How do I do this on a ti 84 calculator?

You might wanna try by changing the base of the variable by writing the new variables as y = x - 57 , so that the new mean is mean(y)= mean(x) - 57
= 58.04 - 57 = 1.04
The new variables are -6, -3,0, 3 , 6 . Moreover ,
∑ f ( x - mean(x) ) ^2 = ∑ f ( y + 57 - mean(y) - 57 )^2 = ∑ f ( y - mean(y) )^2 = ∑ f (y)^2 - { mean(y) ∑ f y } - { mean(y) ∑ f (y - mean(y) ) }
= ∑ f (y)^2 - mean(y)∑ f y ( since ∑ f(y - mean(y) ) = 0 , always)
= 5* (-6)^2 + 8*(-3)^2 + 13*(3)^2 + 11*(6)^2 - 1.04* ( -30-24+39+66)
= 180+72+117+396 - 1.04*51 = 765 - 53.04 = 711.96
( did you approximated your result?)

...what? I'm so lost and confused. Could you dumb it down a bit? what's the base of the variable? What's the variable? How did you get the new variables? How did

SUM OF ALL(x-MEAN)^2*f

become

∑ f ( x - mean(x) ) ^2 = ∑ f ( y + 57 - mean(y) - 57 )^2 = ∑ f ( y - mean(y) )^2 = ∑ f (y)^2 - { mean(y) ∑ f y } - { mean(y) ∑ f (y - mean(y) ) }
= ∑ f (y)^2 - mean(y)∑ f y ( since ∑ f(y - mean(y) ) = 0 , always)
= 5* (-6)^2 + 8*(-3)^2 + 13*(3)^2 + 11*(6)^2 - 1.04* ( -30-24+39+66)
= 180+72+117+396 - 1.04*51 = 765 - 53.04 = 711.96

???

And yes, the answer is approximated, but I got 711.92

Well , the numerical difference between our results is due to the fact that on the first hand you gave an approx. mean , it should be (51/49)+57 which is not exactly 58.04 . Base change is a term we commonly use in statistics , your variables were x= 51,54,57,60,63 I said to change it to y= x - 57 , so the new variables are -6(=51-57) , -3(=54-57) ...etc. Also y= 57 - x → fy= fx - 57f ( for all f ,x,y) summing over all the values we get ∑ fy
= ∑fx - 57 ∑f , dividing by ∑f we get mean(y) = (∑fy)/∑f = ( ∑fx)/∑f - 57
= mean(x) - 57 . And now for the most important part we note that x - mean(x) = y - mean(y) ; so that f ( x - mean(x) )^2 = f (y - mean(y) )^2
= fy^2 - 2f y mean(y) + f(mean(y))^2 , one can not but notice that with respect to summation over all values of f and y , the mean is constant. So summing we get ∑ f(x - mean(x) )^2
= ∑ f (y)^2 - 2 mean(y)∑ fy + (mean(y))^2 ∑ f =∑f (y)^2 - 2 (mean(y))^2 ∑f + (mean(y) )^2 ∑f = ∑f (y)^2 - (mean(y))^2 ∑f ; since (mean(y))^2 ∑f = ( ( ∑fy) / (∑f ) )^2 ∑ f = (∑fy) (∑fy)/(∑f)
=mean(y) ∑fy , we can also write the formula as ∑f (y)^2 - mean(y) ∑fy ; the rest follows . I hope it is clear to you now.