Summarizing Categorical Data

Summarizing Categorical Data

Tools for Summarizing Data

• Tools for calculating numerical and graphical summaries are divided between categorical and numerical data.

• The focus here is on categorical data.

Penguin Data

• Dr. Gorman collected data near Palmer Station, Antarctica.

• Data includes 8 variables on 333 penguins.

• Example of the dataset:

  • species: Adelie, Chinstrap, Gentoo

  • island: Biscoe, Dream, Torgersen

  • billlengthmm, billdepthmm, flipperlengthmm, bodymassg, sex, year

Challenges with Raw Data

• Raw data frames contain too much information to process at a glance.

• Need to consolidate information into numerical summaries.

Contingency Tables

• Categorical variables don’t take numbers as values, so canʼt take an average or a median.

• Count the number of penguins in every combination of levels.

• Display counts in a contingency table.

• Example:

  • Counts of penguin species on different islands.

  • Adelie: Biscoe (44), Dream (55), Torgersen (47)

  • Chinstrap: Biscoe (0), Dream (68), Torgersen (0)

  • Gentoo: Biscoe (119), Dream (0), Torgersen (0)

Definition of Contingency Table

• A table that shows the counts/frequencies of observations in every combination of levels of two categorical variables.

• Used to display the relationship between variables.

Bar Charts

• Present counts in graphical form using bar charts.

• Two common methods:

  • Stacked bar chart

  • Side-by-side (dodged) bar chart

Stacked Bar Chart

• One variable on the x-axis (e.g., species).

• Y-axis filled according to the counts in each level of the other variable (e.g., island).

Side-by-Side (Dodged) Bar Chart

• Similar to stacked, but unstacks the bars.

• Puts the levels of the second variable besides one another.

Choosing a Bar Chart Type

• Stacked bar charts highlight total counts (e.g., total number of Adelie penguins).

• Side-by-side charts make it easier to see the relative sizes of each level of the second variable (e.g., which islands Adelie penguins came from).

From Counts to Proportions

• Emphasize relative magnitude by converting counts into proportions.

• Example table of proportions:

  • Adelie: Biscoe (0.132), Dream (0.165), Torgersen (0.142), Total (0.439)

  • Chinstrap: Biscoe (0.000), Dream (0.204), Torgersen (0.000), Total (0.204)

  • Gentoo: Biscoe (0.357), Dream (0.000), Torgersen (0.000), Total (0.357)

  • Total: Biscoe (0.489), Dream (0.369), Torgersen (0.142), Grand Total (1.000)

Joint Proportion

• Proportion of observations of multiple variables in a combination of levels.

• Example: $44 / 333 = 0.132$, the proportion of all penguins that were Adelie and from Biscoe.

Marginal Proportion

• Proportion of observations in one variable that appear in a single level of that variable.

• Example: $(44 + 119) / 333 = 0.489$, the proportion of all penguins that were from Biscoe.

Conditional Proportion

• Proportion of observations in one level of one variable that appear in a level of a second variable.

• Example: $0.132 / 0.489 = 0.269$, the proportion of penguins from Biscoe that were Adelie.

Normalized Stacked Bar Chart

• Plots proportions instead of raw counts.

• Typically shows conditional proportions.

• Example: Height of a bar indicates the proportion of all penguins of a species that were from Torgersen: $47 / 146 = 0.322$.

• Conditioning on a variable means that variable is in the denominator of the proportion.

Conditioning on Different Variables

• Conditioning on island instead of species gives a different story.

• Example: Proportion of penguins from Biscoe that were Gentoo: $119 / 163 = 0.730$.

• Choosing which variable to condition on is crucial for the message you want to convey.

Importance of Choosing the Right Chart

• Example: To determine species distribution on Dream Island, a chart conditioning on the island is needed.

• Conditioning on species loses the relative numbers of Chinstraps and Adelies.

• It is essential to think carefully about which conditional proportion your chart is displaying if you are making a claim rooted in a conditional proportion and are using a normalized bar chart.

Association

• There is an association between two categorical variables if the conditional proportions vary as you move from one level of the conditioning variable to the next.

• Example: If the distribution of species is unchanged across islands, there is no association.

• Real data shows an association: Biscoe dominated by Gentoos, Dream with Chinstraps and Adelies, Torgersen only Adelies.

Summary

• Categorical summaries involve counts of categories or proportions.

• Proportions can be joint, marginal, or conditional.

• Counts and proportions are displayed in contingency tables or bar charts.

• Subtle choices of which proportion to present results in the telling of dramatically different stories.

• Summarizing categorical data involves deciding what to add and divide.