Solve Tricky Summation Homework Statement

[Saladsamurai]

Homework Statement

I have this nasty summation and I am close to finding a way to calculate it with my graphing calculator. I just need to iron out the details. If I can rewrite the summation on terms of [itex]\bar{x}[/itex], [itex]\bar{y}[/itex] and [itex]\sum x_iy_i[/itex] I will be all set. I will explain these terms in a moment. First, here is the sum:

[tex]S_{xy} = \sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})[/tex]

Where [itex]x_i[/itex] and [itex]y_i[/itex] are the x and y (sample) averages.

If I do out the multiplication I get [tex]S_{xy} =
\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})[/tex]

[tex] = \sum x_iy_i - x_i\bar{y} - y_i\bar{x} + \bar{x}\bar{y}[/tex]

[tex] = \sum x_iy_i - \bar{y}\sum x_i - \bar{x}\sum y_i +\sum\bar{x}\bar{y}[/tex]

Now I it is just the last term [itex]\sum\bar{x}\bar{y}[/itex] that is bothering me. I think the answer is obvious, but I would like confirmation: is the expression [itex]\sum\bar{x}\bar{y}[/itex] identical to [itex]n*(\bar{x}\bar{y})[/itex] ?It must be. I don't know why I am doubting this.

 

[rock.freak667]

Yes

[tex]\sum\bar{x}\bar{y} = n\bar{x}\bar{y}[/tex]

Also remember that

[tex]\sum x_i = n \bar{x}[/tex]

[tex]\sum y_i = n \bar{y}[/tex]

So what does

[tex]- \bar{y}\sum x_i - \bar{x}\sum y_i +\sum\bar{x}\bar{y}[/tex]

become?

 

[Saladsamurai]

Oh neat. Looks like it reduces to [itex]-n\bar{x}\bar{y}[/itex]

Yes?

 

[rock.freak667]

Oh neat. Looks like it reduces to [itex]n\bar{x}\bar{y}[/itex]

Yes?

Yes

Which can also be written as (1/n)∑x_i∑y_i

 

[Saladsamurai]

I like this. If I could write it all on terms of xbar, ybar, and n, that would be best. I think that the sum(xi*yi) term can be written in terms of these. I'll post back I'm a moment. Thanks again!

Hmmm... I am not seeing an easy way to write [itex]\sum x_iy_i[/itex] in terms of [itex]\bar{x}[/itex], [itex]\bar{x}[/itex], and n. Perhaps there isn't one?

 

[statdad]

"I am not seeing an easy way to write in terms of [itex] \bar x[/itex], [itex] \bar y [/itex] , and n. Perhaps there isn't one?"

No, there is not.