AP Stats ~ 3A: Scatterplots and Correlation OBJECTIVES:

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1 OBJECTIVES: IDENTIFY explanatory and response variables in situations where one variable helps to explain or influences the other. MAKE a scatterplot to display the relationship between two quantitative variables. DESCRIBE the direction, form, and strength of a relationship displayed in a scatterplot and identify outliers in a scatterplot. INTERPRET the correlation. UNDERSTAND the basic properties of correlation, including how the correlation is influenced by outliers USE technology to calculate correlation. EXPLAIN why association does not imply causation.

2 Scatterplots are used to compare QUANTITATIVE variables within one group of individuals. (For example, we can plot people s weights and heights in order to compare them.) One of the reasons we like to use scatterplots is that they allow us to explore whether one variable affects another (they have a specific relationship) in addition to just exploring the general relationship between two variables.

3 DEFINITIONS: EXPLANATORY VARIABLE (aka Independent variable): explains the observed outcomes. (Usually plotted as the xvariable.) RESPONSE VARIABLE (aka Dependent variable): measures an outcome of a study. (Usually plotted as the yvariable.) Loading... Please Note!!! Just because we see a relationship between 2 variables, it doesn't mean there is cause & effect happening.

4 time exam Examples: Identify the response and explanatory variables if there are any. a)amount of time studying and the exam grade Explanatory b)weight and height of a person NO clear explanatory 1 response spent studying, response c)yearly rainfall and yield of a crop Explanatory yearly rainfall, response d) student s grades in Statistics and French No explanatory / response crop yield grade

5 GRAPHING SCATTERPLOTS: 1.Place the explanatory variable on the xaxis, and the variable on the yaxis. response 2.Label the axes! Your labels should be specific don t just name what you are counting on the axis, put the unit of measure as well. Loading... 3.Try to find a scale that will fit your whole grid. We don t everything stuck in one corner if we can avoid it. want 4.Make your scatterplot big enough to allow your data to out a bit. spread

6 INTERPRETING A SCATTERPLOT: 1. Look for overall patterns and major deviations (outliers). 1.Describe. the overall pattern. Include a description of the form, the direction, and the strength. All 3 must be discussed! FORM: curved? Are there clusters (groups)? Outliers? Is it linear or DIRECTION: Is the data positively associated, negatively associated, or is there no visible association? STRENGTH: This describes how closely the points follow a clear form. If they are closely clustered and following a pattern we consider it to be a strong relationship. If the points are randomly scattered, it s a weak relationship.

7 values MORE ABOUT "DIRECTION": Positively Associated means that as values increase along the xaxis, the corresponding yvalues also increase. (Above average values or below average values are clustered.) Negatively Associated means that above average eragexxvalues cluster with below average yvalues. (As one value increases, the other decreases.) correspond

8 Example: Manatees are large gentle sea creatures that live along the Florida coast. Many manatees are killed or injured by powerboats. Here are the powerboat registrations (in thousands) and the number of manatees killed by boats in Florida in the years A)Which is the explanatory variable? B) Make a scatterplot of the data. C) Describe the relationship. see # of powerboats see next page next page X X x y

9 Clusters ± : 55 : 50 T z s " o o # 15. :%s to its too ' ho ' too ' also ' too ' to # of registered powerboats 4,000's ) Form : Somewhat linear. in the lower Left corner. Direction : Positively associated strength : Relatively strong.

10 Now use your calculator to make the scatterplot. Is it the same or different? CALCULATOR INSTRUCTIONS: Input explanatory data into L1, and response data into L2. Go into statplot (2nd Y=) and define as follows Use ZoomStat (Zoom 9) to graph. PLEASE NOTE: the calculator doesn t show a scale, however, when you sketch the graph on your homework or the test, you will need to label and scale your axes! Use the trace feature to figure out the scales the calculator used.

11 We ve been describing our scatterplots based on strength, direction, and form. Unfortunately, 2 different people may have differing ideas about what makes a relationship strong, or relatively strong. Depending on the scale of the scatterplot, a relationship may look clustered or very widely spread, which will give the interpreter a different idea of the strength and form of a relationship. In order to be consistent in our interpretation, we measure the correlation of a relationship. NOTE: This only works with linear relationships!

12 DEFINITION: CORRELATION: correlation measures the direction and strength of a LINEAR relationship between 2 quantitative variables. The notation for correlation is Formula for correlation: Loading... This formula is the "average" of the product of the zscores. The reality is that we will never use the formula ~ we will use our calculators.

13 KEY FACTS ABOUT CORRELATION: You absolutely need to know these things! ALWAYS! The closer you get to 1 or 1, the stronger your relationship. 1 means that there is a perfect negative correlation. Every point of data lies on the same line. 0 means there is absolutely no correlation at all (this would be a randomly scattered plot) +1 means that there is a perfect positive correlation.

14 A positive value of r means that you have a positive association and vice versa. This method only works with LINEAR relationships! You may get a good correlation value on a graph that is curved (like an exponential), but the value is meaningless in that case. It doesn t matter which variable is x or y. Explanatory vs. response doesn t matter here. Why?

15 Both variables must be quantitative. The r value doesn t change if the unit measure changes, because standardized values don t have units of measure. Because r is based on the mean and the standard deviation, it is not resistant. Outliers will reduce the value of r. Removal of outliers will increase it. Correlation alone is not considered to be a complete description of twovariable data. Your description must also include form, direction, and strength.

16 CORRELATION DOES NOT IMPLY CAUSATION! A good correlation value does not necessarily mean that there is a cause and effect relationship occurring. Both variables could be related to a 3rd variable that affects them both the same way. For example: Pollen counts and sun screen sales have a positive correlation, but they do not affect each other. They are however both related to the weather. There is no set value of r that signifies a strong relationship, although a good rule of thumb is that the. association is strong if r < 0.8 or r > 0.8

17 Guess the correlation. a o k 0.1 r=o. 3 r= 0.5 r O 7 r 0.9

18 Example: The following scatterplot comparing the average number of points scored per game and the number of wins for college football teams in the Southeastern Conference. For these data, r= Interpret the value of r in context. The highlighted point is Mississippi. What effect does it have on the correlation? There is a strong, positive Correlation between the number of points scored per game and the number of Wins by college football teams. Th point is slightly outside of Fhe pattern, so it would weaken the correlation.

19 Homework: Page 159: #15, 725 odds, 2732 Read pp