Week 09 – Describing Categorical Data

Introduction

This week we explore categorical variables using the fictional dataset of 150 summer-school students introduced in Week 1.

The aim is to learn how to describe categorical data, both:

numerically (using summary measures)
visually (using graphs in R)

Categorical data appear in many subjects, including education, health, psychology, linguistics, and business. In practice, we are often interested in questions such as:

How many individuals fall into each category?
What proportion of the sample belongs to each group?

1. Descriptive Statistics: What does it mean to describe a variable?

We describe a variable by summarising its distribution

We usually want to know:

Where do the values tend to cluster?

→ the centre of the distribution
How are the values distributed?

→ the spread of the distribution

We can do this:

numerically with summary statistics
visually with plots (graphs)

The exact summaries depend on the type of variable

Nominal categorical variables (no natural order)
→ frequencies, percentages, mode, bar chart
- We describe how observations are distributed across categories
- The mode identifies the most common category
Ordinal categorical variables (meaningful order)
→ frequencies, percentages, mode, median, quartiles / IQR, bar chart
- Because categories are ordered, we can identify the median
- We can also describe spread using quartiles and the IQR

Getting started in R

To follow the examples, first load the tidyverse:

library(tidyverse)

Then load the dataset:

students <- read_csv("Lecture1_data.csv", show_col_types = FALSE)

You can view the first few rows:

head(students)

# A tibble: 6 × 14
  ID    Degree        Year Gender     Study_hours Sleep_hours Stress_level
  <chr> <chr>        <dbl> <chr>            <dbl>       <dbl> <chr>       
1 S001  Architecture     2 Female             6.8         7.4 Moderate    
2 S002  Linguistics      1 Male               3.1         5.8 High        
3 S003  English          2 Female             8.2         7.6 Low         
4 S004  Philosophy       1 Male               5.5         6.7 Moderate    
5 S005  Linguistics      2 Non-binary         4           5.9 High        
6 S006  Linguistics      2 Female             7.4         7.8 Low         
# ℹ 7 more variables: Satisfaction_Likert_item <chr>,
#   Satisfaction_Likert_value <dbl>, Coffee_per_day <dbl>,
#   Social_media_hr <dbl>, Exercise <chr>, Overall_mark <dbl>, IQ <dbl>

Data exploration

Once we have a dataset, a natural first step is to explore it:

What variables do we have?
How many students are there?
What sort of values do we see?

We can use summary() to get a quick overview:

summary(students)

      ID               Degree               Year          Gender         
 Length:200         Length:200         Min.   :1.000   Length:200        
 Class :character   Class :character   1st Qu.:1.000   Class :character  
 Mode  :character   Mode  :character   Median :2.000   Mode  :character  
                                       Mean   :1.925                     
                                       3rd Qu.:2.000                     
                                       Max.   :3.000                     
  Study_hours     Sleep_hours    Stress_level       Satisfaction_Likert_item
 Min.   :2.100   Min.   :4.200   Length:200         Length:200              
 1st Qu.:4.500   1st Qu.:6.000   Class :character   Class :character        
 Median :5.900   Median :6.800   Mode  :character   Mode  :character        
 Mean   :5.845   Mean   :6.758                                              
 3rd Qu.:7.200   3rd Qu.:7.600                                              
 Max.   :9.900   Max.   :8.600                                              
 Satisfaction_Likert_value Coffee_per_day  Social_media_hr   Exercise        
 Min.   :1.0               Min.   :0.000   Min.   :0.800   Length:200        
 1st Qu.:2.0               1st Qu.:1.000   1st Qu.:2.300   Class :character  
 Median :4.0               Median :2.000   Median :3.100   Mode  :character  
 Mean   :3.6               Mean   :1.825   Mean   :3.191                     
 3rd Qu.:5.0               3rd Qu.:3.000   3rd Qu.:4.200                     
 Max.   :5.0               Max.   :4.000   Max.   :5.700                     
  Overall_mark         IQ       
 Min.   :43.00   Min.   : 85.0  
 1st Qu.:57.00   1st Qu.: 98.0  
 Median :68.00   Median :104.0  
 Mean   :66.24   Mean   :103.6  
 3rd Qu.:74.25   3rd Qu.:109.2  
 Max.   :86.00   Max.   :126.0

Important

For numeric variables, this shows the minimum, maximum, mean, and so on.
For categorical variables, it shows the number of students in each category.

Important: How categorical variables are stored in R

For categorical variables, summary() behaves differently depending on how the variable is stored in R.

What is a character variable?

A character variable stores data as plain text (strings).

Each value is treated simply as a label
R does not recognise it as a categorical variable
It does not automatically count how many times each category appears

Example:

students$Degree

  [1] "Architecture" "Linguistics"  "English"      "Philosophy"   "Linguistics" 
  [6] "Linguistics"  "Philosophy"   "Education"    "Anthropology" "Sociology"   
 [11] "Design"       "English"      "Psychology"   "Psychology"   "English"     
 [16] "Psychology"   "English"      "Design"       "Anthropology" "Business"    
 [21] "Sociology"    "Linguistics"  "Philosophy"   "Linguistics"  "Anthropology"
 [26] "Business"     "Politics"     "Anthropology" "Linguistics"  "History"     
 [31] "Anthropology" "Psychology"   "Psychology"   "Architecture" "Psychology"  
 [36] "Architecture" "Education"    "Philosophy"   "Linguistics"  "Philosophy"  
 [41] "Politics"     "Anthropology" "English"      "Education"    "Psychology"  
 [46] "Education"    "Design"       "Philosophy"   "Linguistics"  "Politics"    
 [51] "English"      "Philosophy"   "History"      "History"      "Anthropology"
 [56] "History"      "History"      "Sociology"    "Politics"     "Linguistics" 
 [61] "History"      "Psychology"   "Architecture" "Politics"     "Education"   
 [66] "Linguistics"  "Linguistics"  "Education"    "Anthropology" "Education"   
 [71] "Philosophy"   "Psychology"   "Linguistics"  "Education"    "History"     
 [76] "Linguistics"  "History"      "Anthropology" "Linguistics"  "Design"      
 [81] "Politics"     "Politics"     "History"      "Architecture" "Philosophy"  
 [86] "Linguistics"  "Philosophy"   "English"      "Education"    "Linguistics" 
 [91] "Sociology"    "Education"    "Education"    "Business"     "Business"    
 [96] "Psychology"   "History"      "Business"     "Politics"     "Business"    
[101] "Education"    "English"      "Philosophy"   "Anthropology" "Psychology"  
[106] "Politics"     "Philosophy"   "Linguistics"  "Psychology"   "Education"   
[111] "Linguistics"  "Design"       "Politics"     "History"      "History"     
[116] "History"      "Linguistics"  "English"      "Linguistics"  "Psychology"  
[121] "Design"       "Psychology"   "Sociology"    "Business"     "Design"      
[126] "History"      "Linguistics"  "Anthropology" "Architecture" "Politics"    
[131] "Business"     "Psychology"   "History"      "Design"       "History"     
[136] "Anthropology" "Sociology"    "Linguistics"  "English"      "Sociology"   
[141] "Psychology"   "Anthropology" "Linguistics"  "Linguistics"  "Education"   
[146] "Design"       "English"      "Anthropology" "Politics"     "Anthropology"
[151] "Linguistics"  "Linguistics"  "Business"     "Psychology"   "Anthropology"
[156] "History"      "Philosophy"   "Psychology"   "Philosophy"   "Education"   
[161] "Psychology"   "Business"     "Design"       "Linguistics"  "Linguistics" 
[166] "Design"       "Psychology"   "Education"    "Business"     "Anthropology"
[171] "Design"       "Sociology"    "History"      "Business"     "Design"      
[176] "Politics"     "Anthropology" "History"      "Business"     "Anthropology"
[181] "Sociology"    "Business"     "Architecture" "Anthropology" "Psychology"  
[186] "Psychology"   "Psychology"   "Anthropology" "Psychology"   "Anthropology"
[191] "Design"       "Business"     "Education"    "Politics"     "Design"      
[196] "English"      "Linguistics"  "History"      "Design"       "Philosophy"

R sees values like "Education", "Linguistics", "Psychology" as text, not as categories.

summary(students$Degree)

   Length     Class      Mode 
      200 character character

What is a factor?

A factor is R’s way of representing a categorical variable.

It stores categories as a set of levels
R understands that the variable consists of groups
It enables proper categorical analysis

When a variable is a factor:

R can count how many observations are in each category
Functions like summary() return a frequency table

This is why we often convert variables using factor():

students$Degree <- factor(students$Degree)

We check the type of variable of Degree using the function class ()

This tells you the type of the variable, for example:

"character" → text
"factor" → categorical variable
"numeric" → numbers

class(students$Degree)

[1] "factor"

Now we run summary()

summary(students$Degree)

Anthropology Architecture     Business       Design    Education      English 
          22            7           15           16           17           12 
     History  Linguistics   Philosophy     Politics   Psychology    Sociology 
          20           29           15           14           24            9

Key rule

Before analysing a categorical variable, always check:
Is it stored as a factor?

What do we do next?

Now that we understand how categorical variables are stored in R, we can begin to describe them properly.

When working with a categorical variable, we follow a simple process:

Identify the type of variable Is it nominal or ordinal?
Check how it is stored in R. Is it a factor?
Describe the variable numerically
- Frequencies and percentages
- Measures of central tendency (mode, median)
Describe the variable visually Bar charts / Boxplots

We now apply this process step by step.

2. Describing Nominal Categorical Variables

Nominal variables have no natural order.
Examples: Degree, Gender, Exercise.

Because categories are not ordered, we focus on:

how many observations fall into each category
which category is most common

2.1 Frequency distributions

The first step is always to create a frequency table.

students |>
  count(Degree)

# A tibble: 12 × 2
   Degree           n
   <fct>        <int>
 1 Anthropology    22
 2 Architecture     7
 3 Business        15
 4 Design          16
 5 Education       17
 6 English         12
 7 History         20
 8 Linguistics     29
 9 Philosophy      15
10 Politics        14
11 Psychology      24
12 Sociology        9

⚠️ Common error: “could not find function count”

If you get the following error, is because the library (tidyverse) is not loaded

library(tidyverse)

students |>
  count(Degree) |>
  print(n = Inf)

# A tibble: 12 × 2
   Degree           n
   <fct>        <int>
 1 Anthropology    22
 2 Architecture     7
 3 Business        15
 4 Design          16
 5 Education       17
 6 English         12
 7 History         20
 8 Linguistics     29
 9 Philosophy      15
10 Politics        14
11 Psychology      24
12 Sociology        9

Function count() = “how many of each category?”

How to read this table

Each row represents a category of Degree
The column n shows how many students are in each category

For example:

Architecture = 7 → there are 7 students studying Architecture
This table describes the distribution of the variable.

Ordering the table (most to least frequent)

Sometimes we want to see which categories are most common first.

We can reorder the table using arrange():

students |>
  count(Degree) |>
  arrange(desc(n))

# A tibble: 12 × 2
   Degree           n
   <fct>        <int>
 1 Linguistics     29
 2 Psychology      24
 3 Anthropology    22
 4 History         20
 5 Education       17
 6 Design          16
 7 Business        15
 8 Philosophy      15
 9 Politics        14
10 English         12
11 Sociology        9
12 Architecture     7

What does this do?

arrange() sorts the rows
desc(n) means:
→ order by the column n

→ from highest to lowest

Why is this useful?

This makes it easier to:

quickly identify the most common category (the mode)
compare categories
interpret the distribution more clearly

Note about R code: Using the Pipe Operator (|>)

Using the pipe `|>`: reading left to right

Instead, we can write code step by step, using the pipe: |>.

The pipe:

takes the output on the left-hand side
and passes it as the input to what is on the right-hand side

You can read it as “and then”.

For example

1:10 |>
  diff() |>
  cumsum() |>
  log() |>
  mean() |>
  round()

[1] 1

This is easier to read:

Take the numbers 1 to 10 and then take differences and then take cumulative sums and then take logs and then take the mean and then round.

We will use this style a lot in this course, especially with functions from tidyverse.

An example using our dataset

Let’s say we want to:

start with the students dataset
pick the Degree column
count how many students are in each degree

Using the pipe, we can write this clearly:

students |>
  select(Degree) |>
  count(Degree)

# A tibble: 12 × 2
   Degree           n
   <fct>        <int>
 1 Anthropology    22
 2 Architecture     7
 3 Business        15
 4 Design          16
 5 Education       17
 6 English         12
 7 History         20
 8 Linguistics     29
 9 Philosophy      15
10 Politics        14
11 Psychology      24
12 Sociology        9

How to read this

Take the students data
and then ‘select’ the column called Degree
and then ‘count’ how many times each degree appears

This is easier to read than nesting everything inside one line, such as:

count(select(students, Degree), Degree)

# A tibble: 12 × 2
   Degree           n
   <fct>        <int>
 1 Anthropology    22
 2 Architecture     7
 3 Business        15
 4 Design          16
 5 Education       17
 6 English         12
 7 History         20
 8 Linguistics     29
 9 Philosophy      15
10 Politics        14
11 Psychology      24
12 Sociology        9

Another example

Imagine we want to:

take the Year column
count how many students are in each year
add percentages

students |>
  count(Year) |>
  mutate(Percent = round((n / sum(n))*100, 1))

# A tibble: 3 × 3
   Year     n Percent
  <dbl> <int>   <dbl>
1     1    63    31.5
2     2    89    44.5
3     3    48    24

Read it as:

start with students
and then count how many students are in each Year
and then create a new column with percentages

About `%>%`

You may still see the older pipe symbol %>% in online examples:

students %>% select(Degree) %>% count(Degree)

# A tibble: 12 × 2
   Degree           n
   <fct>        <int>
 1 Anthropology    22
 2 Architecture     7
 3 Business        15
 4 Design          16
 5 Education       17
 6 English         12
 7 History         20
 8 Linguistics     29
 9 Philosophy      15
10 Politics        14
11 Psychology      24
12 Sociology        9

It works in exactly the same way, but the newer |> is now preferred.

2.2 Numerical description: Proportions and Percentages

Frequencies tell us how many observations fall into each category.

However, we often want to understand:

How large each category is relative to the whole sample

We do this using proportions and percentages.

What is a proportion?

A proportion is the fraction of observations in a category.

\[ \text{proportion} = \frac{\text{number in category}}{\text{total number of observations}} \]

Values range from 0 to 1
Example: 0.20 means 20% of the sample

What is a percentage?

A percentage is simply a proportion multiplied by 100.

\[ \text{percentage} = \text{proportion} \times 100 \]

Values range from 0 to 100
Example: 20% instead of 0.20

A percentage is just a different way of expressing a proportion.

When do we use proportions and percentages?

A percentage is just a different way of expressing a proportion, but they are used in different contexts.

Use proportions when:

performing calculations
working in statistical formulas
conducting inferential statistics (e.g. hypothesis tests)

Proportions are preferred because they are easier to use mathematically (values between 0 and 1).

Use percentages when:

reporting results
creating graphs and tables
communicating findings to a general audience

Percentages are easier to interpret and more intuitive to read.

Adding proportion and percentages to the frequency table

students |>
  count(Degree) |>
  arrange(desc(n)) |>
  mutate(
    proportion = round(n / sum(n), 3),
    percentage = round(n / sum(n) * 100, 1)
  )

# A tibble: 12 × 4
   Degree           n proportion percentage
   <fct>        <int>      <dbl>      <dbl>
 1 Linguistics     29      0.145       14.5
 2 Psychology      24      0.12        12  
 3 Anthropology    22      0.11        11  
 4 History         20      0.1         10  
 5 Education       17      0.085        8.5
 6 Design          16      0.08         8  
 7 Business        15      0.075        7.5
 8 Philosophy      15      0.075        7.5
 9 Politics        14      0.07         7  
10 English         12      0.06         6  
11 Sociology        9      0.045        4.5
12 Architecture     7      0.035        3.5

What the code does:

1. Start with the dataset

students→ This is our data

Count observations

count(Degree)

→ Counts how many students are in each degree

→ Creates a column called n

Order the table

arrange(desc(n))

→ Sorts the table from most common → least common

Calculate proportion

n / sum(n)

→ Divides each category count by the total number of students

This gives the proportion (a value between 0 and 1)

round(..., 3)

→ Rounds to 3 decimal places

5. Calculate percentage

n / sum(n) * 100

→ Converts the proportion into a percentage

This gives values between 0 and 100

round(..., 1)

→ Rounds to 1 decimal place

Final table:

Example:

Linguistics = 29 students

→ proportion ≈ 0.15 → percentage ≈ 14.5%

We take the count for each category and divide it by the total to understand how large each group is relative to the whole sample.

2.3 Numerical description: Mode (central tendency)

So far, we have described how the data are distributed across categories (spread).

We now focus on the centre of the distribution.

Note on Measures of central tendency

When describing a variable, we often want to identify its centre.

Measures of central tendency are statistics that describe the typical or central value of a distribution.

There are three main measures:

The mean is the sum of all values divided by the number of observations.

\[ \text{mean} = \frac{\text{sum of values}}{\text{number of observations}} \]

Used for numeric variables
Not appropriate for categorical data

The median is the middle value when the data are ordered.

Used for:
- numeric variables
- ordinal categorical variables

The mode is the most frequent value.

Used for:
- numeric variables
- categorical variables (nominal and ordinal)

What is the mode?

The mode is the most frequent category; the category with the highest count.

It represents the central point of a categorical distribution
It tells us which category is most typical

Identifying the mode in our table

Because we ordered the table from most to least frequent, the mode is:

students |>
  count(Degree) |>
  arrange(desc(n))

# A tibble: 12 × 2
   Degree           n
   <fct>        <int>
 1 Linguistics     29
 2 Psychology      24
 3 Anthropology    22
 4 History         20
 5 Education       17
 6 Design          16
 7 Business        15
 8 Philosophy      15
 9 Politics        14
10 English         12
11 Sociology        9
12 Architecture     7

The most common degree is Linguistics, so this is the mode of the variable Degree.

Why do we use the mode?

For nominal variables, the mode is the only measure of central tendency we can use.

This is because:

categories have no natural order
we cannot calculate a median or mean

⚠️ Important: The median is NOT based on frequencies

A common misunderstanding is to think that we can find the median by:

ordering categories by frequency (most common → least common), and
picking the “middle” category

Example (incorrect reasoning)

Suppose we have this frequency table for Degree:

Degree	n
Linguistics	29
Psychology	24
Anthropology	22
History	20
Education	17

You might think:

“Anthropology is in the middle, so it is the median.”

❌ This is incorrect.

Why is wrong?

The table is ordered by frequency, not by the values of the variable.

“Linguistics”, “Psychology”, “Anthropology”
do not have a natural order

What the median actually requires

The median is the middle value of the ordered data.

This means:

for an ordinal categorical variable → we order the categories according to their natural order
for a numerical variable → we order the numeric values

Key rule

If a variable has no natural order, it has no median.

2.4 Visual Description: Bar Charts

So far, we have described categorical variables numerically.

We now describe them visually, using graphs.

Bar charts are the most common way to visualise categorical distributions, as they show how many observations fall into each category.

`Degree` distribution

degree_barplot <- ggplot(students, aes(x = Degree)) +
  geom_bar(fill = "steelblue4") +
  labs(x = "\n Degree subject", y = "Count \n")
degree_barplot

How to read this plot

each bar represents a category of Degree
the height of each bar shows the frequency (count)
taller bars represent more observations
shorter bars represent fewer observations

This gives a quick visual summary of the distribution of the variable.

Changing the colour

We can change the bar colour using the fill = argument inside geom_bar():

degree_barplot <- ggplot(students, aes(x = Degree)) +
  geom_bar(fill = "darkgreen") +
  labs(x = "\n Degree subject", y = "Count \n")
degree_barplot

Changing the order of categories

Sometimes we want to control the order of categories in the plot.

For a nominal variable like Degree, there is no natural order, so we may choose an order that makes the plot easier to read.

Often, it is more helpful to display categories from most common to least common.

We can do this using fct_infreq() from the forcats package, which is included in the tidyverse.

students |>
  mutate(Degree = fct_infreq(Degree)) |>
  ggplot(aes(x = Degree)) +
  geom_bar(fill = "steelblue4") +
  labs(
    x = "\nDegree subject",
    y = "Count\n"
  )

What does `fct_infreq()` do?

fct_infreq(Degree) reorders the categories of Degree from:

most frequent to
least frequent

This makes it easier to:

identify the mode
compare category sizes
interpret the distribution quickly

Reverse the order: least common to most common

Sometimes we want the smallest categories first and the largest categories last.

We can reverse the order using fct_rev():

students |>
  mutate(Degree = fct_rev(fct_infreq(Degree))) |>
  ggplot(aes(x = Degree)) +
  geom_bar(fill = "steelblue4") +
  labs(
    x = "\nDegree subject",
    y = "Count\n"
  )

What does `fct_rev()` do?

fct_rev() reverses the order of the factor levels.

So:

fct_infreq(Degree) gives
most common → least common
fct_rev(fct_infreq(Degree)) gives least common → most common

Why might we reorder a bar chart?

Reordering categories can make the graph easier to read.

For example:

most common → least common helps you spot the mode immediately
least common → most common can make differences between smaller groups easier to see

Want to learn more about ggplot?

For a deeper explanation of ggplot, including the grammar of graphics,
aesthetics, geoms, and themes, see the Plots & Grammar page.

Plots and Grammar – How ggplot Works

3. Describing Ordinal Categorical Variables

Ordinal variables are categorical variables with a meaningful order.

Examples in our dataset:

Year (1, 2, 3, 4)
Stress_level (Low, Moderate, High)
Satisfaction_Likert_value (1–5 scale)

Because the categories are ordered, we can calculate additional summary measures.

We can describe:

distribution (spread across categories) → frequencies, proportions, percentages
centre → mode (nominal & ordinal), median (ordinal only)
dispersion (variability) → range and interquartile range (IQR) (only for ordinal variables)

Important distinction

Frequencies tell us how observations are distributed across categories
→ “how many are in each group”
Dispersion measures tell us how spread out ordered data are
→ “how far apart the values are”

3.1 Frequency distributions

Some categorical variables in the students dataset have a natural order.

For example:

Year

Step 1: Inspect the variable

First, let’s take a quick look at the variable:

summary(students$Year)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  1.000   1.000   2.000   1.925   2.000   3.000

This gives:

minimum = 1
maximum = 3
median and quartiles

At this stage, R is treating Year as a numeric variable.

We can confirm this:

class(students$Year)

[1] "numeric"

Why does this matter?

Although Year is stored as numbers (1, 2, 3), it actually represents categories with order, not true numerical measurements.

This means it is an ordinal categorical variable, not a continuous numeric variable.

Step 2: Convert to a categorical variable

We convert Year into a factor:

students$Year <- factor(students$Year)

Check again:

class(students$Year)

[1] "factor"

Step 3: Look at the summary again

summary(students$Year)

 1  2  3 
63 89 48

Now the output shows:

counts for each category (Year 1, Year 2, Year 3)

This is now a frequency table, which is what we want for categorical data.

Now that we have correctly treated Year as a categorical variable, we can describe its distribution using frequencies, proportions, and percentages.

⚠️ Important: Numbers do NOT always mean numeric data

Just because a variable contains numbers, it does not mean it should be treated as numeric.

For example:

Year uses values 1, 2, 3
Satisfaction_Likert_value uses values 1–5

These are codes for categories, not true measurements.

What’s the difference?

Numeric variables → numbers represent quantities
(e.g. height, income, age)
Ordinal variables → numbers represent ordered categories
(e.g. Year 1 → Year 2 → Year 3)

Why does this matter?

If we treat ordinal variables as numeric:

the mean may not be meaningful
distances between values may be misleading
(is the gap between 1 and 2 the same as 4 and 5?)

What should we do?

Convert to a factor for frequency tables and plots
Use median and quartiles, not the mean, to describe the data

Always think: Do these numbers represent quantities or categories?

Frequency table

students |>
  count(Year)

# A tibble: 3 × 2
  Year      n
  <fct> <int>
1 1        63
2 2        89
3 3        48

How to interpret this table

Each row represents a year of study
The column n shows the number of students in each year

For example:

If Year = 2 and n = 89, this means there are 89 students in Year 2.

What does this tell us?

This table allows us to:

see how students are distributed across the years
identify which years have more or fewer students

However, counts alone can be difficult to compare.

For example, saying “89 students are in Year 2” is less informative than knowing what proportion of the sample this represents.

To better understand the distribution, we convert counts into:

proportions (values between 0 and 1)
percentages (values out of 100)

3.2 Adding relative frequencies (proportions & percentages)

We can also calculate proportions and percentages to better understand how the students are distributed across the years.

students |>
  count(Year) |>
  mutate(
    proportion = round(n / sum(n), 3),
    percentage = round(n / sum(n) * 100, 1)
  )

# A tibble: 3 × 4
  Year      n proportion percentage
  <fct> <int>      <dbl>      <dbl>
1 1        63      0.315       31.5
2 2        89      0.445       44.5
3 3        48      0.24        24

How to interpret this table

n → number of students in each year
proportion → share of the sample (between 0 and 1)
percentage → proportion of the sample (out of 100%)

Example

If Year = 2 has proportion = 0.45, this means that 45% of students are in Year 2.

Why is this important?

It makes comparisons between categories clearer
It allows us to understand how the data are distributed along the ordered scale

Most importantly, it prepares us to calculate cumulative percentages, which we use to identify:

the median
the quartiles

Tip

frequency table → “how many”
proportions → “how much of the sample”
cumulative → “how data build up”
quartiles → “where key thresholds are reached”

3.3 Cumulative percentages

To understand how the data build up across the ordered categories, we calculate cumulative percentages.

Cumulative percentage The percentage of observations in a category or below it.

Calculate cumulative percentages

students |>
  count(Year) |>
  mutate(
    percent = round(n / sum(n) * 100, 1),
    cumulative = cumsum(percent)
  )

# A tibble: 3 × 4
  Year      n percent cumulative
  <fct> <int>   <dbl>      <dbl>
1 1        63    31.5       31.5
2 2        89    44.5       76  
3 3        48    24        100

How to interpret this table

percent → percentage of students in each year
cumulative → percentage of students in that year or any earlier year

What is happening here?

As we move from Year 1 → Year 3, we are accumulating students:

Year 1 → percentage of students in Year 1
Year 2 → percentage in Year 1 and Year 2 combined
Year 3 → percentage in all years

The cumulative column shows how the data build up along the ordered scale

Why is this important?

Cumulative percentages allow us to identify key points in the data by seeing where the distribution crosses certain thresholds.

25% point → first value where cumulative ≥ 25% → Q1
50% point → first value where cumulative ≥ 50% → median
75% point → first value where cumulative ≥ 75% → Q3

Apply this to our table

Year	Percent	Cumulative
1	31.5	31.5
2	44.5	76.0
3	24.0	100.0

Q1 (25% point)

We look for the first cumulative value ≥ 25
Year 1 = 31.5%
Q1 = Year 1

Median (50% point)

First cumulative value ≥ 50%
Year 2 = 76.0%
Median = Year 2

Q3 (75% point)

First cumulative value ≥ 75%
Year 2 = 76.0%
Q3 = Year 2

Quartiles are based on cumulative percentages, not on the size of each category.

Interpretation

Most students are in Year 1 and Year 2
The middle 50% of students are concentrated between Year 1 and Year 2
Very few students are in the highest category (Year 3)

Visualising cumulative percentages

How to read this plot

Bars (blue) → percentage of students in each year
Green line → cumulative percentage
Dashed lines → 25%, 50%, 75% thresholds

What do we look for?

Where the black line crosses the dashed lines

Interpretation

The curve passes 25% at Year 1 → Q1
The curve passes 50% at Year 2 → Median
The curve passes 75% at Year 2 → Q3

Notice that Q2 and Q3 fall in the same category. This is normal when many observations are concentrated in one group.

Interquartile Range (IQR)

The interquartile range (IQR) measures how spread out the middle 50% of the data are.

It focuses only on the central part of the distribution, ignoring the lowest and highest values.

The IQR is calculated as: IQR = Q3 - Q1

Apply this to our example

From our previous results:

Q1 = Year 1
Q3 = Year 2

IQR = 2-1 = 1

In R

IQR(students$Year)

[1] 1

It means the middle 50% of students lie between Year 1 and Year 2, which are one category apart.

so, “1” is a distance between categories

Concept	Meaning
Q1 = 1	starting category
Q3 = 2	ending category
IQR = 1	number of steps between them

The IQR is 1. This does not mean “Year 1”. It means that the middle 50% of students fall within a range that spans one category level, from Year 1 to Year 2.

What does this tell us?

The IQR tells us the range of categories that contain the middle 50% of students.

While the median tells us the centre of the data, the IQR tells us how spread out the typical values are.

In this case:

The middle half of students are between Year 1 and Year 2

The red shaded region shows the interquartile range (IQR): from Q1 to Q3.
This is where the middle 50% of students fall when the data are ordered.

Visualising spread with a boxplot

Because ordinal variables have an order, we can also visualise their centre and spread using a boxplot.

A boxplot is a compact visual summary of a distribution.

The box runs from Q1 to Q3. This is the interquartile range (IQR), so it contains the middle 50% of the data.

The line inside the box is the median. It marks the centre of the ordered data.

The whiskers extend from the box to the most extreme values that are still within 1.5 × IQR from Q1 and Q3.

Points beyond those limits are shown as outliers.

So, in order:

Q1 = lower edge of the box
Median = line inside the box
Q3 = upper edge of the box
IQR = width of the box
Whiskers = most extreme non-outlying values
Outliers = values below Q1 - 1.5 × IQR or above Q3 + 1.5 × IQR

The box shows where the middle 50% of the data lie, the line inside shows the median, the whiskers show the most extreme typical values, and any separate points are potential outliers.

Boxplot `Year`

Because Year only has a few ordered categories, the boxplot will be less detailed than for a continuous variable like IQ, but it still summarises the same ideas:

Q1 = lower edge of the box
Median = line inside the box
Q3 = upper edge of the box
IQR = width of the box

In this example, the median and Q3 fall in the same category, so the median line lies on the upper edge of the box. We highlight it separately in green to make it visible.

3.4 Visual Description

Ordinal variables can be visualised much like nominal variables, but with one important difference:

For ordinal variables, the categories must remain in their natural order.

For example, with Year, we keep the order:

1 = Year 1
2 = Year 2
3 = Year 3

We do not reorder these categories by frequency.

Bar chart using counts

A simple bar chart shows how many students are in each year.

ggplot(students, aes(x = Year)) +
  geom_bar(fill = "steelblue4") +
  labs(
    x = "Year",
    y = "Count"
  ) +
  theme_minimal()

How to read this plot

each bar represents a year category
the height of the bar shows the number of students
because the categories are ordered, the bars move from lower to higher year

This plot gives a quick visual summary of how students are distributed across the ordered categories.

Bar chart using percentages

Sometimes percentages are easier to interpret than raw counts.

First, we create a summary table:

year_percent <-
  students |>
  count(Year) |>
  mutate(
    percent = n / sum(n) * 100
  )

year_percent

# A tibble: 3 × 3
  Year      n percent
  <fct> <int>   <dbl>
1 1        63    31.5
2 2        89    44.5
3 3        48    24

Then we plot the percentages:

ggplot(year_percent, aes(x = Year, y = percent)) +
  geom_col(fill = "steelblue4") +
  geom_text(
    aes(label = paste0(round(percent, 1), "%")),
    vjust = 1.5, colour = "white", size = 4
  ) +
  labs(
    x = "Year",
    y = "Percentage of students"
  ) +
  theme_minimal()

How to interpret this plot

each bar shows the percentage of students in each year
the height represents the relative size of each group
Year 2 has the largest share of students
Year 3 has the smallest share

This gives a clearer understanding of the distribution compared to raw counts.

Week 09 Summary: How to describe categorical variables

What to remember

When we describe a categorical variable, we want to understand:

the distribution → how observations are spread across categories
the centre → the typical or most common category
the spread / dispersion → how spread out the ordered data are (for ordinal variables only)

Step-by-step process

1. Identify the type of variable

Ask:

Is it nominal? → categories with no natural order
Is it ordinal? → categories with a meaningful order

Examples:

Nominal: Degree, Gender, Exercise
Ordinal: Year, Stress_level, Satisfaction_Likert_value

2. Check how the variable is stored in R

Before analysing a categorical variable, check whether it is stored as a factor.

Useful functions:

class() → shows the type of variable
summary() → gives an overview
factor() → converts a variable to a categorical variable

3. Describe the variable numerically

For nominal variables

Use:

frequency tables
proportions / percentages
mode

Do not use:

median
quartiles
IQR

because nominal variables have no natural order.

For ordinal variables

Use:

frequency tables
proportions / percentages
median
quartiles
IQR

because ordinal variables have a meaningful order.

4. Describe the variable visually

For categorical variables

Use:

bar charts to show counts or percentages across categories

For ordered variables

Keep the categories in their natural order.

Do not reorder ordinal categories by frequency.

5. For ordinal variables, use cumulative percentages

Cumulative percentages help us identify:

Q1 → first category where cumulative ≥ 25%
Median → first category where cumulative ≥ 50%
Q3 → first category where cumulative ≥ 75%

Then:

IQR = Q3 - Q1

Key rules

No natural order → no median
Numbers do not always mean numeric data
Quartiles are based on cumulative percentages
IQR measures the spread of the middle 50%
For ordinal variables, the IQR value is a distance between categories, not a category itself

Main visual tools from this week

Bar chart → shows the distribution across categories
Boxplot → summarises median, quartiles, IQR, whiskers, and possible outliers

In one sentence

To analyse a categorical variable, we:

identify whether it is nominal or ordinal
check that it is stored correctly in R
describe it numerically
describe it visually
for ordinal variables, use median, quartiles, and IQR

Introduction

1. Descriptive Statistics: What does it mean to describe a variable?

The exact summaries depend on the type of variable

Getting started in R

Data exploration

Important: How categorical variables are stored in R

What is a character variable?

What is a factor?

What do we do next?

2. Describing Nominal Categorical Variables

2.1 Frequency distributions

How to read this table

Ordering the table (most to least frequent)

What does this do?

Why is this useful?

Using the pipe |>: reading left to right

An example using our dataset

How to read this

Another example

About %>%

2.2 Numerical description: Proportions and Percentages

What is a proportion?

What is a percentage?

When do we use proportions and percentages?

Use proportions when:

Use percentages when:

Adding proportion and percentages to the frequency table

What the code does:

2.3 Numerical description: Mode (central tendency)

Note on Measures of central tendency

What is the mode?

Identifying the mode in our table

Why do we use the mode?

Example (incorrect reasoning)

Why is wrong?

What the median actually requires

Key rule

2.4 Visual Description: Bar Charts

Degree distribution

How to read this plot

Changing the colour

Changing the order of categories

What does fct_infreq() do?

Reverse the order: least common to most common

What does fct_rev() do?

Why might we reorder a bar chart?

3. Describing Ordinal Categorical Variables

3.1 Frequency distributions

Step 1: Inspect the variable

Why does this matter?

Step 2: Convert to a categorical variable

Step 3: Look at the summary again

What’s the difference?

Why does this matter?

What should we do?

Frequency table

How to interpret this table

What does this tell us?

3.2 Adding relative frequencies (proportions & percentages)

How to interpret this table

Example

Why is this important?

3.3 Cumulative percentages

Calculate cumulative percentages

How to interpret this table

What is happening here?

Why is this important?

Apply this to our table

Interpretation

Visualising cumulative percentages

How to read this plot

What do we look for?

Interpretation

Interquartile Range (IQR)

Apply this to our example

In R

What does this tell us?

Visualising spread with a boxplot

Boxplot Year

3.4 Visual Description

Using the pipe `|>`: reading left to right

About `%>%`

`Degree` distribution

What does `fct_infreq()` do?

What does `fct_rev()` do?

Boxplot `Year`