Scatterplots, Association, and Correlation

Scatterplots are the best way to start observing the relationship and picturing the association between two variables. This timeplot is a scatterplot of the average error in nautical miles of the predicted position of Atlantic hurricanes for predictions made by the National Hurricane Center (NHC) of the National Oceanic and Atmospheric Administration (NOAA), plotted against the year in which the predictions were made. We can observe that in recent years, predictions have improved (the errors have reduced).

Roles for Variables Explanatory Variable - x (think independent variable) Response Variable - y (think dependent variable)

Which variable would be the explanatory variable and which would be the response variable? 1)T-shirts at a store: price each, number sold 2)People: age, grip strength 3)Apples: circumference, weight 4)Students: weight, score on a test explanatory: price; response: number sold explanatory: age; response: grip strength explanatory: circumference; response: weight explanatory: weight; score on a test

3 Things to Look for in Scatterplots:  Direction (positive, negative, neither)  Form (straight, curved, no pattern)  Strength of scatter  Unusual Features (outliers, subgroups, etc.)

Direction Positive RelationshipNegative Relationship (As x increases, y increases)(As x increases, y decreases)

Form Straight (Linear) Association Curved Association No Association

Strength Strong Association Moderate Association Weak or No Association

Example This scatterplot shows a moderate, negative, linear association. In general, lower central pressure is found in hurricanes that have higher maximum wind speeds. Maximum wind speeds vs. central pressure for 163 hurricanes that have hit the U.S. since 1851

Example Data collected from students in Statistics classes included their heights (inches) and weights (pounds) There is a fairly straight, positive association. But there is an outlier. In general, the taller students tend to weigh more. What type of association is there between height and weight for these students?

The distribution on the left is in inches and pounds while the distribution on the right is in centimeters and kilograms. Notice that the scatterplots look the same. Changing the units does not change the shape of the pattern.

We can also remove the units altogether by converting each value to a z-score and creating a scatterplot of the standardized scores. Here we have the weights and heights of the students again, with units removed.

Correlation How strong is the association? The correlation coefficient (r) gives the strength of the linear relationship of the two variables. (But we’ll let the calculator do the work)

Correlation r is always a number between -1 and 1 The closer it is to 0, the weaker the linear association

Correlation

Fast food is often considered unhealthy because much of it is high in both fat and sodium. But are the two related? Here are the fat and sodium contents of several brands of burgers. Analyze the association between fat content and sodium. Example Fat (g) Sodium (mg) There does not appear to be a relationship between sodium and fat content in burgers. The correlation of shows a weak relationship.

Be Careful! In order to use correlation… Both variables must be quantitative The form of the scatterplot must be fairly straight If an outlier is present, calculate with and without the outlier

Your teacher tells you that the correlation between the scores on Exam 1 and Exam 2 was )Before calculating the correlation, what should the teacher check? 2)If 10 points are added to each Exam 1 score, how will this change the correlation? 3)If she standardizes scores on each exam, how will it affect the correlation? Both variables are quantitative, so the teacher should check for a linear relationship and check for outliers It won’t change the correlation

Your teacher tells you that the correlation between the scores on Exam 1 and Exam 2 was ) In general, if someone did poorly on Exam 1, are they likely to have done poorly on Exam 2? 2) If someone did poorly on Exam 1, can you be sure that they did poorly on Exam 2 as well? They are likely to have done poorly. The positive correlation means low scores on Exam 1 are associated with low scores on Exam 2. (And high scores on Exam 1 are associated with high scores on Exam 2) The association is positive, but individual performances may vary because r isn’t equal to 1.

Correlation vs. Causation Pay close attention to the variables. Sometimes correlation is present even though one variable does not actually cause the other variable’s response. This can be a coincidence, or it can be due to a lurking variable.

Today’s Assignment:  Read Chapter 7  Homework p.164 #3-8, 10-12, 16, 18, 20, 23-26, 32, 33, 36, 40