Best fit lines can also be drawn approximately by eye. We start by finding the average x and y, giving the point (xˉ,yˉ​). We then take a ruler and place it on this point, and adjust the slope until we find a reasonable best fit line.

Linear regression is a statistical method used to model the relationship between two variables when data is given as pairs of points (x,y). We fit a straight line (called the regression line) that minimizes the average vertical distance from the points:

The general equation of the regression line is:

where a is the slope and b is the y-intercept.

The values of a and b can be found using a calculator:

• Use Stat>Edit to fill in x- and y-values into L1​ and L2​.

• Then, press Stat, right arrow to the CALC menu, and select 4:LinReg(ax+b).

Pearson's product-moment correlation coefficient, denoted by r, measures the strength and direction of a linear relationship between two numerical variables x and y. Its value always lies between −1 and +1:

• r=+1: perfect positive linear relationship

• r=−1: perfect negative linear relationship

• r=0: no linear relationship

A positive value means y generally increases as x increases; a negative value means y generally decreases as x increases. The closer r is to ±1, the stronger the linear relationship.

If you clickmode, scroll to STAT DIAGNOSTICS , hover over ON, and click ENTER, then any time you perform a linear regression, the calculator will provide Pearson's coefficient in addition to the regression line.

Once we have a regression line y=ax+b, we can use it to predict y by plugging in a value of x.

When using a regression line to predict y from x, we need to be aware of the danger of extrapolation. This occurs when we try to predict y for a value of x far outside the range of x values in our data. For such an x, we cannot trust that the relationship is the same.

While it is possible to use a regression line y=ax+b to predict x with

this is not a reliable process. The best fit line is determined to minimize the difference between the real y’s and the predicted y’s,so the difference between real and predicted values for x may be much larger.

Spearman's rank correlation coefficient rs​ measures how close data is to be monotonic - either solely increasing or solely decreasing. Spearman's coefficient is calculated by ranking x- and y-values from least to greatest and performing a linear regression on the ranked lists.

If rs​=1, then the data is perfectly monotonic increasing: as one variable increases, the other does too.

If rs​=−1, then the data is perfectly monotonic decreasing: as one variable increases, the other decreases.

Spearman's coefficient is useful because it captures all monotonic relationships - unlike Pearson's coefficient, which only measures linear monotonic relationships. Consequently, Spearman's coefficient is not as sensitive to outliers as Pearson's coefficient.