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In
probability and
statistics, the '''standard deviation''' is the most commonly used measure of
statistical dispersion. Standard deviation is defined as the
square root of the
variance. It is defined this way in order to give us a measure of dispersion that is 1) a non-negative number; and 2) has the same units as the data.
We distinguish between the standard deviation σ (
sigma) of a whole ''
population'' or of a
random variable, and the standard deviation ''s'' of a subset-population ''
sample''. The formulas are given below.
The term standard deviation was introduced to statistics by
Karl Pearson (''On the dissection of asymmetrical frequency curves'',
1894).
==Interpretation and application==
Simply put, the standard deviation tells us how far a typical member of the population (or sample) is from the
mean value of that population (or sample). A large standard deviation suggests that a typical member is far away from the mean. A small standard deviation suggests that members are clustered closely around the mean.
For example, the sets { 0, 5, 9, 14 } and { 5, 6, 8 ,9 } each have a mean of 7, but the second set has a much smaller standard deviation.
Standard deviation is often thought of as a measure of uncertainty. In physical science for example, when making repeated
measurements the standard deviation of the set of measurements is the
precision of those measurements. When deciding whether measurements agree with a 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. See
prediction interval.
==Definition and shortcut calculation of standard deviation==
Suppose we are given a population ''x''
1,...,''x''
''N'' of values (which are
real numbers). The
mean of this population is defined as
:<math>\overline{x}=\frac{1}{N}\sum_{i=1}^N x_i</math>
(see
summation notation) and the standard deviation of this population is defined as
:<math>\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - \overline{x})^2}</math>
A slightly faster way to compute the same number is given by the formula
:<math>\sigma = \sqrt{{\sum_{i=1}^N{{x_i}^2}\over{N}}-\left({\sum_{i=1}^N{x_i}\over{N}}\right)^2\ } = \sqrt{\frac{N\sum_{i=1}^N{{x_i}^2} - \left(\sum_{i=1}^N{x_i}\right)^2}{N^2}\ }.</math>
The standard deviation of a
random variable ''X'' is defined as
:<math>\sigma = \sqrt{\operatorname{E}((X-\operatorname{E}X)^2)} = \sqrt{\operatorname{E}(X^2) - (\operatorname{E}(X))^2}</math>
Note that not all random variables have a standard deviation, since these
expected values need not exist.
If the random variable ''X'' takes on the values ''x''
1,...,''x''
''N'' with equal probability, then its standard deviation can be computed with the formula given earlier.
Given only a sample of values ''x''
1,...,''x''
''n'' from some larger population, many authors define the ''sample standard deviation'' by
:<math>
s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2}
</math>
The reason for this definition is that ''s''
2 is an
unbiased estimator for the
variance σ
2 of the underlying population. Note however that ''s'' itself is ''not'' an unbiased estimator for the standard deviation σ; it tends to underestimate the population standard deviation.
==Rules for normally distributed data==
In practice, one often assumes that data is approximately
normally distributed. If that assumption can be justified, then 68% of the values are at most 1 standard deviation away from the mean, 95% of the values are at most two standard deviations away from the mean, and 99.7% of the values lie within 3 standard deviations of the mean. This is known as the "68-95-99.7 rule".
==Relation between standard deviation and mean==
The mean and the standard deviation of a data set go hand in hand and are usually reported together. In a certain sense, the standard deviation is the "natural" measure of
statistical dispersion if the center of the data is measured by the mean. The precise statement is the following: suppose ''x''
1,...,''x''
''N'' are real numbers and define the function
:<math>\sigma(r) = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - r)^2}</math>
Using
calculus, it is not difficult to show that σ(''r'') has a unique minimum for
:<math>r = \overline{x}</math>
==Geometric interpretation==
To gain some geometric insights, we will start with a population of three values, ''x''
1, ''x''
2, ''x''
3. This defines a point ''P'' = (''x''
1, ''x''
2, ''x''
3) in '''R'''
3. 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
:<math>R = (\overline{x},\overline{x},\overline{x})</math>
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''.
==Standard deviation as a confidence level==
In experimental science, the confidence one has that a measured event is the result of a signal, rather than just statistical noise. So the higher your sigma confidence level, the less likely it is that the measured event is a result of noise.
== Related articles ==
*
variance
*
Chebyshev's inequality
*
saturation (color theory)
*
root mean square
*
mean
*
skewness
*
kurtosis
*
raw score
*
standard score
*
algorithms for calculating variance
*
an inequality on location and scale parameters
== External links ==
*
Standard Deviation Calculator
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