated by a modified quantity called the sample standard deviation, explained below. Note, however, that for measurements with percentage as unit, the standard deviation will have percentage points as unit. A useful property of standard deviation is that, unlike variance, it is expressed in the same units as the data. This was as a replacement for earlier alternative names for the same idea: for example Gauss used 'mean error'. The term standard deviation was first used in writing by Karl Pearson in 1894, following his use of it in lectures. 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. In science, researchers commonly report the standard deviation of experimental data, and only effects that fall far outside the range of standard deviation are considered statistically significant-normal random error or variation in the measurements is in this way distinguished from causal variation. The reported margin of error is typically about twice the standard deviation РІР‚вЂњ the radius of a 95% confidence interval. 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. In addition to expressing the variability of a population, standard deviation is commonly used to measure confidence in statistical conclusions. Three standard deviations account for 99% of the sample population being studied, assuming the distribution is normal (bell-shaped). If the standard deviation were 20ins, then men would have much more variable heights, with a typical range of about 50to90in. If the standard deviation were zero, then all men would be exactly 70ins high. This means that most men (about 68 percent, assuming a normal distribution) have a height within 3insn}} of the mean (67-73in}) РІР‚вЂњ one standard deviation, whereas almost all men (about 95%) have a height within 6in of the mean 64-76in РІР‚вЂњ 2 standard deviations.

A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values.Р”Р‡or example, the average height for adult men in the United States is about 70ins}}, with a standard deviation of around 3|ins. It shows how much variation there is from the 'average' (mean). Standard deviation is a widely used measure of the variability or dispersion, being algebraically more tractable though practically less robust than the expected deviation or average absolute deviation. In probability theory and statistics, the standard deviation of a statistical population, a data set, or a probability distribution is the square root of its variance. File:Standard deviation illustration.gifР”Р„ data set with a mean of 50 (shown in blue) and a standard deviation (РџС“) of 20. Each colored band has a width of one standard deviation. A plot of a normal distribution (or bell curve).

Error-weighted standard deviation