The standard deviation of a (univariate) likelihood dispersion is equivalent to that of a random variable having that appropriation. Not every single arbitrary variable has a standard deviation since these ordinary qualities need not exist. For instance, the standard deviation of a random variable that follows a Cauchy dispersion is unclear because its normal worth μ is indistinct.

Discrete arbitrary variable

For the situation where X takes arbitrary qualities from a limited informational index x1, x2, …, xN, with each worth having a similar likelihood, the standard deviation is

{\displaystyle \sigma ={\sqrt {{\frac {1}{N}}\left[(x_{1}-\mu )^{2}+(x_{2}-\mu )^{2}+\cdots +(x_{N}-\mu )^{2}\right]}},{\rm {\ \ where\ \ }}\mu ={\frac {1}{N}}(x_{1}+\cdots +x_{N}),}\sigma ={\sqrt {{\frac {1}{N}}\left[(x_{1}-\mu )^{2}+(x_{2}-\mu )^{2}+\cdots +(x_{N}-\mu )^{2}\right]}},{\rm {\ \ where\ \ }}\mu ={\frac {1}{N}}(x_{1}+\cdots +x_{N}),

or on the other hand, utilizing summation documentation,

{\displaystyle \sigma ={\sqrt {{\frac {1}{N}}\sum _{i=1}^{N}(x_{i}-\mu )^{2}}},{\rm {\ \ where\ \ }}\mu ={\frac {1}{N}}\sum _{i=1}^{N}x_{i}.}\sigma ={\sqrt {{\frac {1}{N}}\sum _{i=1}^{N}(x_{i}-\mu )^{2}}},{\rm {\ \ where\ \ }}\mu ={\frac {1}{N}}\sum _{i=1}^{N}x_{i}.

If, rather than having equivalent probabilities, the qualities have various possibilities, let x1 have likelihood p1, x2 have likelihood p2, …, xN have likelihood pN. Right now, the standard deviation will be

{\displaystyle \sigma ={\sqrt {\sum _{i=1}^{N}p_{i}(x_{i}-\mu )^{2}}},{\rm {\ \ where\ \ }}\mu =\sum _{i=1}^{N}p_{i}x_{i}.}\sigma ={\sqrt {\sum _{i=1}^{N}p_{i}(x_{i}-\mu )^{2}}},{\rm {\ \ where\ \ }}\mu =\sum _{i=1}^{N}p_{i}x_{i}.

Consistent irregular variable

Check this out for **standard deviation calculator** of a consistent, genuine esteemed irregular variable X with likelihood thickness work p(x) is

{\displaystyle \sigma ={\sqrt {\int _{\mathbf {X} }(x-\mu )^{2}\,p(x)\,{\rm {d}}x}},{\rm {\ \ where\ \ }}\mu =\int _{\mathbf {X} }x\,p(x)\,{\rm {d}}x,}{\displaystyle \sigma ={\sqrt {\int _{\mathbf {X} }(x-\mu )^{2}\,p(x)\,{\rm {d}}x}},{\rm {\ \ where\ \ }}\mu =\int _{\mathbf {X} }x\,p(x)\,{\rm {d}}x,}

Furthermore, where the integrals are definite integrals taken for x running over the arrangement of potential estimations of the irregular variable X.

On account of a parametric group of circulations, the standard deviation can be communicated as far as the parameters. For instance, on account of the log-ordinary appropriation with settings μ and σ2, the standard deviation calculator is

{\displaystyle {\sqrt {(e^{\sigma ^{2}}-1)e^{2\mu +\sigma ^{2}}}}.}{\displaystyle {\sqrt {(e^{\sigma ^{2}}-1)e^{2\mu +\sigma ^{2}}}}.}