The probability that a measurement will fall within a given Closed Interval
. For a continuous distribution,
![\begin{displaymath}
\mathop{\rm CI}\nolimits (a,b)\equiv\int_b^a P(x)\,dx,
\end{displaymath}](c2_998.gif) |
(1) |
where
is the Probability Distribution Function. Usually, the confidence interval of interest is
symmetrically placed around the mean, so
![\begin{displaymath}
\mathop{\rm CI}\nolimits (x)\equiv \mathop{\rm CI}\nolimits (\mu-x,\mu+x)=\int_{\mu-x}^{\mu+x} P(x)\,dx,
\end{displaymath}](c2_1000.gif) |
(2) |
where
is the Mean. For a Gaussian Distribution, the probability that a measurement falls within
of the mean
is
Now let
, so
. Then
where
is the so-called Erf function. The variate value producing a confidence interval CI is often denoted
, so
![\begin{displaymath}
x_{\rm CI} = \sqrt{2}\,\mathop{\rm erf}\nolimits ^{-1}(\mathop{\rm CI}\nolimits ).
\end{displaymath}](c2_1011.gif) |
(5) |
range |
CI |
![$\sigma$](c2_1012.gif) |
0.6826895 |
![$2\sigma$](c2_1013.gif) |
0.9544997 |
![$3\sigma$](c2_1014.gif) |
0.9973002 |
![$4\sigma$](c2_1015.gif) |
0.9999366 |
![$5\sigma$](c2_1016.gif) |
0.9999994 |
To find the standard deviation range corresponding to a given confidence interval, solve (4) for
.
![\begin{displaymath}
n=\sqrt{2}\,{\rm erf}^{-1}(\mathop{\rm CI}\nolimits )
\end{displaymath}](c2_1017.gif) |
(6) |
CI |
range |
0.800 |
± 1.28155![$\sigma$](c2_1012.gif) |
0.900 |
± 1.64485![$\sigma$](c2_1012.gif) |
0.950 |
± 1.95996![$\sigma$](c2_1012.gif) |
0.990 |
± 2.57583![$\sigma$](c2_1012.gif) |
0.995 |
± 2.80703![$\sigma$](c2_1012.gif) |
0.999 |
± 3.29053![$\sigma$](c2_1012.gif) |
© 1996-9 Eric W. Weisstein
1999-05-26