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.