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 \bar{x}, \bar{y} and \sum x_iy_i I will be all set. I will explain these terms in a moment. First, here is the sum:

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

Where x_i and y_i are the x and y (sample) averages.

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

= \sum x_iy_i - x_i\bar{y} - y_i\bar{x} + \bar{x}\bar{y}

= \sum x_iy_i - \bar{y}\sum x_i - \bar{x}\sum y_i +\sum\bar{x}\bar{y}

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

 

[rock.freak667]

Yes

\sum\bar{x}\bar{y} = n\bar{x}\bar{y}

Also remember that

\sum x_i = n \bar{x}

\sum y_i = n \bar{y}

So what does

- \bar{y}\sum x_i - \bar{x}\sum y_i +\sum\bar{x}\bar{y}

become?

 

[Saladsamurai]

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

Yes?

 

[rock.freak667]

Oh neat. Looks like it reduces to n\bar{x}\bar{y}

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 \sum x_iy_i in terms of \bar{x}, \bar{x}, and n. Perhaps there isn't one?

 

[statdad]

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

No, there is not.