### Video Transcript

If all points on a scatter diagram lie directly on a straight line of negative slope, what is the value of the product-moment correlation coefficient for this data set?

A scatter diagram is used as a tool for visualizing bivariate data, that is, data where two measurements are recorded for each subject, for example, their height and their weight, or maybe the amount of time spent watching TV and the amount of time spent doing homework.

Weโre told that for this data set, every point on a scatter plot lies directly on a straight line with a negative slope. So the scatter plot looks like this. Weโre then asked to determine from this information alone the value of the product-moment correlation coefficient for this data set. The product-moment correlation coefficient, or PMCC, which we often denote using the letter ๐, is a measure for quantifying the strength of linear correlation between two variables and how closely they follow their statistical line of best fit.

๐ can take values in the closed interval from negative one to one. And the value of ๐ determines the strength of the linear correlation. We can visualize this on a number line. A data set for which the value of ๐ is negative one has perfect negative linear correlation. And a data set for which the value of ๐ is positive one has perfect positive correlation. If the value of ๐ is zero, then there is no linear correlation between the two sets of data.

We can go further than this and describe data sets as having either weak, strong, or moderate positive or negative correlation depending on where their values of ๐ fall on the scale. So if, as in this example, points lie directly on a straight line with no deviation from it, then there is perfect linear correlation between the two variables.

If the slope of the line is negative, as we have here, then that perfect correlation is negative. From our number line, this means that the value of ๐ for such a data set will be negative one. It is important to know that this doesnโt mean that the slope of the line itself is negative one. Regardless of how steep or shallow the line is, there will still be perfect negative correlation and an ๐-value of negative one if all points lie directly on the line and it has a negative slope.

Our answer to the question is that if all points on the scatter diagram lie directly on a straight line of negative slope, then the value of the product-moment correlation coefficient is negative one.