UNIT 4 Bivariate Data Scatter Plots and Regression

What is Bivariate Data? Bivariate Data are two quantitative variables.

Example: The percentage of people who would vote for a woman president over the last century

Displaying Bivariate Data Bivariate Data is typically displayed with a Scatter Plot Scatterplots may be the most common and most effective display for data. Scatterplots are the best way to start observing the relationship and the ideal way to picture associations between two quantitative variables. X axis – the explanatory (or predictor) variable. Y axis – the response variable.

Examples: For the following state the explanatory variable and the response variable. Do students learn better with the amount of homework assigned? Does the weight of a car affect the miles per gallon that the car gets? Are SAT math scores and GPA related? Is there an association between a person’s speed and the amount of weight they can squat?

What should you look for in a scatterplot Direction – which way are the points going? positive, negative, neither. Form - linear, quadratic, exponential, logarithmic Strength – how much scatter is there in the plot? Weak, moderate, strong

Example: Peak Period Freeway Speed and cost per person What is the direction, form strength?

What is the direction, form strength?

Outliers You can describe the overall pattern of a scatterplot by Direction, Form, and Strength of the relationship. An important kind of deviation is an outlier, an individual value that falls outside the overall pattern of the relationship.

A scatter plot is a picture of the relationship between two quantitative variables. If a liner relationship exists between two variables the scatter plot will exist as a swarm of points stretched out in a generally consistent manner. If the relationship isn’t straight, we can find ways to make it more nearly straight.

Correlation Calculation The correlation coefficient (r) gives us a numerical measurement of the strength of the linear relationship between the explanatory and response variables. The calculator will do the work.

Correlation Conditions Correlation measures the strength of the linear association between two quantitative variables. Before you use correlation, you must check several conditions: –Quantitative Variables Condition – Correlation is only used for quantitative variables –Straight Enough Condition - But correlation measures the strength only of the linear association, and will be misleading if the relationship is not linear. –Outlier Condition – Outliers can distort the correlation. When you see an outlier, it’s often a good idea to report the correlations with and without the point. Sli de 7- 13

Correlation Correlation describes the direction and strength of a linear relationship -1 ≤ r ≤ 1 r > 0 positive linear association r = 0 no association r < 0 negative linear association

Strength of the relationship

Correlation and Causation We need to be careful about interpreting correlation coefficients. Just because two variables are highly correlated does not mean that one causes the other. In statistical terms we say that correlation does not imply causation. Examples: The number of ice cream sales and the number of shark attacks on swimmers are highly correlated. The increase in stock prices and the length of women’s skirts are highly correlated. The number of cavities in elementary school children and vocabulary size have a strong positive correlation.

Three relationships which can be taken (or mistaken) for causation Causation – Changes in X causes changes in Y Common response – Both X and Y respond to some unobserved variable Confounding – The effect of X on Y is hopelessly mixed up with the effects of other variables on Y.

Correlation and Causation Association, relationship and correlation does not imply causation Causation does imply association, relationship and correlation.