Frequency tables

How often does each value occur? Counting, percentages and classes

Describe the distribution of categorical and discrete variables with absolute, relative and cumulative frequencies, using table(), prop.table(), dplyr::count() and janitor::tabyl(), and group a continuous variable into classes with cut().

Level 2 · Describe & explore ⏱ ~60 min Prerequisites: Factors, Grouped summaries

Learning objectives

By the end of this lesson you will be able to:

build absolute, relative and cumulative frequency tables in base R and with dplyr;
produce publication-ready one-way tables with janitor::tabyl() and the adorn_*() helpers;
find the mode of a categorical or discrete variable;
group a continuous variable into classes with cut() and explain why the choice of classes matters;
read a frequency table critically (denominators, missing values, rounding).

library(dplyr)
library(janitor)

1. Why count?

The penguins data contain 344 birds. If someone asks “which species did the researchers study?”, printing the species column gives you 344 words in a row — correct, but useless. What we really want to know is:

Which values occur, and how often does each one occur?

That list of values together with how often they occur is called the distribution of a variable. A frequency table is the simplest way to show a distribution. It is usually the first thing a statistician looks at for any categorical variable, and it is how every bar chart, histogram and percentage in a report is built.

2. Which table for which variable?

Recall the four types of variables from the Factors lesson:

Type	Example in `penguins`	Values have an order?	Typical summary
Nominal (categorical, unordered)	`species`, `island`, `sex`	No	Counts, percentages, mode
Ordinal (categorical, ordered)	a “small / medium / large” size class	Yes	Counts, percentages, cumulative percentages, median category
Discrete numeric (few distinct values)	`year` (2007, 2008, 2009)	Yes	Same as ordinal
Continuous numeric	`body_mass`, `flipper_len`	Yes	Frequency table only after grouping into classes (Section 8)

Frequency tables work directly for the first three types. For a continuous variable almost every value is different, so a table of individual values would have hundreds of rows with a count of 1 — we first have to group the values.

3. Absolute frequencies with `table()`

The absolute frequency $n_i$ of a value is simply the number of observations that take that value.

table(penguins$species)

#> 
#>    Adelie Chinstrap    Gentoo 
#>       152        68       124

The output lists each level of the factor and how many penguins belong to it. The counts add up to the number of observations:

sum(table(penguins$species))

#> [1] 344

Missing values

By default table() silently drops NA. The variable sex has missing values, and you would never know from the default output:

table(penguins$sex)

#> 
#> female   male 
#>    165    168

table(penguins$sex, useNA = "ifany")

#> 
#> female   male   <NA> 
#>    165    168     11

useNA = "ifany" adds an NA column only when there are missing values. Get into the habit of using it: a table that hides 11 unknown birds tells a different story from one that shows them.

4. Relative frequencies with `prop.table()`

Counts are hard to compare between datasets of different size: 50 out of 100 is a lot, 50 out of 10,000 is not. The relative frequency divides each count by the total:

\[ f_i = \frac{n_i}{n}. \]

tab_species <- table(penguins$species)
prop.table(tab_species)

#> 
#>    Adelie Chinstrap    Gentoo 
#>    0.4419    0.1977    0.3605

round(100 * prop.table(tab_species), 1)

#> 
#>    Adelie Chinstrap    Gentoo 
#>      44.2      19.8      36.0

How to read this output

The first table gives proportions (between 0 and 1); they add up to 1.
The second multiplies by 100 and rounds: percentages. About 44% of the penguins in the data are Adelie.
Because of rounding, rounded percentages do not always add up to exactly 100 (for example 99.9 or 100.1). This is normal; mention it in a footnote rather than “fixing” a number by hand.
These are percentages of the birds in this dataset, not of all penguins in Antarctica. Whether they describe a wider population depends on how the birds were sampled — a question for Level 3.

Going deeper: notation for frequency distributions

Suppose a variable takes $K$ distinct values (or categories) $x_1, \dots, x_K$ in a sample of size $n$. Then

absolute frequencies: $n_1, \dots, n_K$ with $\sum_{i=1}^K n_i = n$;
relative frequencies: $f_i = n_i / n$ with $0 \le f_i \le 1$ and $\sum_{i=1}^K f_i = 1$;
for ordered values $x_1 < x_2 < \dots < x_K$, cumulative frequencies: $N_j = \sum_{i \le j} n_i$ and $F_j = \sum_{i \le j} f_i = N_j / n$, so that $F_K = 1$.

The sequence $F_1, \dots, F_K$ is the empirical cumulative distribution function evaluated at the observed values; you will meet it again as a plot in Distribution plots.

5. The tidy way: `dplyr::count()`

When your data are in a data frame (they usually are), dplyr::count() gives the same information as a data frame you can keep working with:

penguins |>
  count(island) |>
  mutate(prop = n / sum(n),
         percent = round(100 * prop, 1))

#>      island   n   prop percent
#> 1    Biscoe 168 0.4884    48.8
#> 2     Dream 124 0.3605    36.0
#> 3 Torgersen  52 0.1512    15.1

count(island) is shorthand for group_by(island) |> summarise(n = n()). The mutate() adds the relative frequency. Two useful options:

penguins |>
  count(sex, sort = TRUE)   # largest group first; NA is kept as its own row

#>      sex   n
#> 1   male 168
#> 2 female 165
#> 3   <NA>  11

Note a difference from table(): count() keeps missing values as a row. So n / sum(n) uses all birds as the denominator. If you want the percentage among birds whose sex is known, filter the missing values out first (filter(!is.na(sex))). Both are legitimate — but you must say which one you report.

6. Ready-to-publish tables with `janitor::tabyl()`

The janitor package wraps all of this into one function whose output is a data frame with counts and percentages:

penguins |>
  tabyl(sex)

#>     sex   n percent valid_percent
#>  female 165 0.47965        0.4955
#>    male 168 0.48837        0.5045
#>    <NA>  11 0.03198            NA

tabyl() reports percent (denominator: all birds, including NA) and valid_percent (denominator: birds with a known value). The adorn_*() functions then polish the table for a report:

penguins |>
  tabyl(species) |>
  adorn_totals("row") |>
  adorn_pct_formatting(digits = 1)

#>    species   n percent
#>     Adelie 152   44.2%
#>  Chinstrap  68   19.8%
#>     Gentoo 124   36.0%
#>      Total 344  100.0%

adorn_totals("row") adds a Total row and adorn_pct_formatting() turns proportions into formatted percentages. Formatted percentages are text, so do this only as the very last step, after all calculations.

7. The mode

The mode is the most frequent value. It is the only “average” that makes sense for a nominal variable (you cannot compute the mean of "Adelie", "Gentoo" and "Chinstrap"). R has no built-in function for the statistical mode (mode() tells you the storage type of an object!), but it is easy to find from a table:

tab_island <- table(penguins$island)
names(tab_island)[tab_island == max(tab_island)]

#> [1] "Biscoe"

We compare with max() rather than using which.max() because a distribution can have two or more modes (ties); which.max() would silently return only the first one.

8. Ordinal and discrete variables: cumulative frequencies

When the values have a natural order, a new question becomes meaningful: “how many observations are at most this value?” The cumulative frequency answers it by adding up counts from the smallest value upwards.

The number of forward gears in mtcars is a discrete variable:

tab_gear <- table(mtcars$gear)
data.frame(
  gears    = names(tab_gear),
  n        = as.vector(tab_gear),
  percent  = round(100 * as.vector(prop.table(tab_gear)), 1),
  cum_perc = round(100 * cumsum(as.vector(prop.table(tab_gear))), 1)
)

#>   gears  n percent cum_perc
#> 1     3 15    46.9     46.9
#> 2     4 12    37.5     84.4
#> 3     5  5    15.6    100.0

How to read this output

Read the last column as “at most this many gears”: 84.4% of the cars have 4 gears or fewer. The last cumulative percentage is always 100. A cumulative column would be meaningless for species, because “at most Chinstrap” has no meaning when the categories have no order.

The order of the levels matters

cumsum() adds the counts in the order in which they appear in the table, and for a factor this is the order of its levels. For text data that order is alphabetical, which is usually wrong for ordinal answers. Here is a small survey:

answers <- c("Agree", "Neutral", "Disagree", "Agree", "Strongly agree",
             "Agree", "Neutral", "Strongly agree", "Disagree", "Agree")
table(answers)    # alphabetical: Agree, Disagree, Neutral, Strongly agree

#> answers
#>          Agree       Disagree        Neutral Strongly agree 
#>              4              2              2              2

A cumulative percentage of this table would mix up the scale. Declare the order explicitly with an ordered factor:

likert <- c("Strongly disagree", "Disagree", "Neutral",
            "Agree", "Strongly agree")
answers_f <- factor(answers, levels = likert, ordered = TRUE)

tab_ans <- table(answers_f)
round(100 * cumsum(prop.table(tab_ans)), 1)

#> Strongly disagree          Disagree           Neutral             Agree 
#>                 0                20                40                80 
#>    Strongly agree 
#>               100

Notice that the empty category “Strongly disagree” now appears with a count of 0: it is a possible answer that nobody chose, and that is information too. From the cumulative percentages we can read the median category: the first category whose cumulative percentage reaches 50% — here Agree.

9. Continuous variables: grouping into classes with `cut()`

To tabulate body_mass we must group values into classes (also called bins or intervals). The function cut() does this:

mass_class <- cut(penguins$body_mass,
                  breaks = seq(2500, 6500, by = 500),
                  dig.lab = 4)
table(mass_class, useNA = "ifany")

#> mass_class
#> (2500,3000] (3000,3500] (3500,4000] (4000,4500] (4500,5000] (5000,5500] 
#>          11          67          92          57          54          33 
#> (5500,6000] (6000,6500]        <NA> 
#>          26           2           2

How to read this output

(3000,3500] means “more than 3000 g and up to and including 3500 g”. By default classes are open on the left and closed on the right (right = TRUE); use right = FALSE for [3000,3500).
breaks must cover the whole range of the data, otherwise values outside become NA. Add include.lowest = TRUE if the minimum coincides with the first break.
dig.lab = 4 stops R from writing the limits in scientific notation (3e+03).

You can give the classes readable names with labels, which also creates an ordinal variable:

size_class <- cut(penguins$body_mass,
                  breaks = c(-Inf, 3500, 4500, Inf),
                  labels = c("small", "medium", "large"),
                  ordered_result = TRUE)
table(size_class, useNA = "ifany")

#> size_class
#>  small medium  large   <NA> 
#>     78    149    115      2

Why the choice of classes matters

Grouping always throws information away, and different classes can tell different stories. Compare the 500 g classes above with wider 900 g classes:

table(cut(penguins$body_mass,
          breaks = seq(2700, 6300, by = 900),
          include.lowest = TRUE, dig.lab = 4))

#> 
#> [2700,3600] (3600,4500] (4500,5400] (5400,6300] 
#>          97         130          81          34

With 500 g classes you can see that the most common masses are between 3500 and 4000 g and that there is a long “shoulder” of heavier birds (these are mainly Gentoo penguins). With four wide classes that detail disappears and the distribution just looks like a smooth hill. A few practical rules:

Use round, equal-width limits that are easy to read (500, 1000, …).
Aim for roughly 5–15 classes for a table; more for a histogram with lots of data.
Decide the classes before looking for a particular result. Choosing limits to make a pattern appear or disappear is a quiet form of cheating.
If you must use unequal widths (e.g., income bands), compare classes using density = relative frequency ÷ width, not raw counts — a wider class collects more observations simply because it is wider.

You will see the same issue, in graphical form, with histogram bin widths in Distribution plots.

Going deeper: rules of thumb for the number of classes

Several formulas suggest a number of classes $k$ for $n$ observations: Sturges $k = \lceil \log_2 n \rceil + 1$ (R’s default for hist(), via nclass.Sturges()), Scott (width $h = 3.49\, s\, n^{-1/3}$) and Freedman–Diaconis (width $h = 2\,\mathrm{IQR}\, n^{-1/3}$, robust to outliers). For our data:

mass <- penguins$body_mass[!is.na(penguins$body_mass)]
c(sturges = nclass.Sturges(mass), scott = nclass.scott(mass),
  fd = nclass.FD(mass))

#> sturges   scott      fd 
#>      10       9      11

They are starting points, not laws: always look at a few alternatives.

10. Reading a frequency table correctly

Before you trust (or publish) a frequency table, ask:

What is the denominator? All observations, or only those with a known value? Are missing values shown?
How many observations are behind the percentages? “67% of respondents” means something very different when $n = 3$ and when $n = 3000$. Always report $n$.
Are the categories exhaustive and mutually exclusive? Each observation should fall in exactly one row.
For ordinal data, are the categories in the right order? Otherwise cumulative columns are nonsense.
For grouped continuous data, what are the class limits and are they closed on the left or on the right?

Common pitfalls

Forgetting useNA = "ifany". table() drops missing values without warning, so percentages are silently computed on a smaller denominator.
Cumulating a nominal variable or an ordinal variable whose levels are in alphabetical order. Set the levels explicitly with factor(..., levels = ...).
Using mode() to get the statistical mode. mode(x) returns "numeric" or "character" — the storage type, not the most frequent value.
Formatting too early. After adorn_pct_formatting() or paste0(x, "%") the numbers are text; you can no longer sum or plot them.
cut() with breaks that do not cover the data. Values outside the breaks become NA; check with table(..., useNA = "ifany").

11. Check your understanding

Q1. In a table of eye colour, “Brown” has a relative frequency of 0.37. What does this number mean?

37% of the observations in the data have brown eyes; it is the count of “Brown” divided by the total number of observations used in the table (check whether missing values were included in that total).

Q2. For which of these variables does a cumulative percentage make sense: favourite colour, education level (primary / secondary / degree), number of siblings?

Education level (ordinal) and number of siblings (discrete). Favourite colour is nominal: its categories have no order, so “at most blue” means nothing.

Q3. What is the difference between percent and valid_percent in a tabyl()?

percent divides by all observations, including NA; valid_percent divides only by the observations with a known value. They differ only when there are missing values.

Q4. Two colleagues group the same income data into classes and get tables with different shapes. Can both be “right”?

Yes — both describe the same data at different resolutions. That is exactly why class limits must be reported, chosen in advance and, ideally, checked with more than one choice.

12. Exercises

Interactive (run in your browser)

★ Exercise 1 — counts and percentages. Build the frequency table of the number of cylinders (cyl) in mtcars, then the percentages rounded to one decimal.

★★ Exercise 2 — the tidy way, with and without NA. Using dplyr, compute the percentage of male and female penguins (a) out of all birds, (b) out of the birds whose sex is known.

penguins |>
  count(sex) |>
  mutate(percent = 100 * n / sum(n))

penguins |>
  filter(!is.na(sex)) |>
  count(sex) |>
  mutate(percent = 100 * n / sum(n))

In (a) the percentages for female and male are a little below 50% each because 11 birds are NA. In (b) they are close to 50%. Neither is wrong; a report must say which denominator was used.

★★ Exercise 3 — classes and cumulative percentages. The faithful data record the waiting time (minutes) between eruptions of the Old Faithful geyser. Group waiting into 10-minute classes from 40 to 100, build the table, and add the cumulative percentages. What percentage of waiting times are 70 minutes or less?

w_class <- cut(faithful$waiting, breaks = seq(40, 100, by = 10))
tab <- table(w_class, useNA = "ifany")
tab
round(100 * cumsum(prop.table(tab)), 1)

About 39% of waiting times are at most 70 minutes. Notice the shape of the counts: there are two peaks (around 50–60 and 70–90 minutes) with fewer values in between — a bimodal distribution, which a single average would hide.

★★ Exercise 4 — debug the cumulative table. A colleague reports that “50% of customers rated the service Fair or worse”. Run the code and find the bug.

ratings_f <- factor(ratings,
                    levels = c("Poor", "Fair", "Good", "Excellent"),
                    ordered = TRUE)
tab <- table(ratings_f)
cumsum(prop.table(tab))

With character data the categories are sorted alphabetically (Excellent, Fair, Good, Poor), so the colleague’s 50% is the share of “Excellent” plus “Fair” — a meaningless sum. With the correct order, only 30% of ratings are “Fair” or worse, and 70% are “Good” or worse.

Practice on your own computer

★★ Exercise 5 — a report-ready table. With janitor, build a one-way table of island in penguins with a total row and percentages formatted with one decimal. Then do the same for year. Which island is the mode? Is the number of birds similar across years?

Solution

penguins |>
  tabyl(island) |>
  adorn_totals("row") |>
  adorn_pct_formatting(digits = 1)

penguins |>
  tabyl(year) |>
  adorn_totals("row") |>
  adorn_pct_formatting(digits = 1)

Biscoe is the mode (168 birds, about 49%). The three years have similar numbers of birds (110, 114 and 120).

★★★ Exercise 6 — how classes change the story. Group penguins$flipper_len into classes of width 5 mm, 10 mm and 25 mm (choose sensible start and end points that cover the range of the data). For each choice, describe the shape of the distribution in one sentence. Which choice would you use in a report, and why?

Solution

range(penguins$flipper_len, na.rm = TRUE)   # 172 to 231

for (w in c(5, 10, 25)) {
  br <- seq(170, 235, by = w)
  if (max(br) < 231) br <- c(br, max(br) + w)
  print(table(cut(penguins$flipper_len, breaks = br)))
}

With narrow classes the table shows two groups of values (around 190 mm and around 215 mm — Adelie/Chinstrap versus Gentoo). With 25 mm classes there are only three classes and the two groups disappear. Classes of about 10 mm keep the main feature while staying readable.

Summary

Task	Base R	Tidyverse / janitor
Absolute frequencies	`table(x, useNA = "ifany")`	`count(df, x)`
Relative frequencies	`prop.table(table(x))`	`mutate(prop = n / sum(n))`
Percentages for a report	`round(100 * prop.table(t), 1)`	`tabyl(df, x) \\|> adorn_pct_formatting()`
Totals	`addmargins(t)`	`adorn_totals("row")`
Cumulative frequencies	`cumsum(prop.table(t))`	`mutate(cum = cumsum(prop))`
Mode	`names(t)[t == max(t)]`	`count(df, x, sort = TRUE)`
Classes for a continuous variable	`cut(x, breaks = ..., labels = ...)`	`mutate(cls = cut(x, ...))`

A frequency table shows the distribution of a variable: which values occur and how often.
Always know (and report) your denominator and how many observations are behind each percentage.
Cumulative frequencies only make sense for ordered values — check the order of the factor levels.
Grouping a continuous variable loses information; the choice of classes can change the impression, so choose it sensibly and in advance.

1. Why count?

2. Which table for which variable?

3. Absolute frequencies with table()

Missing values

4. Relative frequencies with prop.table()

5. The tidy way: dplyr::count()

6. Ready-to-publish tables with janitor::tabyl()

7. The mode

8. Ordinal and discrete variables: cumulative frequencies

The order of the levels matters

9. Continuous variables: grouping into classes with cut()

Why the choice of classes matters

10. Reading a frequency table correctly

11. Check your understanding

12. Exercises

Interactive (run in your browser)

Practice on your own computer

Summary

3. Absolute frequencies with `table()`

4. Relative frequencies with `prop.table()`

5. The tidy way: `dplyr::count()`

6. Ready-to-publish tables with `janitor::tabyl()`

9. Continuous variables: grouping into classes with `cut()`