Scatterplots By Wendy Knight

Review of Scatterplots  Scatterplots – Show the relationship between 2 quantitative variables measured on the same individual  Can not be done with categorical data  Explanatory Variable – Explains or causes the changes in the response variable (plotted on the x-axis)  Response Variable – measures an outcome or results of a study (plotted on the y-axis)  If there is no explanatory-response distinction, you can put the variables on either axis

Example Label the graph Label the Axis No breaks in the graph Plot the points

Example 2 Label the graph Label the Axis No breaks in the graph Plot the points

Interpreting Scatterplots  Look for the overall pattern  Describe the overall pattern using DIRECTION, FORM, and STRENGTH of the relationship  Look for outliers

Direction  Positively Associated: slopes upward from left to right  Negatively Associated: slopes downward from left to right  No Association PositiveNegativeNo CorrelationPositive

Form Strength  Linear  Non-linear  Quadratic  Exponential  Trigonometric  Determines how closely the points follow the form

Example  The scatter plot below shows a relationship between hours worked and money earned. Which best describes the relationship between the variables?

Review  Scatterplots display directions, form and strength of the relationship between two variables  Straight-line relationships are simple patterns and common  A straight-line relation is strong if the points lie close to the line; weak if they are widely scattered.

Thought Question 1:  Use following two pictures to speculate on what  influence outliers have on correlation.  For each picture, do you think the correlation is  higher or lower than it would be without the outlier?  (Hint: Correlation measures how closely points fall to a straight line.)

Thought Question 2:  A strong correlation has been found in a certain city in the northeastern United States between weekly sales of hot chocolate and weekly sales of facial tissues.  Would you interpret that to mean that hot chocolate causes people to need facial tissues? Explain.

Thought Question 3:  Researchers have shown that there is a positive correlation between the average fat intake and the breast cancer rate across countries. In other words, countries with higher fat intake tend to have higher breast cancer rates.  Does this correlation prove that dietary fat is a contributing cause of breast cancer? Explain.

Thought Question 4:  If you were to draw a scatterplot of number of women in the work force versus number of Christmas trees sold in the United States for each year between 1930 and the present, you would find a very strong correlation.  Why do you think this would be true?  Does one cause the other?

How can we counteract this?  We can standardize the correlation with a numerical value  Find the R value with the outlier and without

Correlation “r”  Correlation describes the direction and strength of a straight-line relationship between two quantitative variables. Correlation is usually written as r.  Positive r indicates positive association between variables  Negative r indicates negative association  r always falls between -1 and 1  Because r uses standardized scores, the correlation does not change when we change units of measurement  Correlation ignores distinction between explanatory and response variables  Correlation measures the strength of only straight-line association between two variables  Correlation is strongly affected by a few outlying observations

Example 1: Highway Deaths and Speed Limits  Correlation between death rate and speed limit is  If Italy removed, correlation drops to  If then Britain removed, correlation jumps to 0.70