8.+Scatterplots

A scatter plot is a type of mathematical diagram using Cartesian coordinates to display values for two variables for a set of data.¹ Scatterplots use two axes (vertical and horizontal) to plot points of data. By using this type of graph, one can distinguish how one variable is affected by another. This is called correlation.

There are seven general types of Scatter plots, those with a: • Perfect positive correlation • Perfect negative correlation • High positive correlation • High negative correlation • Low positive correlation • Low negative correlation • No evident correlation

The R value (see Regression) represents the preciseness of the graph. If a graph had a perfect positive correlation (meaning it had a perfect slope of the correlation of its points increasing over values of x), then it is said to have an R-value of 1. If it has a perfect negative correlation (same as positive except correlation of points decreases over values of x), it is said to have a value of -1. If there is no correlation present, it has a value of 0. For a positive correlation, the closer the number is to 1, the more closely it resembles a perfect positive graph; closer to 0 resembles a non-evident graph. For a negative correlation, the closer the R value is to -1, the more closely it resembles a perfect negative correlation; again, the closer it is to 0, the more it resembles no evident correlation.

An example of a perfect positive correlation could be the time spent on a treadmill and the distance covered (assuming a constant speed). The time intervals would be on the x-axis while the distance covered would be on the y-axis. The distance would be dependent on the time meaning that the more time spent on the treadmill, the more distance would be covered.² An example of a perfect negative correlation could be the length of a pencil and the number of letters or numbers that the pencil wrote. This time, the number of symbols written would be on the x-axis and the length of the pencil would be on the y-axis because the length of the pencil is dependent on the number of marks made by that pencil.² The two graphs below show a perfect positive correlation and a perfect negative correlation.

An example of a strong positive correlation, but not a perfect positive correlation, could be the number of hours spent studying and the grades received on the tests. Even though an increase in the number of hours spent studying will most likely result in better test grades, it might not necessary be the case. An example of a strong negative correlation, but not a perfect negative correlation, could be the amount of sunlight and the amount of carbon dioxide in an area. As the amount of sunlight increases in an area, the plants will most likely create more glucose since they are going through photosynthesis constantly and therefore decrease the amount of carbon dioxide in an area. However, there are times when an increase amount sunlight might not necessarily result in a direct decrease of carbon dioxide. There still might be a high level of carbon dioxide even after the plant goes through photosynthesis. Here are some examples of High positive, low negative, high negative, low positive, and no correlation:





SAT Questions:³

1) r=0.63 2) r=0.94 _ 3) r= -0.4 _ 4) r=-0.78 _

5) A computer lab has 20 computers, Usually there are a number of computers out of order. What is the correlation coefficient between the number of computers out of order and the number of computers working? A) 0 B) 2 C) 0.5 D) -0.5 E) 1 F) -1

6) The average yearly wine consumption and average life span in 50 countries was computed. The correlation for the 50 pairs of averages was r= 0.8. Can you conclude that the correlation between wine consumption and life span for the millions of individuals in these 50 countries is also 0.8? A) Yes B) No, r would probably be higher than 0.8 C) No, r would probably be lower than 0.8

Answers: 1) C 2) B 3) D 4) A 5) F 6) C

¹ http://en.wikipedia.org/wiki/Scatter_plot ² http://mste.illinois.edu/courses/ci330ms/youtsey/scatterinfo.html ³ http://www.stat.uiuc.edu/courses/stat100/Exams/F09Keys/Exam2m.pdf