Math 1153 - 22 September 2026 - Ch3

Overview

Mike Jacobsen analyzes a 626-person University of Texas Southwestern Medical Center study of tattoo history and hepatitis C using a two-way table, marginal and conditional distributions, and row- and column-percentage displays. The percentages show a strong association—hepatitis C prevalence is 22.1% among participants with tattoos versus 4.3% among those without—but the observational study cannot establish that tattoos cause hepatitis C; Jacobsen then previews box plots and five-number summaries for Chapter 4.

Key takeaways

In the 626-person study, hepatitis C prevalence was 22.1% among participants with tattoos (25/113), compared with 4.3% among those without tattoos (22/513).
The overall hepatitis C rate of 7.5% differs from the subgroup rates, illustrating why conditional percentages can reveal relationships hidden by marginal totals.
For a percentage involving multiple categories, combine the relevant counts and group the numerator with parentheses—for example, (52 + 61)/626 × 100 = 18.1%.
Row percentages compare hepatitis C outcomes within tattoo groups; column percentages compare tattoo status within hepatitis C groups, and both are valid when interpreted with the correct denominator.
The table supports an association between tattoo status and hepatitis C, not a causal claim, because the researchers observed and interviewed participants rather than conducting a controlled experiment.
A box plot summarizes quantitative data through the minimum, Q1, median, Q3, and maximum; the interquartile range is calculated as Q3 − Q1.

Chapters

0:00 Week 5 Logistics and the Three-Exam Course Schedule

Mike Jacobsen notes that Monday was reserved for the first exam and says grading and Canvas feedback will begin after quiz grading.
The course has three exams plus a final, with the next exam planned for week 9 rather than midterm week.
Chapter 3 focuses on relationships between two categorical variables.

2:25 Tattoo and Hepatitis C Counts in the Southwestern Study

A University of Texas Southwestern Medical Center study examines whether hepatitis C is associated with having tattoos and where they were obtained.
The 626 participants are classified by tattoo status in rows and hepatitis C status in columns.
The table records 47 participants with hepatitis C and 579 without it.

5:20 Marginal Distributions Ignore the Other Variable

The tattoo marginal distribution uses row totals: 52 parlor tattoos, 61 tattoos obtained elsewhere, and 513 participants with no tattoos.
The hepatitis C marginal distribution uses column totals: 47 positive cases and 579 participants without hepatitis C.
Each marginal distribution describes one variable alone, ignoring the other variable in the table.

7:00 Conditional Distributions Restrict the Study to a Group

For the tattoo distribution given no hepatitis C, the relevant group is the 579 participants in the no-hepatitis-C column.
Within that restricted group, the tattoo counts are 35 parlor, 53 elsewhere, and 491 no tattoos.
A conditional distribution changes the denominator from all 626 participants to the count meeting the stated condition.

12:30 Percent Questions: Separate Whole-Study Counts from Cell Counts

For the percentage of all participants with hepatitis C, use 47/626 × 100, which is 7.5%.
For participants who both have hepatitis C and have no tattoos, use the single table cell 22/626 × 100, or 3.5%.
A cell count represents two conditions at once, while a row or column total represents a broader group.

18:04 Add Multiple Tattoo Categories Before Dividing

The percentage of all participants with tattoos combines the parlor and elsewhere row totals: (52 + 61)/626 × 100 = 18.1%.
Jacobsen warns that a calculator evaluates 52 + 61/626 × 100 in the wrong order unless the numerator is parenthesized.
Use parentheses or calculate 52 + 61 first, then divide by 626 and multiply by 100.

24:06 Compare Hepatitis C Rates Among Tattooed and Untattooed Participants

Among the 113 participants with tattoos, 25 had hepatitis C: (17 + 8)/(52 + 61) × 100 = 22.1%.
Among the 513 participants with no tattoos, 22 had hepatitis C: 22/513 × 100 = 4.3%.
The overall rate is 7.5%, so the subgroup rates suggest a marked relationship between tattoo status and hepatitis C.

30:42 Build Row Percentages to Compare Tattoo Groups Fairly

Row percentages use each tattoo category's row total as the denominator, making every row sum to 100%.
Among participants with parlor tattoos, 17/52 = 32.7% had hepatitis C and 67.3% did not.
Among participants tattooed elsewhere, 8/61 = 13.1% had hepatitis C and 86.9% did not.
Jacobsen notes that the 100% total lets students find one row percentage by subtracting the other from 100%.

38:30 Complete the Row-Percentage Display and Read Its Conditional Rates

For participants with no tattoos, 22/513 = 4.3% had hepatitis C and 95.7% did not.
The overall row is 7.5% with hepatitis C and 92.5% without it, based on 47/626 and 579/626.
Each row percentage describes the hepatitis C distribution within a particular tattoo group.

43:34 Column Percentages Reverse the Comparison

Column percentages use hepatitis C outcomes as the groups and describe tattoo status within each outcome.
Among hepatitis C cases, 36.2% had parlor tattoos, 17.0% had tattoos obtained elsewhere, and 46.8% had no tattoos.
Among participants without hepatitis C, 6.0% had parlor tattoos, 9.2% had tattoos obtained elsewhere, and 84.8% had no tattoos.
Either row or column percentages can be used; the choice depends on which comparison best answers the research question.

52:50 Association Is Not Causation: Assessing Dependence

The different tattoo distributions across hepatitis C outcomes indicate that the two categorical variables are associated, or dependent.
Independence would mean the distribution of tattoo status stayed roughly the same for hepatitis C-positive and hepatitis C-negative groups.
Because researchers interviewed participants rather than controlling an experiment, the data cannot establish that tattoos cause hepatitis C.

59:52 Chapter 4 Preview: Comparing Quantitative and Categorical Variables

Chapter 4 will compare a quantitative measure of lung strength with categorical groupings such as age or smoking status.
Jacobsen introduces the box plot, also called a box-and-whisker display, as a tool for comparing distributions across groups.
The five-number summary and an outlier rule will underpin the upcoming box-plot work.

1:02:57 Five-Number Summary Preview with Radioactive Tuna Data

A five-number summary consists of the minimum, lower quartile Q1, median, upper quartile Q3, and maximum.
Q1, the median, and Q3 correspond to the 25th, 50th, and 75th percentiles; the interquartile range is Q3 − Q1.
For the radioactive tuna dataset, the sorted observations show a minimum of 4.6 and a maximum of 15.6 becquerels per kilogram.
Jacobsen postpones importing the data and calculating the remaining summary values until the next class; Quiz 3 is due then and Quiz 4 will be posted.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Mike Jacobsen.