How do you calculate r squared by hand?

2 Answers
May 31, 2015

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:

#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.