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I'm taking an online stats class. Currently we are on the topic of a correlation coefficient. Several formulas for $r$ were given in the reading material. The formulas are below. Unfortunately it was not explained how these formulas were derived.

Formula 1:

$$r_{XY}=\frac{\sum{z_X z_Y}}{n-1}$$

Formula 2:

$$r_{XY}=\frac{SP}{(n-1)(s_x)(s_y)},$$

where $SP$ is the sum of products and calculated as

Formula 3: $$SP=\sum{xy} - \frac{\sum{x} \sum{y}}{n}$$

Formula 4: $$r_{XY} = \frac{n \sum{xy} - \sum{x}\sum{y}}{\sqrt{[n\sum{x^2} - (\sum{x})^2]} \sqrt{[n\sum{y^2} - (\sum{y})^2]}}$$

That's a lot of formulas with no explanations.

  1. How is formula 1 derived?

  2. Can someone explain how we arrived to Formula 2 from Formula 1?

  3. How did we arrive to formula 3 from formula 1?

  4. What do $s_x$ and $s_y$ mean in the denominator of formula 2? Nowhere in the previous readings as well as lecture videos it was explained?

  5. Sum of products to me would more look like $SP = \sum{xy}$, so where does $- \frac{(\sum{x}) (\sum{y})}{n}$ in Formula 3 comes from?

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    $\begingroup$ You may find the answer to this question of some value; it may also help you to focus your question further. $\endgroup$ – Glen_b Feb 26 '16 at 8:19
  • $\begingroup$ What is $z_X$? The formulas 3 and 4 are derived from formula 2 by substituting the definitions of $SP$ and $s_x$ and $s_y$. Note that you need to give at least one formula for correlation coefficient as given, since you need to define it. When you have a definition then all the other formulas are simply equivalent mathematical transformations of the original formula. $\endgroup$ – mpiktas Feb 26 '16 at 8:26
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    $\begingroup$ Taking the first formula as the definition, the rest are obtained by straightforward algebra. I am sure you have seen expressions like "$s_x$" before: I would hunt for them in the materials where the standardized values $z_x$ and $z_y$ were defined. $\endgroup$ – whuber Feb 26 '16 at 15:37
  • $\begingroup$ @mpiktas That's the problem, it is not explained what $z_X$, the formula is just given. The same goes for formula 2, it doesn't explain what $s_x$ and $s_y$ are. If you know what they are, do you mind filling in the blanks? $\endgroup$ – flashburn Feb 26 '16 at 18:46

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