Where does slope collide?

The number that measures how steep a straight line is, found as the change in the vertical coordinate divided by the change in the horizontal coordinate between any two points on the line. The idea belongs to the coordinate geometry that René Descartes set out in La géométrie (1637), which represented lines and curves by equations on a pair of axes and so let a line's inclination be read off as a fixed ratio. In its modern form the value is the coefficient m in the equation y = mx + b, a single dimensionless number that is the same everywhere along the line. Here the referent is a fixed property of an abstract line, not a quantity estimated from noisy data or the physical tilt of a hillside.[1]

Unit

A dimensionless ratio, rise over run, the same at every point on the line

Recovery means

Not applicable

Measured by

Computed as the difference in y-values divided by the difference in x-values between two points, or read as the coefficient m in y = mx + b

The coefficient that gives the average change in an outcome for a one-unit change in a predictor, estimated by fitting a line to scattered observations. Francis Galton's 1886 Regression Towards Mediocrity in Hereditary Stature introduced the fitted line whose gradient described how children's heights related to their parents', and Karl Pearson's 1896 Regression, Heredity, and Panmixia gave it its formal mathematical treatment. Unlike a fixed geometric slope, this value is an estimate that carries uncertainty: it has a standard error, is tested against zero, and can shift with the sample and the model. The referent is a parameter of a statistical relationship, not the steepness of a single drawn line or the tilt of a physical hillside.[2][3]

Unit

An estimated coefficient, in the outcome's units per one unit of the predictor

Recovery means

Not applicable

Measured by

Estimated by least squares or related fitting, reported with a standard error and a confidence interval and tested against a null of zero