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Re: Visualizing where to draw the standard deviation line
Posted:
Jun 15, 2012 12:01 PM
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On Jun 15, 11:43 am, Kaba <k...@nowhere.com> wrote:
> 15.6.2012 6:29, Onion Knight kirjoitti:
>
> > There has been a debate in COLA as to if the distance from the mean to
> > the inflection point is where the standard deviation should be. One
> > person is saying that this distance is where the line should always be
> > drawn while one other is saying the distance from the mean is
> > irrelevant and it is only the area under the curve that matters. The
> > first agrees the area under the curve is also always the same but
> > insists the inflection point is where the line should be drawn. He
> > even produced a video to show his ideashttp://www.youtube.com/watch?v=MoW3hMq-eIc
> > and he showed what he claimed was an incorrectly depicted image and
> > showed where he says it should behttp://tmp.gallopinginsanity.com/sd.png
> > I admit this goes over my head. Is he correct? Is it really that easy
> > that you can just look at the inflection point and see where the
> > standard deviation should be drawn? I was never taught that in school.
>
> Hi,
>
> That's correct. We are given the probability density function of the
> normal distribution
>
> f : R --> R : f(x) = C exp(-(x - m)^2 / (2s^2)),
>
> where C in R is the normalization constant which makes f integrate to 1,
> m is the mean, and s is the standard deviation. The claim is that the
> inflection points of f are exactly at a distance of one standard
> deviation from the mean, i.e. at m +- s. To prove this claim, compute
> the first and second derivative of f:
>
> f'(x) = C exp(-(x - m)^2 / (2s^2)) (-(x-m)/s^2), and
>
> f''(x) = (C / s^2) exp(-(x - m)^2 / (2s^2)) [(x - m)^2 / s^2 - 1].
>
> Then the inflection points of f are given by solving the equation
>
> f''(x) = 0.
>
> This simplifies to
>
> (x - m)^2 / s^2 - 1 = 0
> <=>
> (x - m)^2 = s^2
> <=>
> |x - m| = s.
>
> QED :)
>
> --http://kaba.hilvi.org
While a lot of that still goes over my head, if I get the gist of it,
the person who said you can visualize the standard deviation based on
the distance from the mean (to the inflection point) was correct at
least in the case of a normal distribution. That is what I assumed
but wanted a different opinion.
Do you know much about linear trendlines? The same debate included the
drawing of those in Excel. The same person who spoke of the standard
deviation also showed how to make a linear trendline,
http://tmp.gallopinginsanity.com/LinearTrendLineCreation.mov
Others in the same group insisted he was missing steps but if you look
at the Microsoft site it offers instructions and it seems his process
is just fine. What steps if any did he skip? Would love to get some
other input from people who are not involved.
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