Numerical Calculation of \Gamma = 2.354 \sigma

[tony873004]

For starters... What does "data reduction" mean? The book starts throwing around this term (which is also part of the book's title) without even defining it.

homework question:

Show by numerical calculation that, for the Gaussian probability distribution, the full-width at half maximum $$\Gamma$$ is related to the standard deviation by $$\Gamma=2.354 \sigma$$.

What is a numerical calculation? Does this just mean "do the math", or does it mean numerical as in numerical vs. analytic methods, where I'm supposed to make the computer crunch a whole bunch of otherwise-unmanagable numbers to get an answer?

Any idea how to do this problem? I know that the standard deviation is equal to the square root of the mean. And I know that the full-width, half max is equal to the width of the curve halfway to the top of the curve. But I don't see a formula for computing it. When describing it, the book gives $$p_G \left( {\mu \pm 1/2\Gamma ,\mu ,\sigma } \right) = 1/2p_G \left( {\mu ;\mu ,\sigma } \right)$$

There's a nice graph in the book showing standard deviation and FWHM, where eye-balling it, 2.354 seems like reasonable number.

 

[link2001]

data reduction is just the process of transforming data to a more usable form (binning, averaging, etc.).

Here I think the numerical calculation part is due to the fact that 2.354 is only an approximation.

The proof for the relationship between the FWHM and the std. dev. isn't overly difficult. As a starter, what is the x value for which the gaussian reaches half maximum?

 

[tony873004]

data reduction is just the process of transforming data to a more usable form (binning, averaging, etc.).

Here I think the numerical calculation part is due to the fact that 2.354 is only an approximation.

The proof for the relationship between the FWHM and the std. dev. isn't overly difficult. As a starter, what is the x value for which the gaussian reaches half maximum?

Thanks. We talked about this in class today. I'm starting to get it.