t-statistic for correlation, $r$ \[ t = r \sqrt{\frac{n-2}{1-r^2}} \]

The reverse equation is \[ r = \frac{t}{\sqrt{t^2 + n-2}} \]

Derivation of the reverse equation: \begin{align*} & t = r \sqrt{\frac{n-2}{1-r^2}} & \\ & \frac{t^2}{r^2} = \frac{n-2}{1-r^2} & \\ & \frac{t^2}{n-2} = \frac{r^2}{1-r^2} & \\ & \frac{t^2}{t^2 + n-2} = r^2 & \\ & r = \sqrt{\frac{t^2}{t^2 + n-2}} & \end{align*}

See also:

• https://stackoverflow.com/questions/2632628/left-align-block-of-equations - shows a trick to left align an equation array

\begin{align*} \sec x + \tan x & = \tan \left( \frac{x}{2} + \frac{\pi}{4} \right) \\ & = \sqrt{\frac{1 + \sin x}{1 - \sin x}} \end{align*}

I came across this simplification while reading https://liorsinai.github.io/mathematics/2020/08/27/secant-mercator.html which talks about the integral of the secant which in turn has applications in Mercator map.

• fibonacci sequence
• Leibniz integral rule

• What is the formula for sum of the p-th powers of the first n positive integers?
  • Use Faulhaber's formula in described in https://en.wikipedia.org/wiki/Faulhaber%27s_formula
  • The formula involves binomial coefficients and Bernoulli numbers.
  • See also: https://www.johndcook.com/blog/2016/12/31/sums-of-consecutive-powers/

use cases | to copy paste syntax

• https://liorsinai.github.io/mathematics/2020/08/27/secant-mercator.html
• https://en.wikipedia.org/wiki/Fibonacci_sequence
• https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

• https://www.mathcon.org/
  • weekly practice tests - https://www.mathcon.org/apps/pages/WeeklyPracticeTests
  • grade 4 through 12
• Khan academy
• https://www.cmleague.com/
• https://www.mathcounts.org/
• https://bestbrains.com/