It is not known why the letter m was chosen for slope; the choice may have been arbitrary. John Conway has suggested m could stand for “modulus of slope.” One high school algebra textbook says the reason for m is unknown, but remarks that it is interesting that the French word for “to climb” is monter.

In the equation of a straight line (when the equation is written as “y = mx + b”), the slope is the number “m” that is multiplied on the x, and “b” is the y-intercept (that is, the point where the line crosses the vertical y-axis).

The slope of a line is constant because it doesn’t depend on where you measure it. This is due to how we define the slope of a curve at some point, ie the slope of the tangent line at that point. So the tangent line at every point along a straight line is always the same.

In the equation y = mx + c the value of m is called the slope, (or gradient), of the line. It can be positive, negative or zero. Lines with a positive gradient slope upwards, from left to right. It is the value of y when x = 0. When drawing a line, c gives the position where the line cuts the vertical axis.

When two lines are perpendicular, the slope of one is the negative reciprocal of the other. Try this Drag points C or D. Note the slopes when the lines are at right angles to each other. So for example in the figure above, the line AB has a slope of 0.5, meaning it goes up by a half for every one across.

Slope of Line AB = (Rise/Run) = -1.

Slope is the ‘steepness’ of the line, also commonly known as rise over run. We can calculate slope by dividing the change in the y-value between two points over the change in the x-value.

When you look at the two lines, you can see that the blue line is steeper than the red line. It makes sense the value of the slope of the blue line, 4, is greater than the value of the slope of the red line, . The greater the slope, the steeper the line.

A positive value indicates a positive slope, while a negative value indicates a negative slope. In the function y = 3x, for example, the slope is positive 3, the coefficient of x. In statistics, a graph with a negative slope represents a negative correlation between two variables.

A higher positive slope means a steeper upward tilt to the curve, which you can see at higher output levels. A negative slope that is larger in absolute value (that is, more negative) means a steeper downward tilt to the line. A slope of zero is a horizontal line.

If the slope is negative, then there is a negative linear relationship, i.e., as one increases the other variable decreases. If the slope is 0, then as one increases, the other remains constant, i.e., no predictive relationship.

A regression coefficient is the same thing as the slope of the line of the regression equation. The equation for the regression coefficient that you’ll find on the AP Statistics test is: B1 = b1 = Σ [ (xi – x)(yi – y) ] / Σ [ (xi – x)2].

Interpreting the slope of a regression line The slope is interpreted in algebra as rise over run. If, for example, the slope is 2, you can write this as 2/1 and say that as you move along the line, as the value of the X variable increases by 1, the value of the Y variable increases by 2.

In this context, correlation only makes sense if the relationship is indeed linear. Second, the slope of the regression line is proportional to the correlation coefficient: slope = r*(SD of y)/(SD of x) Third: the square of the correlation, called “R-squared”, measures the “fit” of the regression line to the data.

The correlation, denoted by r, measures the amount of linear association between two variables. r is always between -1 and 1 inclusive. The R-squared value, denoted by R 2, is the square of the correlation. Correlation r = 0.9; R=squared = 0.81. Small positive linear association.

While for exploratory research, using cross sectional data, values of 0.10 are typical. In scholarly research that focuses on marketing issues, R2 values of 0.75, 0.50, or 0.25 can, as a rough rule of thumb, be respectively described as substantial, moderate, or weak.

linear correlation coefficient

A linear regression line has an equation of the form Y = a + bX, where X is the explanatory variable and Y is the dependent variable. The slope of the line is b, and a is the intercept (the value of y when x = 0).

Another way to calculate the correlation coefficient (r) is to multiply the slope of the regression line by the standard deviation of X and then divide by the standard deviation of Y.

r = 0 means there is no correlation. r = 1 means there is perfect positive correlation. r = -1 means there is a perfect negative correlation.