The standard deviation of a ( univariate) probability distribution is the same as that of a random variable having that distribution. That is, the standard deviation σ ( sigma) is the square root of the variance of X, i.e., it is the square root of the average value of ( X − μ) 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 μ: Three standard deviations account for 99.7% of the sample population being studied, assuming the distribution is normal (bell-shaped). If the standard deviation were 20", then men would have much more variable heights, with a typical range of about 50"–90". If the standard deviation were zero, then all men would be exactly 70" tall. This means that most men (about 68%, assuming a normal distribution) have a height within 3" of the mean (67"–73") - one standard deviation - and almost all men (about 95%) have a height within 6" of the mean (64"–76") - two standard deviations. See the section Estimation below for more details.Ī slightly more complicated real life example, the average height for adult men in the United States is about 70", with a standard deviation of around 3". If they instead were a random sample, drawn from some larger, "parent" population, then we should have used 7 (which is n − 1) instead of 8 (which is n) in the denominator of the last formula, and then the quantity thus obtained would have been called the sample standard deviation. The formula is valid only if the eight values we began with form the complete population. This quantity is the population standard deviation it is equal to the square root of the variance. Next compute the average of these values, and take the square root: To calculate the population standard deviation, first compute the difference of each data point from the mean, and square the result of each: These eight data points have the mean (average) of 5: Basic examplesĬonsider a population consisting of the following eight values: Which can be verbally stated as The variance is the mean of the squares minus the square of the mean. Īnd the general formula for the standard deviation becomes. These expressions are generalized to the case where there are n numbers involved: a 1, a 2. Their mean number μ is the mid point and the standard deviation σ is the distanceįrom each of the numbers to μ. The population standard deviation can be estimated by a modified quantityĬalled the sample standard deviation, explained below. When only a sample of data from a population is available, 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,Īnd 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 – the radius of a 95 percent confidence interval. The reported margin of error is typically about twice the standard deviation If the same poll were to be conducted multiple times. Standard deviation is commonly used to measure confidence in statistical conclusions.įor example, the margin of error in polling data is determinedīy calculating the expected standard deviation in the results In addition to expressing the variability of a population, It is expressed in the same units as the data. A useful property of standard deviation is that, unlike variance, It is algebraically simpler though practically less robust than the average absolute deviation. The standard deviation of a random variable, statistical population, data set, or probability distribution is the square root of its variance. Or " dispersion " exists from the average ( mean ,Ī low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data points are spread out over a large range of values. Standard deviation (represented by the symbol σ ) shows how much variation Cumulative probability of a normal distribution with expected value 0 and standard deviation 1