Standard deviation

In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. It is defined as the square root of the variance.

The standard deviation is the root mean square (RMS) deviation of the values from their arithmetic mean. For example, in the population {4, 8}, the mean is 6 and the standard deviation is 2. This may be written: {4, 8} ≈ 6±2. In this case 100% of the values in the population are at one standard deviation of the mean.

Standard deviation is the most common measure of statistical dispersion, measuring how widely spread the values in a data set are. If the data points are all close to the mean, then the standard deviation is close to zero. If many data points are far from the mean, then the standard deviation is far from zero. If all the data values are equal, then the standard deviation is zero.

The standard deviation (σ) of a population can be estimated by a modified standard deviation (s) of a sample. The formulae are given below.

Definition and calculation

Standard deviation of a random variable

The standard deviation of a random variable X is defined as:

$\sigma ={\sqrt {\operatorname {E} ((X-\operatorname {E} (X))^{2})}}={\sqrt {\operatorname {E} (X^{2})-(\operatorname {E} (X))^{2}}}$

where E(X) is the expected value of X.

Not all random variables have a standard deviation, since these expected values need not exist. For example, the standard deviation of a random variable which follows a Cauchy distribution is undefined.

If the random variable X takes on the values x₁,...,x_N (which are real numbers) with equal probability, then its standard deviation can be computed as follows. First, the mean of X, ${\overline {x}}$, is defined as:

${\overline {x}}={\frac {1}{N}}\sum _{i=1}^{N}x_{i}={\frac {x_{1}+x_{2}+\cdots +x_{N}}{N}}$

(see sigma notation) where N is the number of samples taken. Next, the standard deviation simplifies to:

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

In other words, the standard deviation of a discrete uniform random variable X can be calculated as follows:

1. For each value $x_{i}$ calculate the difference $x_{i}-{\overline {x}}$ between $x_{i}$ and the average value ${\overline {x}}$.
2. Calculate the squares of these differences.
3. Find the average of the squared differences. This quantity is the variance $\sigma ^{2}$.
4. Take the square root of the variance.

Estimating population standard deviation from sample standard deviation

In the real world, finding the standard deviation of an entire population is unrealistic except in certain cases, such as standardized testing, where every member of a population is sampled. In most cases, sample standard deviation ($s$) is used to estimate population standard deviation ($\sigma$). Given only a sample of values x₁,...,x_N from some larger population, many authors define the sample (or estimated) standard deviation by

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

The reason for this definition is that s² is an unbiased estimator for the variance σ² of the underlying population, if it is uncorrelated and has uniform variance of σ². However, s is not an unbiased estimator for the standard deviation σ; it tends to underestimate the population standard deviation. 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; see bias of an estimator. Some have argued that even the difference between N and N − 1 in the denominator is overly complex and insignificant. The necessity of the N − 1 (instead of N) can be rationalized if one realizes that the $x_{i}-{\overline {x}}$ vector lies in an N − 1 dimensional space. Without that term, what is left is the simpler expression:

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

form has a uniformly smaller mean squared error than does the unbiased estimator, and is the maximum-likelihood estimate when the population (or the random variable X) is normally distributed.

Example

We will show how to calculate the standard deviation of a population. Our example will use the ages of four young children: { 5, 6, 8, 9 }.

Step 1. Calculate the mean average, ${\overline {x}}$:

${\overline {x}}={\frac {1}{N}}\sum _{i=1}^{N}x_{i}$

We have N = 4 because there are four data points:

$x_{1}=5\,\!$

$x_{2}=6\,\!$

$x_{3}=8\,\!$

$x_{4}=9\,\!$

${\overline {x}}={\frac {1}{4}}\sum _{i=1}^{4}x_{i}$       Replacing N with 4

${\overline {x}}={\frac {1}{4}}\left(x_{1}+x_{2}+x_{3}+x_{4}\right)$

${\overline {x}}={\frac {1}{4}}\left(5+6+8+9\right)$

${\overline {x}}=7$   This is the mean.

Step 2. Calculate the standard deviation $\sigma \,\!$:

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

$\sigma ={\sqrt {{\frac {1}{4}}\sum _{i=1}^{4}(x_{i}-{\overline {x}})^{2}}}$       Replacing N with 4

$\sigma ={\sqrt {{\frac {1}{4}}\sum _{i=1}^{4}(x_{i}-7)^{2}}}$       Replacing ${\overline {x}}$ with 7

$\sigma ={\sqrt {{\frac {1}{4}}\left[(x_{1}-7)^{2}+(x_{2}-7)^{2}+(x_{3}-7)^{2}+(x_{4}-7)^{2}\right]}}$

$\sigma ={\sqrt {{\frac {1}{4}}\left[(5-7)^{2}+(6-7)^{2}+(8-7)^{2}+(9-7)^{2}\right]}}$

$\sigma ={\sqrt {{\frac {1}{4}}\left((-2)^{2}+(-1)^{2}+1^{2}+2^{2}\right)}}$

$\sigma ={\sqrt {{\frac {1}{4}}\left(4+1+1+4\right)}}$

$\sigma ={\sqrt {\frac {10}{4}}}$

$\sigma ={\sqrt {\frac {5}{2}}}$

$\sigma =1.5811\,\!$   This is the standard deviation.

Were this set a sample drawn from a larger population of children, and the question at hand was the standard deviation of the population, convention would replace the N (or 4) here with N−1 (or 3).

Interpretation and application

A large standard deviation indicates that the data points are far from the mean and a small standard deviation indicates that they are clustered closely around the mean.

For example, each of the three data sets (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 set 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 four siblings, the standard deviation is 5 years.

As another example, the data set (1000, 1006, 1008, 1014) may represent the distances traveled by four athletes in 3 minutes, measured in meters. It has a mean of 1007 meters, and a standard deviation of 5 meters.

In the age example above, a standard deviation of 5 may be considered large; in the distance example above, 5 may be considered small.

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 the precision 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 we consider the measurements as contradicting the prediction. 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. See prediction interval.

Real life examples

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

Weather

As a simple example, consider average temperatures for cities. While the average for all cities may be 60°F, it's helpful to understand that the range for cities near the coast is smaller than for cities inland, which clarifies that, while the average is similar, the chance for variation is greater inland than near the coast.

So, an average of 60 occurs for one city with highs of 80°F and lows of 40°F, and also occurs for another city with highs of 65 and lows of 55. The standard deviation allows us to recognize that the average for city with the wider variation, and thus a higher standard deviation will not offer as reliable a prediction of temperature as the city with the smaller variation and lower standard deviation.

Sports

Another way of seeing it is to consider sports teams. In any set of categories, there will be teams that rate highly at some things and poorly at others. Chances are, the teams that lead in the standings will not show such disparity, but will be pretty good in most categories. The lower the standard deviation of their ratings in each category, the more balanced and consistent they might be. So, a team that is consistently bad in most categories will have a low standard deviation indicating they will probably lose more often than win. A team that is consistently good in most categories will also have a low standard deviation and will therefore end up winning more than losing. A team with a high standard deviation might be the type of team that scores a lot (strong offense) but gets scored on a lot too (weak defense); or vice versa, might get scored on, but compensate with higher scoring - teams with a higher standard deviation will be more unpredictable.

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 racing, a driver is timed on successive laps. A driver with a low standard deviation of lap times is more consistent than a driver with a higher standard deviation. This information can be used to help understand where opportunities might be found to reduce lap times.

Geometric interpretation

To gain some geometric insights, we will start with a population of three values, x₁, x₂, x₃. This defines a point P = (x₁, x₂, x₃) in R³. Consider the line L = {(r, r, r) : r in 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. Moving orthogonally from P to the line L, one hits the point:

$R=({\overline {x}},{\overline {x}},{\overline {x}})$

whose coordinates are the mean of the values we started out with. A little algebra shows that the distance between P and R (which is the same as the distance between P and the line L) is given by σ√3. An analogous formula (with 3 replaced by N) is also valid for a population of N values; we then have to work in R^N.

Rules for normally distributed data

Dark blue is less than one standard deviation from the mean. For the normal distribution, this accounts for 68.27% of the set; while two standard deviations from the mean (blue and brown) account for 95.45%; and three standard deviations (blue, brown and green) account for 99.73%.

In practice, one often assumes that the data are from an approximately normally distributed population. If that assumption is justified, then about 68% of the values are within 1 standard deviation of the mean, about 95% of the values are within two standard deviations and about 99.7% lie within 3 standard deviations. This is known as the "68-95-99.7 rule", or "the empirical rule"

The confidence intervals are as follows:

σ	68.26894921371%
2σ	95.44997361036%
3σ	99.73002039367%
4σ	99.99366575163%
5σ	99.99994266969%
6σ	99.99999980268%
7σ	99.99999999974%

For normal distributions, the two points of the curve which are one standard deviation from the mean are also the inflection points.

Chebyshev rules

If it is not known whether the distribution is normal, one can always use Chebyshev's inequality:

At least 50% of the values are within 1.4 standard deviations from the mean.

At least 75% of the values are within 2 standard deviations from the mean.

At least 89% of the values are within 3 standard deviations from the mean.

At least 94% of the values are within 4 standard deviations from the mean.

At least 96% of the values are within 5 standard deviations from the mean.

At least 97% of the values are within 6 standard deviations from the mean.

At least 98% of the values are within 7 standard deviations from the mean.

At least 100 * (1 - 1/k²) percent of the values are within k standard deviations from the mean.

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, ..., x_n are real numbers and define the function:

$\sigma (r)={\sqrt {{\frac {1}{N}}\sum _{i=1}^{N}(x_{i}-r)^{2}}}$

Using calculus, it is possible to show that σ(r) has a unique minimum at the mean:

$r={\overline {x}}$

(this can also be done with fairly simple algebra alone, since, as a function of r, it is a quadratic polynomial).

The coefficient of variation of a sample is the ratio of the standard deviation to the mean. It is a dimensionless number that can be used to compare the amount of variance between populations with different means.

Chebyshev's inequality proves that in any data set, nearly all of the values will be nearer to the mean value, where the meaning of "close to" is specified by the standard deviation.

Rapid calculation methods

A slightly faster (significantly for running standard deviation) way to compute the population standard deviation is given by the following formula (though this can exacerbate round-off error):

$\sigma \ ={\sqrt {{\frac {1}{N}}\left(\sum _{i=1}^{N}{{x_{i}}^{2}}-{\frac {\left(\sum _{i=1}^{N}{x_{i}}\right)^{2}}{N}}\right)}}={\sqrt {{\frac {1}{N}}\left(\sum _{i=1}^{N}{{x_{i}}^{2}}\right)-{\overline {x}}^{2}}}$

Similarly for sample standard deviation:

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

Or from running sums:

$s={\sqrt {\frac {N\sum _{i=1}^{N}{{x_{i}}^{2}}-\left(\sum _{i=1}^{N}{x_{i}}\right)^{2}}{N(N-1)}}}$

See also algorithms for calculating variance.

An axiomatic approach

It is a nice fact that the mean value μ and the standard deviation σ is completely characterized by the simple algebraic properties a+(μ±σ) = (a+μ)±σ and a(μ±σ) = aμ±aσ , together with a symmetry condition and the initial condition (+1,−1) ≈ ±1 .

The set of two numbers,

X = (X₁, X₂) = 2⁻¹(X₁+X₂) + (+2⁻¹(X₁−X₂), −2⁻¹(X₁−X₂)) = 2⁻¹(X₁+X₂) + 2⁻¹(X₁−X₂)(+1,−1) ≈ 2⁻¹(X₁+X₂) + 2⁻¹(X₁−X₂)(±1) = 2⁻¹(X₁+X₂) ± 2⁻¹(X₁−X₂) = μ±σ

so that

μ = 2⁻¹(X₁+X₂)

and

σ = 2⁻¹(X₁−X₂)

Consider the power sums:

s₀ = X₁⁰+X₂⁰ = 1+1 = 2

s₁ = X₁¹+X₂¹ = X₁+X₂

s₂ = X₁²+X₂²

The power sums s_j are symmetric functions of the vector X, and the symmetric functions μ and σ² are written in terms of these like this:

μ = s₀⁻¹s₁

σ² = s₀⁻²(s₀s₂−s₁²)

(because σ² = (2⁻¹)²(X₁² − 2X₁X₂ + X₂²) = 2⁻²(2(X₁² + X₂²) − (X₁² + 2X₁X₂ + X₁²)), by polynomial expansion and rearrangement)

or

X ≈ μ±σ = s₀⁻¹(s₁±(s₀s₂−s₁²)^1/2)

This formula for the special case n=2 generalizes to n=1,2,3,4,..., preserving the rules. The general power sums are

$\ s_{j}=\sum _{k=1}^{n}{X_{k}^{j}}$

See also

• Algorithms for calculating variance
• An inequality on location and scale parameters
• Chebyshev's inequality
• Confidence interval
• Cumulant
• Deviation (statistics)
• Kurtosis
• Mean absolute error
• Mean
• Raw score
• Root mean square
• Sample size
• Saturation (color theory)
• Six Sigma
• Skewness
• Standard error
• Standard score
• Variance
• Volatility

External links

• Standard Deviation, an elementary introduction
• Standard Deviation, a simpler explanation for writers and journalists
• Standard Deviation Calculator
• Standard Deviation in Excel Spreadsheets