Normal distribution If the considered function is the density of a
normal distribution of the form f(x) = \frac{1}{\sigma \sqrt{2 \pi} } \exp \left[ -\frac{(x-x_0)^2}{2 \sigma^2} \right] where
σ is the
standard deviation and
x0 is the
expected value, then the relationship between FWHM and the
standard deviation is \mathrm{FWHM} = 2\sqrt{2 \ln 2 } \; \sigma \approx 2.355 \; \sigma. The FWHM does not depend on the expected value
x0; it is invariant under translations. The area within this FWHM is approximately 76% of the total area under the function.
Other distributions In
spectroscopy half the width at half maximum (here
γ), HWHM, is in common use. For example, a
Lorentzian/Cauchy distribution of height can be defined by f(x) = \frac{1}{\pi\gamma \left[1 + \left(\frac{x - x_0}{\gamma}\right)^2\right]} \quad \text{ and } \quad \mathrm{FWHM} = 2 \gamma. Another important distribution function, related to
solitons in
optics, is the
hyperbolic secant: f(x) = \operatorname{sech} \left( \frac{x}{X} \right). Any translating element was omitted, since it does not affect the FWHM. For this impulse we have: \mathrm{FWHM} = 2 \operatorname{arcsch} \left(\tfrac{1}{2}\right) X = 2 \ln (2 + \sqrt{3}) \; X \approx 2.634 \; X where is the
inverse hyperbolic secant. == See also ==