Stat 1510:Statistical Thinking and Concepts

Scatterplots and Correlation

Agenda2

Explanatory and Response Variables

Displaying Relationships: Scatterplots

Interpreting Scatterplots

Adding Categorical Variables to Scatterplots

Measuring Linear Association: Correlation

Facts About Correlation

Objectives3

Define explanatory and response variables Construct and interpret scatterplots Add categorical variables to scatterplots Calculate and interpret correlation Describe facts about correlation

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The most useful graph for displaying the relationship between two quantitative variables is a scatterplot.

Scatterplot

A scatterplot shows the relationship between two quantitative variables measured on the same individuals. The values of one variable appear on the horizontal axis, and the values of the other variable appear on the vertical axis. Each individual in the data appears as a point on the graph.

A scatterplot shows the relationship between two quantitative variables measured on the same individuals. The values of one variable appear on the horizontal axis, and the values of the other variable appear on the vertical axis. Each individual in the data appears as a point on the graph.

1. Decide which variable should go on each axis. If a distinction exists, plot the explanatory variable on the x-axis and the response variable on the y-axis.

2. Label and scale your axes.

3. Plot individual data values.

1. Decide which variable should go on each axis. If a distinction exists, plot the explanatory variable on the x-axis and the response variable on the y-axis.

2. Label and scale your axes.

3. Plot individual data values.

How to Make a ScatterplotHow to Make a Scatterplot

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ScatterplotExample: Make a scatterplot of the relationship between body weight and pack weight for a group of hikers.

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

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

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Interpreting Scatterplots

To interpret a scatterplot, follow the basic strategy of data analysis discussed earlier. Look for patterns and important departures from those patterns.

As in any graph of data, look for the overall pattern and for striking departures from that pattern.

•You can describe the overall pattern of a scatterplot by the direction, form, and strength of the relationship.

•An important kind of departure is an outlier, an individual value that falls outside the overall pattern of the relationship.

As in any graph of data, look for the overall pattern and for striking departures from that pattern.

•You can describe the overall pattern of a scatterplot by the direction, form, and strength of the relationship.

•An important kind of departure is an outlier, an individual value that falls outside the overall pattern of the relationship.

How to Examine a ScatterplotHow to Examine a Scatterplot

Interpreting Scatterplots7

Two variables have a positive association when above-average values of one tend to accompany above-average values of the other, and when below-average values also tend to occur together.

Two variables have a negative association when above-average values of one tend to accompany below-average values of the other.

Two variables have a positive association when above-average values of one tend to accompany above-average values of the other, and when below-average values also tend to occur together.

Two variables have a negative association when above-average values of one tend to accompany below-average values of the other.

DirectionDirection FormFormStrengthStrength

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 hikers are carrying lighter backpacks.

Adding Categorical Variables

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Consider the relationship between mean SAT verbal score and percent of high-school grads taking SAT for each state.

To add a categorical variable, use a different plot color or symbol for each category.

Southern states

highlighted

Measuring Linear Association

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A scatterplot displays the strength, direction, and form of the relationship between two quantitative variables.

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 toward -1 or 1.

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

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 toward -1 or 1.

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

Correlation10

Facts About Correlation

1. Correlation makes no distinction between explanatory and response variables.

2. r has no units and does not change when we change the units of measurement of x, y, or both.

3. Positive r indicates positive association between the variables, and negative r indicates negative association.

4. The correlation r is always a number between -1 and 1.

Cautions:• Correlation requires that both variables be quantitative.

• Correlation does not describe curved relationships between variables, no matter how strong the relationship is.

• Correlation is not resistant. r is strongly affected by a few outlying observations.

• Correlation is not a complete summary of two-variable data.

Cautions:• Correlation requires that both variables be quantitative.

• Correlation does not describe curved relationships between variables, no matter how strong the relationship is.

• Correlation is not resistant. r is strongly affected by a few outlying observations.

• Correlation is not a complete summary of two-variable data.

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Correlation Practice12

For each graph, estimate the correlation r and interpret it in context.

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Case Study

Per Capita Gross Domestic Productand Average Life Expectancy for

Countries in Western Europe

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Case StudyCountry Per Capita GDP (x) Life Expectancy (y)

Austria 21.4 77.48

Belgium 23.2 77.53

Finland 20.0 77.32

France 22.7 78.63

Germany 20.8 77.17

Ireland 18.6 76.39

Italy 21.5 78.51

Netherlands 22.0 78.15

Switzerland 23.8 78.99

United Kingdom 21.2 77.37

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Case Study

x y

21.4 77.48 -0.078 -0.345 0.027

23.2 77.53 1.097 -0.282 -0.309

20.0 77.32 -0.992 -0.546 0.542

22.7 78.63 0.770 1.102 0.849

20.8 77.17 -0.470 -0.735 0.345

18.6 76.39 -1.906 -1.716 3.271

21.5 78.51 -0.013 0.951 -0.012

22.0 78.15 0.313 0.498 0.156

23.8 78.99 1.489 1.555 2.315

21.2 77.37 -0.209 -0.483 0.101

= 21.52 = 77.754sum = 7.285

sx =1.532 sy =0.795

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Case Study

0.809

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