Calculating the Mean using change of origin

[Tascja]

I dont really understand what this question is asking me... like what the variables correspond with.

Question:
Let there be a new variable (mu)i = (xi-a)/20. (Let 'a' be the midpoint corresponding to the largest frequency). Calculate the mean weight of apples using this change of origin.

a = 112.5

 

[tkhunny]

8 9 10 9 8 8 11 7

Standard Method:

(8+9+10+9+8+8+11+7)/8 = 70/8 = 8.75

Change of Origin

Just looking at the data, I think the mean is around 9. I then assume the mean is 9 and let the differences from 9 prove otherwise.

(8,9,10,9,8,8,11,7) - 9 = (-1,0,1,0,-1,-1,2,-2)

The mean of these deviations, then (-1+0+1+0+(-1)+(-1)+2+(-2))/8 = -0.25 Since we started with 9, this gives 9 - 0.25 = 8.75

Of course, the astute observer might do it this way

(-1,0,1,0,-1,-1,2,-2)

Who cares about the zeros?

(-1,1,-1,-1,2,-2) <== This is why the mode is a good choice of Origin change. More zeros!

The first two offset each other

(-1,-1,2,-2)

The last two offset each other

(-1,-1)

This gives (-1-1)/8 = -0.25 and the mean is 8.75.