Standard deviation

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A plot of a normal distribution (or bell curve). Each colored band has a width of one standard deviation.


A data set with a mean of 50 (shownin blue) and a standard deviation (σ) of 20.


Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or “dispersion” there is from the average (mean, or expected value). 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 points are spread out over a large range of values.

The standard deviation of a statistical population, data set, or probability distribution is the square root of its variance. It isalgebraically simpler though practically less robust than the average absolute deviation.[1][2] A useful property of standard deviation is that, unlike variance, it is expressed in the same units as the data.



Basic examples

Consider a population consisting of the following eight values:





  \begin{array}{lll}  (2-5)^2 = (-3)^2 = 9   &&  (5-5)^2 = 0^2 = 0 \\  (4-5)^2 = (-1)^2 = 1  &&  (5-5)^2 = 0^2 = 0 \\  (4-5)^2 = (-1)^2 = 1  &&  (7-5)^2 = 2^2 = 4 \\  (4-5)^2 = (-1)^2 = 1  &&  (9-5)^2 = 4^2 = 16 \\  \end{array}

Next compute the average of these values, and take the square root:


This quantity is the population standard deviation; it is equal to the square root of the variance. The formula is valid only if the eight values we began with form the complete population. 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. See the section Estimation below for more details.

A 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″. 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. If the standard deviation were zero, then all men would be exactly 70″ tall. If the standard deviation were 20″, then men would have much more variable heights, with a typical range of about 50″–90″. Three standard deviations account for 99.7% of the sample population being studied, assuming the distribution is normal (bell-shaped).

[edit]Definition of population values

Let X be a random variable with mean value μ:



\sigma = \sqrt{\operatorname{E}\left[(X - \mu)^2\right]}.

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.


[edit]Discrete random variable

In the case where X takes random values from a finite data set x1x2, …, xN, with each value having the same probability, the standard deviation is


or, using summation notation,


If, instead of having equal probabilities, the values have different probabilities, let x1 have probability p1x2 have probability p2, …, xN have probability pN. In this case, the standard deviation will be


[edit]Continuous random variable







[edit]With standard deviation of the sample

An estimator for σ sometimes used is the standard deviation of the sample, denoted by sN and defined as follows:


This estimator has a uniformly smaller mean squared error than the sample standard deviation (see below), and is the maximum-likelihood estimate when the population is normally distributed. But this estimator, when applied to a small or moderately sized sample, tends to be too low: it is a biased estimator.

The standard deviation of the sample is the same as the population standard deviation of a discrete random variable that can assume precisely the values from the data set, where the probability for each value is proportional to its multiplicity in the data set.

[edit]With sample standard deviation

The most common estimator for σ used is an adjusted version, the sample standard deviation, denoted by s and defined as follows:


where \scriptstyle\{x_1,\,x_2,\,\ldots,\,x_N\} are the observed values of the sample items and \scriptstyle\overline{x} is the mean value of these observations. This correction (the use of N − 1 instead of N) is known as Bessel’s correction. The reason for this correction is that s2 is an unbiased estimator for the variance σ2 of the underlying population, if that variance exists and the sample values are drawn independently with replacement. However, s is not an unbiased estimator for the standard deviation σ; it tends to overestimate the population standard deviation.

The term standard deviation of the sample is used for the uncorrected estimator (using N) while the term sample standard deviation is used for the corrected estimator (using N − 1). The denominator N − 1 is the number of degrees of freedom in the vector of residuals\scriptstyle(x_1-\overline{x},\; \dots,\; x_n-\overline{x}).

Difference between standard deviation population vs Sample:

When all available values are used, it is called a population; when only a subset of available values are used, it is called a sample.

Things to remember:

σ is the population standard deviation which is usually unknown.
s is the sample standard deviation which is an estimate of the unknown population standard deviation.
The sum of squares is divided by 1 less than the sample size to account for the error in estimation from the sample standard deviation (called the degrees of freedom).[3]

[edit]Other estimators

Although an unbiased estimator for σ is known when the random variable is normally distributed, the formula is complicated and amounts to a minor correction. Moreover, unbiasedness (in this sense of the word) is not always desirable.[citation needed]

[edit]Identities and mathematical properties


 \operatorname{stdev}(c) = 0 \,
 \operatorname{stdev}(cX) = \|c\| \operatorname{stdev}(X). \,





  \operatorname{stdev}(X) = \sqrt{E[X-E(X)]^2} = \sqrt{E[X^2] - (E[X])^2}.

The sample standard deviation can be computed as:

  \operatorname{stdev}(X) = \sqrt{\frac{N}{N-1}} \sqrt{E[X-E(X)]^2}.



Thus, the standard deviation is equal to the square root of (the average of the squares less the square of the average). See computational formula for the variance for a proof of this fact, and for an analogous result for the sample standard deviation.

[edit]Interpretation and application


For example, each of the three populations {0, 0, 14, 14}, {0, 6, 8, 14} and {6, 6, 8, 8} has a mean of 7. Their standard deviations are 7, 5, and 1, respectively. The third population has a much smaller standard deviation than the other two because its values are all close to 7. In a loose sense, the standard deviation tells us how far from the mean the data points tend to be. It will have the same units as the data points themselves. If, for instance, the data set {0, 6, 8, 14} represents the ages of a population of four siblings in years, the standard deviation is 5 years.


Standard deviation may serve as a measure of uncertainty. In physical science, for example, the reported standard deviation of a group of repeated measurements should give theprecision of those measurements. When deciding whether measurements agree with a theoretical prediction, the standard deviation of those measurements is of crucial importance: if the mean of the measurements is too far away from the prediction (with the distance measured in standard deviations), then the theory being tested probably needs to be revised. This makes sense since they fall outside the range of values that could reasonably be expected to occur if the prediction were correct and the standard deviation appropriately quantified. Seeprediction interval.

[edit]Application examples

The practical value of understanding the standard deviation of a set of values is in appreciating how much variation there is from the “average” (mean).





Trying to predict which teams, on any given day, will win, may include looking at the standard deviations of the various team “stats” ratings, in which anomalies can match strengths vs. weaknesses to attempt to understand what factors may prevail as stronger indicators of eventual scoring outcomes.



In finance, standard deviation is a representation of the risk associated with price-fluctuations of a given asset (stocks, bonds, property, etc.), or the risk of a portfolio of assets (actively managed mutual funds, index mutual funds, or ETFs). Risk is an important factor in determining how to efficiently manage a portfolio of investments because it determines the variation in returns on the asset and/or portfolio and gives investors a mathematical basis for investment decisions (known as mean-variance optimization). The fundamental concept of risk is that as it increases, the expected return on an investment should increase as well, an increase known as the “risk premium.” In other words, investors should expect a higher return on an investment when that investment carries a higher level of risk or uncertainty. When evaluating investments, investors should estimate both the expected return and the uncertainty of future returns. Standard deviation provides a quantified estimate of the uncertainty of future returns.




[edit]Geometric interpretation

To gain some geometric insights and clarification, we will start with a population of three values, x1x2x3. This defines a point P = (x1x2x3) in R3. Consider the line L = {(rrr) : r ∈ R}. This is the “main diagonal” going through the origin. If our three given values were all equal, then the standard deviation would be zero and P would lie on L. So it is not unreasonable to assume that the standard deviation is related to the distance of P to L. And that is indeed the case. To move orthogonally from L to the point P, one begins at the point:


whose coordinates are the mean of the values we started out with. A little algebra shows that the distance between P and M (which is the same as the orthogonal distance between Pand the line L) is equal to the standard deviation of the vector x1x2x3, multiplied by the square root of the number of dimensions of the vector (3 in this case.)

[edit]Chebyshev’s inequality

An observation is rarely more than a few standard deviations away from the mean. Chebyshev’s inequality ensures that, for all distributions for which the standard deviation is defined, the amount of data within a number of standard deviations of the mean is at least as much as given in the following table.

Minimum population Distance from mean
50% √2
75% 2
89% 3
94% 4
96% 5
97% 6
\scriptstyle 1-\frac{1}{k^2}[4] \scriptstyle k

[edit]Rules for normally distributed data



\frac{1}{\sigma\sqrt{2\pi}} \exp\!\left(-\frac{[x-\mu]^2}{2\sigma^2} \right)

where μ is the expected value of the random variables, σ equals their distribution’s standard deviation divided by n1/2, and n is the number of random variables. The standard deviation therefore is simply a scaling variable that adjusts how broad the curve will be, though it also appears in the normalizing constant.

If a data distribution is approximately normal then the proportion of data values within z standard deviations of the mean is defined by:




zσ Percentage within CI Percentage outside CI Fraction outside CI
0.674490σ 50% 50% 1 / 2
0.994458σ 68% 32% 1 / 3.125
68.2689492% 31.7310508% 1 / 3.1514872
1.281552σ 80% 20% 1 / 5
1.644854σ 90% 10% 1 / 10
1.959964σ 95% 5% 1 / 20
95.4499736% 4.5500264% 1 / 21.977895
2.575829σ 99% 1% 1 / 100
99.7300204% 0.2699796% 1 / 370.398
3.290527σ 99.9% 0.1% 1 / 1,000
3.890592σ 99.99% 0.01% 1 / 10,000
99.993666% 0.006334% 1 / 15,787
4.417173σ 99.999% 0.001% 1 / 100,000
4.891638σ 99.9999% 0.0001% 1 / 1,000,000
99.9999426697% 0.0000573303% 1 / 1,744,278
5.326724σ 99.99999% 0.00001% 1 / 10,000,000
5.730729σ 99.999999% 0.000001% 1 / 100,000,000
99.9999998027% 0.0000001973% 1 / 506,797,346
6.109410σ 99.9999999% 0.0000001% 1 / 1,000,000,000
6.466951σ 99.99999999% 0.00000001% 1 / 10,000,000,000
6.806502σ 99.999999999% 0.000000001% 1 / 100,000,000,000
99.9999999997440% 0.0000000002560% 1 / 390,682,215,445

[edit]Relationship between standard deviation and mean

The mean and the standard deviation of a set of data are usually reported together. In a certain sense, the standard deviation is a “natural” measure of statistical dispersion if the center of the data is measured about the mean. This is because the standard deviation from the mean is smaller than from any other point. The precise statement is the following: suppose x1, …,xn are real numbers and define the function:

\sigma(r) = \sqrt{\frac{1}{N-1} \sum_{i=1}^N (x_i - r)^2}.


r = \overline{x}.\,


Often we want some information about the precision of the mean we obtained. We can obtain this by determining the standard deviation of the sampled mean. The standard deviation of the mean is related to the standard deviation of the distribution by:

\sigma_{\text{mean}} = \frac{1}{\sqrt{N}}\sigma

where N is the number of observation in the sample used to estimate the mean. This can easily be proven with:

\begin{align}  \operatorname{var}(X) &\equiv \sigma^2_X\\  \operatorname{var}(X_1+X_2) &\equiv \operatorname{var}(X_1) + \operatorname{var}(X_2)\\  \operatorname{var}(cX_1) &\equiv c^2 \, \operatorname{var}(X_1)  \end{align}


  \begin{align}  \operatorname{var}(\text{mean})  &= \operatorname{var}\left (\frac{1}{N} \sum_{i=1}^N X_i \right)  = \frac{1}{N^2}\operatorname{var}\left (\sum_{i=1}^N X_i \right ) \\  &= \frac{1}{N^2}\sum_{i=1}^N \operatorname{var}(X_i)  = \frac{N}{N^2} \operatorname{var}(X)  = \frac{1}{N} \operatorname{var} (X).  \end{align}

Resulting in:

\sigma_\text{mean} = \frac{\sigma}{\sqrt{N}}.

[edit]Rapid calculation methods

The following two formulas can represent a running (continuous) standard deviation. A set of three power sums s0s1s2 are each computed over a set of N values of x, denoted as x1, …,xN:

\ s_j=\sum_{k=1}^N{x_k^j}.

Note that s0 raises x to the zero power, and since x0 is always 1, s0 evaluates to N.

Given the results of these three running summations, the values s0s1s2 can be used at any time to compute the current value of the running standard deviation:

\sigma = \frac{\sqrt{s_0s_2-s_1^2} }{s_0}

Similarly for sample standard deviation,

s = \sqrt{\frac{s_0s_2-s_1^2}{s_0(s_0-1)}}.

In a computer implementation, as the three sj sums become large, we need to consider round-off errorarithmetic overflow, and arithmetic underflow. The method below calculates the running sums method with reduced rounding errors:

\begin{align}  A_0 &= 0\\  A_i &= A_{i-1}+\frac{x_i-A_{i-1}}{i}  \end{align}

where A is the mean value.

\begin{align}  Q_0 &= 0\\  Q_i &= Q_{i-1}+\frac{i-1}{i} (x_i-A_{i-1})^2 = Q_{i-1}+ (x_i-A_{i-1})(x_i-A_i)  \end{align}

Sample variance:


Standard variance:


[edit]Weighted calculation

When the values xi are weighted with unequal weights wi, the power sums s0s1s2 are each computed as:

\ s_j=\sum_{k=1}^N{w_k x_k^j}.\,



A running sum of weights must be computed:

\begin{align}  W_0 &= 0\\  W_i &= W_{i-1} + w_i  \end{align}

and places where 1/i is used above must be replaced by wi/Wi:

\begin{align}  A_0 &= 0\\  A_i &= A_{i-1}+\frac{w_i}{W_i}(x_i-A_{i-1})\\  Q_0 &= 0\\  Q_i &= Q _{i-1} + \frac{w_i W_{i-1}}{W_i}(x_i-A_{i-1})^2 = Q_{i-1}+w_i(x_i-A_{i-1})(x_i-A_i)  \end{align}

In the final division,



s^2_n = \frac{n'}{n'-1}\sigma^2_n\,

where n is the total number of elements, and n’ is the number of elements with non-zero weights. The above formulas become equal to the simpler formulas given above if weights are taken as equal to one.

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