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I've seen several videos, Khan Academy included, explaining the correlation coefficient formula but none explain the "logic" behind the formula, not to my satisfaction anyways.

The Formula: r = 1 / n-1 ∑(x z-score)(y z-score) condensed the formula a bit for the z-scores

As I have it understood, the r coefficient is the average of the product of the z-scores

Questions

  1. But why? What is the logic behind finding the average product of the z-scores?

  2. What does the first part of the equation mean? 1 / n-1 the denominator looks like the sample population but why have 1 divided by the sample population?

Summary

The answer(s) I'm looking for would be something like, "We need to calculate the correlation between x and y and in order to do that we need to..." In other words, in intuitive approach to explaining the why of the formula.

Thank you all in advance.

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  • $\begingroup$ I think an honest explanation has to have more than one English sentence with no symbols. I have tried to give an intuitive account of this in my Answer. $\endgroup$ – BruceET Dec 29 '18 at 23:41
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Here is a scatterplot of 100 points, both x-values and y-values are roughly standard normal (that is, they are z-scores).

enter image description here

Notice in particular the first point, which is circled in red, roughly at $(0.41, 0.18)$ in the upper-right quadrant. It's contribution to the total $T = \sum_i x_iy_1$ is $x_1y_1 = 0.41(0.18) = +0.0738.$

In upper-right quadrant, all points have $x_i > 0$ and $y_i > 0$ and so they will all make positive contributions to $T.$

Similarly, points in the lower-left quadrant will all have $x_i < 0$ and $y_i < 0$ (both coordinates negative), so they will also all make positive contributions to $T.$

By contrast, the points in the other two quadrants will have one positive and one negative coordinate, so they will make negative contributions to the sum $T.$

In our sample, the correlation is positive. That is when $x$-values increase than corresponding $y$-values also tend to increase. Specifically, for the points in the figure $r = +0.567.$ The total $T$ has both positive and negative contributions, but the positive contributions win out because there are many more points in the upper-right and lower-left quadrants than in the other two quadrants.

If all 100 points were on an upward-sloping diagonal line through the origin, then all points would make positive contributions towards the correlation. The largest possible result.

One can prove mathematically that the regression coefficient $r = \frac{1}{n-1}T$ is always between $-1$ and $+1.$ That is, $ -1 \le r \le 1.$

One reason for dividing $T$ by $n - 1$ is so that this convenient rule will hold true. [The denominator has to depend on $n$ because as we add more points to the plot, the total $T$ tends to become increasing large in absolute value.]

Note: Technically, the reason it's $n-1$ instead of $n$ is that in computing $z$-scores we divide by the the two sample standard deviations, both of which have $n-1$ in their denominators: $$r = \frac{1}{n=1}T = \frac{1}{n-1}\sum_i \left(\frac{X_i-\bar X}{S_x}\right) \left(\frac{Y_i -\bar Y}{S_y}\right),$$ where $S_x^2 = \frac{1}{n-1}\sum_i (X_i - \bar X)^2,$ and similarly for $S_y^2.$

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