Boxplots

SDS 192: Introduction to Data Science

Author

Professor Lindsay Poirier

For Today

Measures of Central Tendency and Dispersion
Boxplots
Project 1 Assigned

A measure of central tendency is a single numeric quantity describing data by identifying a central position.

Let’s create the following data frame to motivate today’s lecture.

library(tidyverse)

── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr     1.1.4     ✔ readr     2.1.5
✔ forcats   1.0.0     ✔ stringr   1.5.1
✔ ggplot2   3.5.1     ✔ tibble    3.3.0
✔ lubridate 1.9.3     ✔ tidyr     1.3.1
✔ purrr     1.1.0     
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors

counties <- read_csv("https://raw.githubusercontent.com/sds-192-intro-fall22/sds-192-public-website-quarto/a8b64e3070ca2543b904d4d92780b09e6062ced6/website/data/nbi_counties.csv")

Rows: 14 Columns: 2
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (1): COUNTY_CODE_003_L
dbl (1): COUNTY_CODE_003

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

route_prefixes <- read_csv("https://raw.githubusercontent.com/sds-192-intro-fall22/sds-192-public-website-quarto/a8b64e3070ca2543b904d4d92780b09e6062ced6/website/data/nbi_route_pre.csv")

Rows: 8 Columns: 2
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (1): ROUTE_PREFIX_005B_L
dbl (1): ROUTE_PREFIX_005B

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

maintenance <- read_csv("https://raw.githubusercontent.com/sds-192-intro-fall22/sds-192-public-website-quarto/a8b64e3070ca2543b904d4d92780b09e6062ced6/website/data/nbi_maintenance.csv")

Rows: 29 Columns: 2
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (1): MAINTENANCE_021_L
dbl (1): MAINTENANCE_021

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

kinds <- read_csv("https://raw.githubusercontent.com/sds-192-intro-fall22/sds-192-public-website-quarto/a8b64e3070ca2543b904d4d92780b09e6062ced6/website/data/nbi_kind.csv")

Rows: 10 Columns: 2
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (1): STRUCTURE_KIND_043A_L
dbl (1): STRUCTURE_KIND_043A

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

nbi_ma <- read.delim("https://www.fhwa.dot.gov/bridge/nbi/2022/delimited/MA22.txt", sep = ",") |>
  left_join(counties) |>
  left_join(route_prefixes) |>
  left_join(maintenance) |>
  left_join(kinds) |>
  filter(SERVICE_ON_042A == 1) |>
  select(STRUCTURE_NUMBER_008, COUNTY_CODE_003_L, ROUTE_PREFIX_005B_L, MAINTENANCE_021_L, YEAR_BUILT_027, ADT_029, STRUCTURE_KIND_043A_L, STRUCTURAL_EVAL_067, BRIDGE_IMP_COST_094) |>
  mutate(STRUCTURE_KIND_043A_L = 
           case_when(
             STRUCTURE_KIND_043A_L == "Concrete continuous" ~ "Concrete",
             STRUCTURE_KIND_043A_L == "Steel continuous" ~ "Steel",
             STRUCTURE_KIND_043A_L == "Prestressed concrete continuous" ~ "Prestressed concrete",
             TRUE ~ STRUCTURE_KIND_043A_L)) |>
  mutate(BRIDGE_IMP_COST_094 = BRIDGE_IMP_COST_094 * 1000)

Joining with `by = join_by(COUNTY_CODE_003)`
Joining with `by = join_by(ROUTE_PREFIX_005B)`
Joining with `by = join_by(MAINTENANCE_021)`
Joining with `by = join_by(STRUCTURE_KIND_043A)`

nbi_hampshire <- nbi_ma |> filter(COUNTY_CODE_003_L == "Hampshire")

rm(counties, kinds, maintenance, route_prefixes)

Mean

Sum of values divided by number of values summed
Takes every value into consideration
Model of entire dataset
Heavily influenced by outliers

Warning in geom_label(aes(x = mean(STRUCTURAL_EVAL_067), y = 100, label = "Mean"), : All aesthetics have length 1, but the data has 2854 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Median

Middle value(s) of the dataset when all values are lined from smallest to largest
Does not model entire dataset
Limited influence from outliers

Warning in geom_label(aes(x = mean(STRUCTURAL_EVAL_067), y = 25, label = "Mean"), : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Warning in geom_label(aes(x = median(STRUCTURAL_EVAL_067), y = 40, label = "Median"), : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Learning Check: How many variables from the dataset are represented on the previous plot?

Normal Distributions

More values huddle around some center line and taper off as we move away from center
Histogram is symmetrical with a perfectly normal distribution
Median and mean should be about the same; mean is a good measure of central tendency

Warning in geom_label(aes(x = mean(STRUCTURAL_EVAL_067), y = 25, label = "Mean"), : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Warning in geom_label(aes(x = median(STRUCTURAL_EVAL_067), y = 40, label = "Median"), : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Skew

Histogram is non-symmetrical when there is skew
Long trail to the right of center indicates a right skew
Median becomes more representative measure of central tendency than mean

Warning in geom_label(aes(x = mean(BRIDGE_IMP_COST_094), y = 25, label = "Mean"), : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Warning in geom_label(aes(x = median(BRIDGE_IMP_COST_094), y = 40, label = "Median"), : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Summarizing Data

Measures of central tendency summarize swaths of information into single value
Can be reductionist
- Example: Measures of central tendency related to wealth in the US only tell us about those in the middle
- Hide the experiences of the most impoverished communities.
Degree of spread or dispersion is just as important as center

Range

Maximum value minus the minimum value
Evaluates the spread of the entire dataset

Warning in geom_label(aes(x = min(STRUCTURAL_EVAL_067), y = "", label = "Min"), : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Warning in geom_label(aes(x = max(STRUCTURAL_EVAL_067), y = "", label = "Max"), : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Warning in geom_label(aes(x = min(STRUCTURAL_EVAL_067), y = 15, label = "Min"), : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Warning in geom_label(aes(x = max(STRUCTURAL_EVAL_067), y = 15, label = "Max"), : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Interquartile Range

1st quartile is middle value between minimum and median
3rd quartile is middle value between median and maximum
IQR is the difference between the 1st and 3rd quartile
Represents the middle 50% of values

Warning in geom_label(aes(x = min(STRUCTURAL_EVAL_067), y = "", label = "Min"), : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Warning in geom_label(aes(x = max(STRUCTURAL_EVAL_067), y = "", label = "Max"), : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Warning in geom_label(aes(x = quantile(STRUCTURAL_EVAL_067)[2], y = "", : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Warning in geom_label(aes(x = quantile(STRUCTURAL_EVAL_067)[4], y = "", : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Warning in geom_label(aes(x = median(STRUCTURAL_EVAL_067), y = "", label = "Median"), : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Warning in geom_label(aes(x = min(STRUCTURAL_EVAL_067), y = 15, label = "Min"), : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Warning in geom_label(aes(x = max(STRUCTURAL_EVAL_067), y = 15, label = "Max"), : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Warning in geom_label(aes(x = quantile(STRUCTURAL_EVAL_067)[2], y = 15, : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Warning in geom_label(aes(x = quantile(STRUCTURAL_EVAL_067)[4], y = 15, : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Warning in geom_label(aes(x = median(STRUCTURAL_EVAL_067), y = 10, label = "Median"), : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Boxplot

Warning in geom_label(aes(x = min(STRUCTURAL_EVAL_067), y = 0.2, label = "Min"), : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Warning in geom_label(aes(x = max(STRUCTURAL_EVAL_067), y = 0.2, label = "Max"), : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Warning in geom_label(aes(x = quantile(STRUCTURAL_EVAL_067)[2], y = 0.1, : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Warning in geom_label(aes(x = quantile(STRUCTURAL_EVAL_067)[4], y = -0.1, : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Warning in geom_label(aes(x = median(STRUCTURAL_EVAL_067), y = 0, label = "Median"), : All aesthetics have length 1, but the data has 171 rows.
ℹ Please consider using `annotate()` or provide this layer with data containing
  a single row.

Grouped Boxplots

ggplot(nbi_hampshire, aes(x = ADT_029, y = ROUTE_PREFIX_005B_L)) +
  geom_boxplot() +
  labs(title = "Distribution in the Average Daily Traffic of Hampshire County, MA Bridges, 2021", 
       x = "Average Daily Traffic",
       y = "Route Prefix") +
  theme_minimal()

Interpreting Boxplots Step 1: Check for Outliers

How many are there? What do they indicate? Do you assume they are errors in teh data? Or do they represent extremes that are important for us to take into consideration?

Interpreting Boxplots Step 2: Compare Medians

Do the medians line up? If not, in which groups are the medians higher and in which are they lower?

Interpreting Boxplots Step 3: Compare Range

Do certain groups have a wider range of values represented than others? In other words, are the values more distributed for certain groups than for others? This might indicate a greater degree of disparity in some groups than others.

Interpreting Boxplots Step 4: Compare IQR

In which groups do the middle 50% of values tend to huddle around a central value? In which are they more spread out from the center?

Interpreting Boxplots Step 5: Compare Symmetry

Does the median appear to be in the center of the range and IQR? Is the median closer to the minimum – or the bottom whisker? Or the top whisker?