Question

How do you calculate r squared by hand?

Answer 1

Assuming this is a general question and not a reference to some undeclared statistical equation,

and assuming you know how to multiply two numbers together by hand,

then `r` squared (often written `r^2`) is simply
`color(white)("XXXXX")` `r xx r` for whatever the value of `r` is

For example if `r =16`
then `r` squared (or `r^2`) `= 16 xx 16 = 256`

However I suspect you had some specific statistical relationship in mind; please resubmit with explicit references if this is the case.

Answer 2

`r^2 = 1- (SS_(Err))/(SS_(Tot))`

Explanation:

The `SS_(Err)` or the sum of squares residuals is:
`\sum y_i^2 - B_0\sumy_i-B_1\sum x_iy_i`
or
simply the square of the value of the residuals. The residual value is difference between the obtained y-value and the expected y-value. The expected y-value is the calculated value from the equation of line/plane.

For example, for a system with 1 unknown parameter/variable x, the calculated y-value would be the sum of `B_0 and B_1x` (i.e. `Y=B_0+B_1x`).

For a system with 2 unknown parameters/variables, `x_1` and `x_2`, the calculated y-value would be the sum of `B_0`, `B_1x`, and `B_2x_2` (i.e. `Y=B_0+B_1x_1+B_2x_2`).

And in general, `Y=B_0+B_1x_1+B_2x_2+B_3x_3+B_4x_4+...+B_nx_n`

Furthermore, the `SS_(Tot) = \sumy_i^2 - ((sumy_i)^2)/(n)` where `n` is the number of observations or trials.