Difference between directly proportional positive correlation?

Directly proportional is written as Y ~ X; where Y, the independent variable, is directly proportional to X, the dependent variable. Whenever we write something like this, we can make the ~ into an = by the so-called constant of proportionality. We can use any symbol we wish for this constant, but k is a popular one. So we can write Y = kX and be mathematically correct. Directly proportional implies a deterministic (not statistical) relationship.

Do you recognize this form Y = kX? What if I write F = ma or maybe F = kN, or how about F = -kX? What these commonplace equations say is the dependent force is proportional to the independent acceleration a, normal weight N, or spring compression X. And if we were to graph these out, each one would form a straight line with m, k, and - k as the respective slopes.

Positive correlation is a term reserved for statistical data. The coefficient of correlation r, is specifically defined and it is a measure of how well the data explain a straight line. r^2 = 1.000 when all the data points fall exactly on top of the straight line the data define. r^2 = 0 when the data points are all over the place (like a circles around the line). That is, in this last case, the data don't define a straight line at all.

The correlation is positive when the defined so-called trend line slants upward from zero and higher. It is negative if the line slants downward with increasing values of the independent variable.

Difference between directly proportial positive correlation and WHAT?

maybe if you picture it as a graph? directly proportional PC would be a straight line, PC that isnt directly proportional would be a curve. As one variable increases, so does the other, but by differing amounts