## Further Calculator Statistics

This is further to my post on calculator statistics.

If the calculator also stores the six values $\Sigma x^3$, $\Sigma x^4$ and $\Sigma x^2y$, then it can calculate the linear regression to a parabola.

The initial state (for the empty data set) has:

$\Sigma x^3 = \Sigma x^4 = \Sigma x^2y = 0$.

Given a new point $\langle f,x,y \rangle$ where $f$ is the frequency and/or weight, these values are updates thus:

 $\Sigma x^3$ :+= $f x^3$ $\Sigma x^4$ :+= $f x^4$ $\Sigma x^2y$ :+= $f x^2 y$

Best fit (minimum square vertical offsets) parabola or quadratic $y = c x^2 + b x + a$:

$c = c_{\rm num}/d$, $b=b_{\rm num}/d$ and $a=a_{\rm num}/d$;

where

$c_{\rm num} = (+ \Sigma x \Sigma x^3 - (\Sigma x^2)^2 ) \Sigma y$
$+ (- n \Sigma x^3 + \Sigma x \Sigma x^2) \Sigma xy$
$+ (+ n \Sigma x^2 - (\Sigma x)^2 ) \Sigma x^2y$;

$b_{\rm num} = (- \Sigma x \Sigma x^4 + \Sigma x^2 \Sigma x^3) \Sigma y$
$+ (+ n \Sigma x^4 - (\Sigma x^2)^2 ) \Sigma xy$
$+ (- n \Sigma x^3 + \Sigma x \Sigma x^2) \Sigma x^2y$;

$a_{\rm num} = (+ \Sigma x^2 \Sigma x^4 - (\Sigma x^3)^2 ) \Sigma y$
$+ (- \Sigma x \Sigma x^4 + \Sigma x^2 \Sigma x^3) \Sigma xy$
$+ (+ \Sigma x \Sigma x^3 - (\Sigma x^2)^2 ) \Sigma x^2y$;

$d = n \Sigma x^2 \Sigma x^4 - n (\Sigma x^3)^2 - (\Sigma x)^2 \Sigma x^4 + 2 \Sigma x \Sigma x^2 \Sigma x^3 - (\Sigma x^2)^3$.