If a straight line passes through two points (x_1, y_1) and (x_2, y_2) where x_2 > x_1. then the run is (x_2 - x_1), the rise is (y_2 - y_1) and the slope m of the line is defined as:
m = (Delta y)/(Delta x) = run / rise = (y_2 - y_1)/(x_2 - x_1)
If instead of a straight line, we have a function f(x) which is continuous and otherwise well-behaved over the interval [x_1, x_2] and f(x_1) = y_1 and f(x_2) = y_2 then the average slope of f(x) over the interval [x_1, x_2] is also m = (y_2 - y_1)/(x_2 - x_1)
If you are familiar with the terminology, we are basically evaluating the integral of the derivative of f(x) over the interval [x_1, x_2], then dividing by the length of the interval. This is like adding up the slopes at each point and dividing by the number of measurements to get the average.
