Joint and marginal frequencies

Why this exists

A frequency table counts one variable. Ask the same people a second question and a second table counts that, but two separate tables lose which answers came from the same person.

At a made-up camp, 12 campers each chose the lake or the forest trail for an afternoon walk. 5 chose the lake, and 6 are under 12. How many campers under 12 chose the lake? Any number from 0 to 5 fits those two counts. To answer, you must count the campers by both answers at once, which is what a two-way table does. It is where every question about whether two variables are connected starts.

Definition

The data are two categorical variables recorded for each member of a group of individuals (people, orders, animals). Each individual has exactly one category of each variable.

A two-way table has one row for each category of one variable and one column for each category of the other. The cell where a row meets a column counts the individuals with both that row’s category and that column’s category. A column on the right holds the row totals, and a row along the bottom holds the column totals. For the 12 campers (made-up data):

• A joint frequency is the number in one cell of the body: how many individuals have one category of the first variable and one of the second. The 4 counts campers under 12 who chose the lake.
• A marginal frequency is a row or column total: how many individuals have one category of one variable, whatever their other answer. The 5 counts all campers who chose the lake. These totals sit in the margins, hence the name.
• The grand total, in the bottom right corner, is the number of individuals.

A joint relative frequency is a joint frequency divided by the grand total, and a marginal relative frequency is a marginal frequency divided by the grand total. Here \tfrac{4}{12} = \tfrac{1}{3} of the campers are under 12 and chose the lake, and \tfrac{5}{12} chose the lake.

Which variable runs down the side is a free choice; swapping them changes none of the counts.

Intuition

Picture each camper as a dot, dropped into the one cell that matches both answers.

Counting across a row forgets the walk question and counts by age alone; counting down a column forgets age. So the right margin is the frequency table for age, the bottom margin the one for the walk, and the body the joint table. The margins can be found from the body by adding, but not the other way round, which is what the opening question ran into.

Results

Worked examples

Boundaries

• Each individual is in exactly one category of each variable. Categories must not overlap and must cover everyone: age groups “10 to 13” and “13 to 16” both claim a 13-year-old.
• The table counts individuals, not answers. If one person may tick several boxes, the cells add to more than the number of people and the totals above fail.
• Relative frequencies here are over the grand total. Over a row or column total they are conditional relative frequencies, a different question.
• The margins do not determine the body. Filling in a table needs enough cells to start from.
• A table describes the group counted. Claims about a larger group need a well-chosen sample: populations and samples.

Where this goes next

Comparing conditional relative frequencies row by row decides whether two variables are connected, which is association in a two-way table. For one individual picked at random, a joint relative frequency is the probability of a compound event. The chi-square statistic compares each joint frequency with what the margins alone would lead you to expect.

Exercises

Reading a two-way table

Exercises 1 to 3 use this made-up survey of 120 students.

1. How many students (a) are in Grade 8 and do a club, (b) do no activity, (c) are in Grade 7? Which answers are joint frequencies and which marginal?
2. How many students are in Grade 8 or do a sport, or both?
3. How many students (a) do a sport or a club, (b) are in Grade 7 and do a sport or a club?

Building a table from a list

4. Fifteen made-up campers gave their age and their walk, L for lake and F for forest trail: 8 L, 13 F, 11 L, 15 F, 9 L, 10 F, 14 L, 12 F, 7 F, 16 F, 13 L, 9 L, 11 F, 15 L, 10 L. Build a two-way table with age groups under 10, 10 to 13, and 14 and over.
5. A made-up café’s 12 orders, coded by drink (T tea, C coffee) and size (S small, M medium, L large), were: TS, CM, CL, TM, CM, CS, TL, CM, TS, CL, CM, TM. Build the two-way table.

Filling in and correcting

6. At a made-up camp of 90 campers, 40 are under 12, 51 chose the lake, and 23 are under 12 and chose the lake. Complete the table of age (under 12, 12 and over) against walk.
7. In a made-up survey of 110 students, 64 are in Grade 8, 45 do a sport and 25 do none. In Grade 7, 18 do a sport and 9 do none; in Grade 8, 21 do a club. Complete the table.
8. A made-up café served 60 drinks: tea, coffee and hot chocolate, small or large. There were 16 teas, 7 of them small, and 5 small and 9 large hot chocolates. It sold four times as many large coffees as small ones. Complete the table.
9. This made-up camp table has the right totals, but one joint frequency was copied wrongly. Find and correct it.

Joint and marginal relative frequencies

10. From the table for exercises 1 to 3, give to three places: (a) the joint relative frequency of Grade 8 and none, (b) the marginal relative frequency of sport, (c) that of Grade 7.
11. Turn your table from exercise 5 into joint and marginal relative frequencies, as fractions in lowest terms.
12. A made-up café sells only tea and coffee, small or large. Its joint relative frequencies are tea and small 0.18, tea and large 0.27, coffee and small 0.35. (a) Find coffee and large. (b) Find the marginal relative frequencies of small and of tea. (c) Of 400 drinks, how many were large coffees?
13. A cell holds 27 students, and its joint relative frequency is 0.15. How many students are in the whole table?

Explaining and reasoning

14. Answers from 30 people are tallied into a two-way table. The row totals add to 30 but the column totals add to 29. What is certain, and how can the faulty entry be found?
15. A table with two rows and two columns has every row and column total equal to 10. Give two different such tables, and say what this shows about marginal frequencies.
16. Which of these is true, and why? (a) The joint relative frequencies in a table add to 1. (b) All the marginal relative frequencies, right side and bottom together, add to 1.
17. A survey asks 40 students “Tick all that apply: sport, club.” The answers are sorted into a table of grade against sport, club and none, and with every tick counted correctly the cells add to 47. How can that happen, and how should the table be set up?
18. All the totals of a table with two rows and three columns are known. Explain why one joint frequency does not, in general, fix the table, and say which pairs of joint frequencies would.