Eighth Grade Mathematics: Marginal and Joint Relative Frequencies
Raw frequency counts tell you how many individuals fall into specific categories, but when comparing groups of different sizes, raw numbers can be misleading. If 40 students at a school of 100 students play sports, that represents 40 percent of the school; if 40 students at a school of 1,000 students play sports, that represents only 4 percent. To make fair, meaningful comparisons, statisticians convert raw frequency counts into relative frequencies by calculating proportions or percentages relative to the grand total. In this chapter, you will master calculating joint relative frequencies and marginal relative frequencies, and understand their probabilistic meanings.
Defining Joint and Marginal Relative Frequencies
Relative frequencies express counts as fractions, decimals, or percentages of the entire survey sample:
The Fundamental Relative Frequency Formulas (Relative to Grand Total N):
1. Joint Relative Frequency:
The ratio of a specific joint frequency (an inner cell) to the GRAND TOTAL N:
Joint Relative Frequency = (Cell Count) / (Grand Total N)
2. Marginal Relative Frequency:
The ratio of a marginal total (row total or column total) to the GRAND TOTAL N:
Marginal Relative Frequency = (Row Total or Column Total) / (Grand Total N)
The word marginal comes from the fact that these totals reside in the margins (outer edges) of the table, while joint frequencies reside at the internal joints where rows and columns intersect.
Converting a Raw Frequency Table to a Relative Frequency Table
To convert a raw table into a relative frequency table, divide every single cell—including all joint cells, row totals, and column totals—by the grand total N:
Raw Table (Grand Total N = 200):
Prefers Math Prefers Science Total
7th Graders 60 40 100
8th Graders 50 50 100
Total 110 90 200
Relative Frequency Table (Dividing every value by 200):
Prefers Math Prefers Science Marginal Row Total
7th Graders 60/200 = 0.30 (30%) 40/200 = 0.20 (20%) 100/200 = 0.50 (50%)
8th Graders 50/200 = 0.25 (25%) 50/200 = 0.25 (25%) 100/200 = 0.50 (50%)
Marginal Col Tot 110/200 = 0.55 (55%) 90/200 = 0.45 (45%) 200/200 = 1.00 (100%)
Notice that the sum of all joint relative frequencies equals 1.00 (100%): 0.30 + 0.20 + 0.25 + 0.25 = 1.00. The sum of the marginal row totals equals 1.00 (0.50 + 0.50 = 1.00), and the sum of the marginal column totals equals 1.00 (0.55 + 0.45 = 1.00).
Interpreting Probabilities from Relative Frequencies
Joint and marginal relative frequencies correspond directly to basic probabilities when selecting a subject at random from the entire surveyed population:
Probabilistic Meaning:
- Joint Relative Frequency of (7th Grade & Math) = 0.30:
If a student is chosen at random from the ENTIRE school, there is a 30% probability
that the student is a 7th grader who prefers math.
- Marginal Relative Frequency of Math = 0.55:
If a student is chosen at random from the ENTIRE school, there is a 55% probability
that the student prefers math, regardless of grade level.
Joint relative frequencies always answer questions regarding the conjunction and (Student is in Category A AND Category B), while marginal relative frequencies answer questions about a single general trait across the entire group.
Chapter Exercises
Exercise 1: A survey of 250 pet owners asked whether they own a dog and whether they live in an apartment. The raw data shows that 35 respondents own a dog and live in an apartment; 115 own a dog and live in a house; 65 do not own a dog and live in an apartment; and 35 do not own a dog and live in a house. Calculate the joint relative frequency of dog owners living in apartments, the marginal relative frequency of all apartment dwellers, and the marginal relative frequency of all dog owners. Express each as both a decimal and a percentage.
Exercise 2: In a two-way relative frequency table of 500 voters, the joint relative frequency for Candidate A supporters under age 30 is 0.18. Calculate the exact number of individuals who fall into this joint category.
Exercise 3: Explain in paragraph form the mathematical distinction between a joint relative frequency and a marginal relative frequency, describing where each appears in a table and what each reveals about the surveyed sample.
Solutions and Explanations
Solution 1: The grand total of surveyed pet owners is N = 250. To find the joint relative frequency of dog owners who live in an apartment, divide their specific joint cell count (35) by the grand total: 35 / 250 = 7/50 = 0.14, or 14 percent. To find the marginal relative frequency of all apartment dwellers, first compute the total number of apartment dwellers by summing the column: 35 (dog owners) + 65 (non-dog owners) = 100 apartment dwellers. Divide this marginal total by the grand total: 100 / 250 = 2/5 = 0.40, or 40 percent. To find the marginal relative frequency of all dog owners, sum the dog owner row: 35 + 115 = 150 dog owners. Divide this marginal total by the grand total: 150 / 250 = 3/5 = 0.60, or 60 percent. In summary: the joint relative frequency is 0.14 (14%), the marginal frequency of apartment dwellers is 0.40 (40%), and the marginal frequency of dog owners is 0.60 (60%).
Solution 2: To find the exact number of individuals represented by a joint relative frequency, multiply the relative frequency decimal by the grand total sample size N: Count = (Joint Relative Frequency) * N. Given that the joint relative frequency is 0.18 and the grand total is N = 500 voters, multiply: 0.18 * 500 = 90. Therefore, exactly 90 individuals in the survey are under age 30 and support Candidate A.
Solution 3: In a two-way table, the mathematical distinction between a joint relative frequency and a marginal relative frequency lies in whether you are measuring the intersection of two traits or a single overarching characteristic relative to the grand total. Joint relative frequencies reside in the internal cells of the table where a specific row and column intersect. They represent the proportion of the entire sample that possesses both characteristics simultaneously, describing the joint occurrence of two traits (such as being an 8th grader and participating in sports). In contrast, marginal relative frequencies reside on the outer margins of the table in the total row and total column. They represent the proportion of the entire sample that possesses a single general category, completely ignoring the other variable (such as the proportion of all students who participate in sports, regardless of grade level). While both are calculated by dividing by the grand total N, joint frequencies isolate specific intersections, whereas marginal frequencies summarize broad totals.