1 STAT 110 – Introduction to Descriptive Statistics Chapter 14 Describing Relationships2 STAT 110 – Introduction to Descriptive Statistics Plots and Numerical Summaries Most statistical studies examine data on more than one variable. The steps when trying to talk about two variables at once are the same as what we used earlier in the semester with just one variables: • Plot the data. • Look for overall patterns and deviations from those patterns. • Use numerical summaries.3 STAT 110 – Introduction to Descriptive Statistics Scatterplot scatterplot – shows the relationship between two quantitative variables measured on the same individuals •Values of one variable appear on the x-axis. This is typically the one doing the explaining – the explanatory, predictor, or independent variable. • Values of the other variable appear on the y-axis. This is typically the one being explained – called the response or dependent variable.4 STAT 110 – Introduction to Descriptive Statistics Scatterplot Example When water flows across farmland, some of the soil is washed away, resulting in erosion. An experiment was conducted to investigate the effect of the rate of water flow on the amount of soil washed away. Flow is measured in liters/second and the eroded soil is measured in kilograms. flow rate .31 .85 1.26 2.47 3.75 eroded soil .82 1.95 2.18 3.01 6.075 STAT 110 – Introduction to Descriptive Statistics Scatterplot • Is there an explanatory variable? • What’s the response variable? • Which variable should be on the x-axis? Flow Rate vs Eroded Soil012345670 1 2 3 4Flow Rate (liters/sec)Eroded Soil (kg)6 STAT 110 – Introduction to Descriptive Statistics Interpreting Scatterplots 1. Look for the overall pattern and for striking deviations from the pattern. 2. Look for outliers. 3. Describe the overall pattern by the form, direction, and strength of the relationship. form – clustered, curved, linear direction – positive or negative strength – determined by how closely the points follow a clear form7 STAT 110 – Introduction to Descriptive Statistics Clustered8 STAT 110 – Introduction to Descriptive Statistics Curved9 STAT 110 – Introduction to Descriptive Statistics Linear A Linear Graph-6-4-20246-6 -4 -2 0 2 4 610 STAT 110 – Introduction to Descriptive Statistics Association positively associated – when above-average values and below-average values tend to occur together scatterplot slopes upward as you move from left to right negatively associated – when above-average values of one variable tend to accompany below-average values of the other scatterplot slopes downward as you move from left to right11 STAT 110 – Introduction to Descriptive Statistics Positive Association A Linear Graph-6-4-20246-6 -4 -2 0 2 4 612 STAT 110 – Introduction to Descriptive Statistics Negative Association A Linear Graph-6-4-20246-6 -4 -2 0 2 4 613 STAT 110 – Introduction to Descriptive Statistics Correlation correlation – (r) describes the direction and strength of a straight-line relationship between two quantitative variables • sign indicates negative or positive association • r = 0 no linear association • r = -1 or r = 1 perfect straight line14 STAT 110 – Introduction to Descriptive Statistics Correlation15 STAT 110 – Introduction to Descriptive Statistics Notes About r •r has no units and won’t change if we change the units of measurement • r ignores the distinction between explanatory and response variables • r is strongly affected by outliers
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