Section 3.1Section 3.1ScatterplotsScatterplots

Two-Variable Two-Variable Quantitative DataQuantitative Data

Most statistical studies involve more than one Most statistical studies involve more than one variable.variable.

We may believe that some of the variables We may believe that some of the variables explain or even cause changes in the explain or even cause changes in the variables. Then we have explanatory and variables. Then we have explanatory and response variables.response variables. Explanatory—like the independent variable, it Explanatory—like the independent variable, it

attempts to explain the observed outcomes.attempts to explain the observed outcomes. Response—like the dependent variable, it measures Response—like the dependent variable, it measures

an outcome of a study.an outcome of a study.

ExamplesExamplesIdentify the explanatory and response variables:Identify the explanatory and response variables:• Alcohol causes a drop in body temperature. Alcohol causes a drop in body temperature.

To measure this, researchers give several To measure this, researchers give several different amounts of alcohol to mice, then different amounts of alcohol to mice, then measure the change in their body temperature measure the change in their body temperature after 15 minutes.after 15 minutes.

• If an object is dropped from a height, then its If an object is dropped from a height, then its downward speed theoretically increases over downward speed theoretically increases over time due to the pull of gravity. To test this, a time due to the pull of gravity. To test this, a ball is dropped and at certain intervals of time, ball is dropped and at certain intervals of time, the speed of the ball is measured.the speed of the ball is measured.

ScatterplotsScatterplots Used for Used for two-variable quantitativetwo-variable quantitative data! data! Explanatory variable goes on the x-axisExplanatory variable goes on the x-axis Response variable goes on the y-axisResponse variable goes on the y-axis The explanatory variable does not The explanatory variable does not

necessarily “necessarily “CAUSECAUSE” the change in the ” the change in the response variable.response variable.

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Make a scatterplot of the relationship between body weight and pack Make a scatterplot of the relationship between body weight and pack weight.weight.

Since Body weight is our eXplanatory variable, be Since Body weight is our eXplanatory variable, be sure to place it on the X-axis!sure to place it on the X-axis!

Body weight (lb) 120 187 109 103 131 165 158 116

Backpack weight (lb) 26 30 26 24 29 35 31 28

Interpreting GraphsInterpreting Graphs

One Variable One Variable Quantitative DataQuantitative Data

Two-Variable Two-Variable Quantitative DataQuantitative Data

CenterCenter FormFormLinear? Clusters? Gaps?Linear? Clusters? Gaps?

ShapeShape DirectionDirectionPositive? Negative?Positive? Negative?

SpreadSpread StrengthStrengthStrong? Weak? Moderate?Strong? Weak? Moderate?

OutliersOutliers OutliersOutliers

In sentence form…In sentence form…

There is a (There is a (strong/weakstrong/weak), ), ((positive/negativepositive/negative), (), (linear/non-linearlinear/non-linear) ) relationship between (relationship between (your two your two variablesvariables).).

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DirectionDirection FormFormStrengthStrength

Outlier There is one possible outlier, the hiker

with the body weight of 187 pounds seems to be carrying relatively less weight than are the other group members.

There is a moderately strong, positive, linear relationship between body weight and pack weight.

It appears that lighter students are carrying lighter backpacks.

Adding Categorical Adding Categorical Variables to ScatterplotsVariables to Scatterplots

You can use different plotting symbols or You can use different plotting symbols or different colors to designate a categorical different colors to designate a categorical variable.variable.

You still have two quantitative variables, You still have two quantitative variables, but you can add a “category” to these but you can add a “category” to these variables.variables.

Some quick tips for Some quick tips for drawing scatterplotsdrawing scatterplots

Choose an appropriate scale for the Choose an appropriate scale for the axes. Use a break if appropriate.axes. Use a break if appropriate.

Label, Label, Label…Label, Label, Label… If you are given a grid, try to use a scale If you are given a grid, try to use a scale

that will make the scatterplot use the that will make the scatterplot use the whole grid.whole grid.

Section 3.2 CorrelationSection 3.2 CorrelationWe are not good judges!We are not good judges! We shouldn’t just rely on our eyes to tell We shouldn’t just rely on our eyes to tell

us how strong a linear relationship is.us how strong a linear relationship is. We have a numerical indication for how We have a numerical indication for how

strong that linear relationship is – it’s strong that linear relationship is – it’s called CORRELATION.called CORRELATION.

Scatterplots and CorrelationScatterplots and Correlation

Definition:

The correlation r measures the strength of the linear relationship between two quantitative variables.

•r is always a number between -1 and 1

•r > 0 indicates a positive association.

•r < 0 indicates a negative association.

•Values of r near 0 indicate a very weak linear relationship.

•The strength of the linear relationship increases as r moves away from 0 towards -1 or 1.

•The extreme values r = -1 and r = 1 occur only in the case of a perfect linear relationship.

Facts About CorrelationFacts About Correlation

It does not require a response and explanatory It does not require a response and explanatory variable. Ex. How are SAT math and verbal variable. Ex. How are SAT math and verbal scores related?scores related?

If you switch the x and the y variables, the If you switch the x and the y variables, the correlation doesn’t changecorrelation doesn’t change..

If you change the units of measurement for x If you change the units of measurement for x and/or y, the and/or y, the correlation doesn’t changecorrelation doesn’t change..

Positive Positive rr values indicate a positive values indicate a positive relationship; negative values indicate a relationship; negative values indicate a negative relationship. Remember… not cause.negative relationship. Remember… not cause.

More FactsMore Facts

Correlation measures the strength of the Correlation measures the strength of the LINEARLINEAR relationship. It doesn’t measure relationship. It doesn’t measure curved relationships.curved relationships.

Correlation is strongly affected by Correlation is strongly affected by outliers.outliers.

r does not have a unit.r does not have a unit.

HomeworkHomework

Chapter 3 Chapter 3

#11, 13, 14, 15, 17, 20, 22, 26#11, 13, 14, 15, 17, 20, 22, 26