MATH 2560 C F03 Elementary Statistics I LECTURE 6: Scatterplots (Continuation).

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1 MATH 2560 C F03 Elementary Statistics I LECTURE 6: Scatterplots (Continuation). 1 Outline. adding categorical variables to scatterplots; more examples of scatterplots; categorical explanatory variables;

2 2 Adding Categorical Variables to Scatterplots We can introduce a third variable into the scatterplot, namely, categorical variable. For example, the Census Bureau groups the states into four broad regions, named Midwest, Northeast, South, and West. The question is: how about regional patterns in SAT exam scores? Figure 2.2 repeats part of Figure 2.1, with an important difference: the Northeast and Midwest groups of states have been plotted only, using the plot symbol e for the northeastern states and the symbol m for the midwestern states.

3 Analysing the Scatterplot 9 Northeastern states ( e ) are all SAT states, at least 68 percents of high school graduates in each of these states take the SAT; 12 Midwestern states ( m ) are mostly ACT states; 10 of these states have the percent taking the SAT between 4 pewrcents and 12 percents; 2 Midwesten states are outliers within the region; Indiana ( m )(60 percents take the SAT) falls close to the northestern ( e ) cluster; Ohio ( m ), where 26 percents take the SAT, lies outside the Midwestern cluster. = we introduced a third variable into the scatterplot, dividing the states into regions. = Region is a categorical third variable. The two regions are displayed by the two different plotting symbols. How to Add Categorical Variables in Scatterplots To add a categorical variable to a scatterplot, use a different plot color or symbol for each category

4 3 More Examples of Scatterplots The foundation for more detailed study the relationships among quantitative variables is experience in examining scatterplots. Let us consider more Examples of scatterplots. Example 2.4. The lengths of two bones: Femur and Humerus Data Femur: 38, 56, 59, 64, 74 Humerus: 41, 63, 70, 72, 84 There is no explanatory-response distinction, so we can put either measurement on the x axis of a scatterplot.

5 The plot appears in Figure 2.3. Individuals: Five (5) species; Two Variables: two (2) bones: femus and humerus. = Association: strong positive association, because points lie close to a stright line; = Form: linear; = Positive association: length of 1 bone increase, so does the length of the other bone.

6 Example 2.5. Corn Yields (Table 2.1) = How much corn per acre should a farmer plant to obtain the highest yield? Explanatory variable: planting rate (horizontally plotted, x variable); Response variable: yield (vertically plotted, y variable);

7 The scatterplot in Figure 2.4 displays the results of this experiment. indicates mean yield. Form: nonlinear; Association: neither positive nor negative; = Incomplete data often complicate a statistical analysis: all five planting rates had to be used in all four years; = Important fact: the relationship between two variables often cannot be fully understood without knowledge about other variables (moisture, fertilizer, etc.).

8 4 Categorical Explanatory Variables = To display a relationship between a categorical explanatory variable and quantitative response variable, make a side-by-side comparison of the distributions of the response for each category. Previous Examples. 1. A back-to back stemplot compares two distributions (comparison of home run counts (the quantitative response) hit by Babe Ruth and Mark McGwire (the categories); 2. Side-byside boxplots compare any number of distributions (comparison of gas milage (the quantitative response) for minicompact and two-seater cars on the highway and in the city (four categories)and. Example 2.7. Education vs Personal Income = Categorical variables: education 1) No Higher School (HS); 2) Some HS; 3) HS grad; 4) Some college; 5) BS degree; 6) Higher degree. = Quantitative variables: personal income (dollars/year) The side-by-side boxplots in Figure 2.6 show the distributions of personal income in March 2000 for people between the ages 25 and 65 years at each level of education.

9 Association: strong positive between education and ernings; Figure 2.6 Summary: 1) the 5th percentile is low in all groups; 2) the median and quartiles increase with more education; 3) Q 1 (Some college)> Q 3 (No HS); 4) the 95th percentile shoots up in the college-educated groups (contain many people with high income).

10 5 Summary Look for the form, direction, and strength of the relationship, and then for outliers. 1. Form: Linear relationships, where the points show a straight-line pattern, are an important form of relationship between two variables. Curved relationship and clusters are other forms to watch for. 2. Direction: If the relationship has a clear direction, we speak of either positive association (high values of the two variables tend to occur together) or negative association (high values of one variable tend to occur with low values of the other variable). 3. Strength: The strength of a relationship is determined by how close the points in the scatterplot lie to a simple form such as a line. 4. To display the relationship between a categorical explanatory variable and a quantitative response variable, make a graph that compares the distributions of the response for each category of the explanatory variable.