Track 2 · Describing Data · Lesson 1 of 11

Frequency and relative frequency

Build and read frequency, relative-frequency and cumulative-frequency tables, including simple two-way tables.

25 minFoundation8 graded exercisesFree

A retail operations manager asks: how do our customers pay? Before any chart or average, the honest first answer is a count — how many baskets were paid by card, by cash, by mobile wallet — and what share of the total each one is.

A frequency table lists each possible value (or category) of a variable and how many observations take it. That count is the frequency. It is the simplest summary there is, and for categorical data it is often the only one you need.

Counts depend on how much data you have: 300 card payments means something different out of 400 baskets than out of 4 000. Dividing each count by the total gives the relative frequency — a proportion between 0 and 1, or a percentage when multiplied by 100. Relative frequencies always add up to 1 (100%), apart from rounding.

\text{relative frequency}_i = \frac{f_i}{n}, \qquad F_k = \sum_{i=1}^{k} f_i, \qquad \text{cumulative relative frequency}_k = \frac{F_k}{n}

Relative and cumulative frequency

$f_i$: the frequency (count) of the $i$ -th value or category
$n$: the total number of observations, $n = \sum f_i$
$F_k$: the cumulative frequency: how many observations fall in the first $k$ categories, counting in order

Cumulative frequency adds the counts up as you move down the table, so it answers how many are at most this value? That only makes sense when the values have a natural order — numbers of items, ratings, age groups. For unordered categories such as payment method or city, a running total depends on the arbitrary order you happened to list them in, so it has no meaning.

Worked example

Items per basket

A shop recorded the number of items in 20 baskets: 1, 3, 2, 2, 4, 1, 3, 2, 5, 2, 3, 1, 2, 4, 2, 3, 6, 2, 3, 4. Build the frequency table and find the percentage of baskets with at most 3 items.

List the values in order and tally: 1 item → 3 baskets, 2 → 7, 3 → 5, 4 → 3, 5 → 1, 6 → 1. Check: $3 + 7 + 5 + 3 + 1 + 1 = 20$ ✓.
Relative frequencies divide by $n = 20$ : 0.15, 0.35, 0.25, 0.15, 0.05, 0.05 (they add to 1).
Cumulative frequencies add as you go: 3, 10, 15, 18, 19, 20. The last one must equal $n$ .
At most 3 items is the cumulative frequency at 3: $F = 15$ , so $15/20 = 0.75$ , i.e. 75% of baskets.

75% of the baskets contain at most 3 items; the most common basket has 2 items (35%).

Main travel mode of 240 employees (commute dataset). A bar chart is simply a frequency table drawn to scale.

Show the data as a table

Travel mode	Count
car	105
bus	35
metro	57
bike	20
walk	23

Mode	Frequency	Relative frequency	Percentage
car	105	0.4375	43.8%
metro	57	0.2375	23.8%
bus	35	0.1458	14.6%
walk	23	0.0958	9.6%
bike	20	0.0833	8.3%
Total	240	1.0000	100%

Frequency table of travel mode in the (synthetic) commute dataset, ordered by frequency

Dataset

Retail transactions

1,500 rows · one row = one transaction (basket) · synthetic data

About this dataset Open in the lab

Store	Customer type	Payment method	Items (items)	Basket value (SAR)
University	returning	card	6	129.86
Harbour Mall	returning	mobile_wallet	2	60.64
Downtown	new	cash	1	14.65
Airport Road	returning	card	1	6.95
Airport Road	returning	mobile_wallet	2	63.04
Airport Road	new	card	1	34.84

Each row is one basket. Count the rows in each category of payment_method to build its frequency table.

Two-way tables. To see how two categorical variables relate, count every combination. The table below cross-tabulates shift and travel mode for the 240 employees. Each cell is a joint frequency; the totals in the margins are the ordinary one-way frequency tables (marginal frequencies).

Shift	car	metro	bus	walk	bike	Total
morning	44	30	18	9	8	109
day	48	17	15	11	9	100
evening	13	10	2	3	3	31
Total	105	57	35	23	20	240

Shift × travel mode, commute dataset (counts)

A cell can be turned into three different percentages, and they answer three different questions. Take the 44 morning-shift drivers:

of everyone: $44/240 = 18.3\%$ of all employees are morning-shift drivers (joint percentage);
of the row: $44/109 = 40.4\%$ of morning-shift employees drive (row percentage);
of the column: $44/105 = 41.9\%$ of drivers work the morning shift (column percentage).

To compare groups, divide by the size of each group: to compare how the shifts travel, use row percentages.

Quick check

Of 109 morning-shift employees, 30 use the metro; 57 employees use the metro overall. Which statement is correct?

Exercises

The lesson is complete when every exercise is answered correctly. Hints are free; viewing a worked solution before solving means the exercise won't count towards mastery.

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Exercise 1 · Guided, difficulty 1 of 3

Build a frequency table

Not started

Twenty patients rated a clinic's waiting room on a scale from 1 (very poor) to 5 (very good):

4, 1, 4, 3, 3, 4, 4, 1, 5, 1, 2, 5, 4, 5, 2, 3, 3, 2, 4, 4

Find (a) the frequency of the rating 4, (b) the relative frequency of the rating 4 as a percentage, and (c) the cumulative percentage of ratings at most 3.

Exercise 2 · Practice, difficulty 1 of 3

Reading a cumulative share

Not started

An HR team recorded how many days of sick leave each employee in a department took last quarter:

Days of sick leave	Employees
0	12
1	12
2	13
3	4
4	4
5	2

What percentage of the employees took at most 2 days of sick leave? Give the percentage to 1 decimal place.

Exercise 3 · Application, difficulty 2 of 3

Mobile wallet share

Not started

In the transactions dataset, the payment method is missing for some baskets. Among the transactions whose payment method was recorded, what percentage were paid by mobile_wallet? Give the percentage to 2 decimal places.

Data: transactions · Open in the lab to explore and calculate.

Exercise 4 · Application, difficulty 2 of 3

Row percentage in a two-way table

Not started

Using the commute dataset, cross-tabulate shift by mode. What percentage of morning-shift employees travel by car? Give the percentage to 1 decimal place.

Data: commute · Open in the lab to explore and calculate.

Exercise 5 · Interpretation, difficulty 2 of 3

Which percentage is it?

Not started

A clinic cross-tabulates 400 patients by smoking status (rows: smoker, non-smoker) and hypertension (columns: yes, no). The report says: “In the smoker row, the hypertension cell shows 30% (row percentages).” Which statement does this support?

Exercise 6 · Practice, difficulty 1 of 3

When does cumulative frequency make sense?

Not started

Sort each variable: is a cumulative frequency (a running total down the table) meaningful for it?

Exercise 7 · Practice, difficulty 1 of 3

Steps to a frequency table

Not started

Put the steps for building a frequency table with relative and cumulative frequencies of an ordered variable in the right order.

Exercise 8 · Interpretation, difficulty 2 of 3

Comparing branches fairly

Not started

Branch A processed 2 000 baskets last month, 300 of them paid in cash. Branch B processed 600 baskets, 150 of them in cash. A manager concludes that “Branch A has the bigger cash problem.” In two or three sentences, explain how the branches should be compared and what the comparison shows.

What you can now do

Build a frequency table with relative and cumulative frequencies
Choose the right denominator, including when values are missing
Read joint, row and column percentages in a two-way table
Compare groups of different sizes with relative frequencies