4.3 Categorical Data4.3 Categorical Data

RelationshipsRelationships

AnalysisAnalysis

• Analyze categorical data by looking at the counts or percents of individuals that fall into various categories.

Marginal DistributionsMarginal Distributions

• Totals that appear at the bottom and right margins of a two – way table.

Seat Belts and ChildrenSeat Belts and Children Do child restraints and seat belts prevent injuries to young

passengers in automobile accidents? Here are data on the 26,971 passengers under the age of 15 in accidents reported in North Carolina during two years before the law required restraints. Page 254: 4.69

Restrained Unrestrained Totals

Injured 197 3,844 4,041

Uninjured 1,749 21,181 22,930

Totals 1,946 25,025 26,971

Marginal DistributionsMarginal Distributions

• Sometimes the totals may contain roundoff errors.

• Often percents give more information about the data than the actual counts.– What is the percent of injured children who

wore a restraint? 197/1946 = 10%– What is the percent of uninjured children who

wore a restraint? 1749/1946 = 90%

Distribution of Young Passengers with Restraints

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To ensure we get the answer to the question we want, ask …

“What group represents the total that we want a percent of?”

Calculating the appropriate percents will help describe the relationships being depicted by the data. A graphical display (bar graph) conveys the information plainly for the reader.

Seat Belts and RestraintsSeat Belts and Restraints

• What is the percent of children who weren’t restrained and injured? 15%

• What is the percent of children who weren’t restrained and uninjured? 85%

Looking at these comparisons it seems like there was only a slight difference between being injured and restrained (10%) versus injured unrestrained (15%)

Distribution of Young Passengers Not Wearing a Restraint

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Distribution of Young Passengers' Injuries

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Conditional DistributionsConditional Distributions

• When the distribution refers to a specific people/group who satisfies the condition.

To draw a more accurate conclusion from the data, we should look at the tableHorizontally. What percentage of children were injured who wore a restraint versus the percentage of children who were injured and not wearing a restraint?

Injured and restrained – 4.9%Injured and unrestrained – 95.1%

Therefore supporting the need for a state law to require child restraints.

Now for the rest of the story …Now for the rest of the story …

For those who are not in favor of child restraint laws, they would use the bottom half conditional data.

Uninjured and restrained – 7.6%

Uninjured and unrestrained – 92.4%

Can statistics support whatever we want?

Distributions of Young Passengers in Auto Accidents

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Seat Belt and ChildrenSeat Belt and Children

• Does the data provide evidence that young passengers are less likely to be injured in an accident if they wear restraints? Calculate and compare percents.

Restrained and injured – 10.1%Unrestrained and injured – 15.6%

Finale Seat Belts & ChildrenFinale Seat Belts & Children

Distribution of Young Passengers involved in Auto Accidents

10.10% 15.40%

89.90% 84.60%

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Restrained Unrestrained

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CautionsCautions

• No single graph summarizes the relationship between categorical variables.

• No single numerical measure summarizes the strength of an association.

• Visuals (bar graph) are helpful in making comparisons.

• Percents are more useful if they are well chosen based upon the need of the question being asked.

Simpson’s ParadoxSimpson’s Paradox

• Reversal of the direction of a comparison or an association when data from several groups are combined to form a single group.

College Admissions ParadoxCollege Admissions Paradox

• Page 251: 4.60

• Exercises 4.53-4.55, 4.60, 4.67