Marginal Distributions
Ms. Patel gave her class an Algebra quiz and a Science quiz, both scored from 5 to 7. Her two-way table shows how many students got each pair of scores. Now she asks: how many students got a 6 in Algebra, no matter what they got in Science? To answer, you ignore one variable and look at the other one alone. That is a marginal distribution.
What a marginal distribution is
A two-way table mixes two kinds of information: how $X$ and $Y$ are spread out separately, and how they are related. If you ignore one variable, what is left is an ordinary one-variable distribution.
Given the following two-way table for the variables $X$ and $Y$: | $X \setminus Y$ | 3 | 4 | |---|---|---| | 1 | 2 | 5 | | 2 | 4 | 1 | Write down the table of the…
Given the following two-way table for the variables $X$ and $Y$:
| $X \setminus Y$ | 3 | 4 |
|---|---|---|
| 1 | 2 | 5 |
| 2 | 4 | 1 |
Write down the table of the marginal distribution of the variable $X$.
The relationship between two statistical variables $X$ and $Y$ has been studied, obtaining the following table of joint frequencies: | $X \setminus Y$ | 10 | 20 |…
The relationship between two statistical variables $X$ and $Y$ has been studied, obtaining the following table of joint frequencies:
| $X \setminus Y$ | 10 | 20 |
|---|---|---|
| 0 | 3 | 2 |
| 1 | 1 | 6 |
| 2 | 4 | 0 |
Determine the marginal distribution of the variable $Y$ and calculate the total population size $N$.
Marginal totals and notation
These totals are called marginal because we usually write them in the margins of the table: a "Total" column on the right and a "Total" row at the bottom. The corner cell is $N$.
| Symbol | Meaning |
|---|---|
| $n_{ij}$ | joint frequency: individuals with $X = x_i$ and $Y = y_j$ |
| $n_{i\bullet}$ | total of row $i$: individuals with $X = x_i$ |
| $n_{\bullet j}$ | total of column $j$: individuals with $Y = y_j$ |
| $N$ | total of everything |
In the following two-way table, the marginal totals (the sums of the rows and columns) are missing. | $X \setminus Y$ | 1 | 2 | 3 | Total $X$ | |---|---|---|---|---| | 5…
In the following two-way table, the marginal totals (the sums of the rows and columns) are missing.
| $X \setminus Y$ | 1 | 2 | 3 | Total $X$ |
|---|---|---|---|---|
| 5 | 2 | 0 | 4 | ? |
| 6 | 1 | 3 | 1 | ? |
| Total $Y$ | ? | ? | ? | $N = ?$ |
Copy the table and complete the missing marginal values.
Observe the following table of joint frequencies: | $X \setminus Y$ | 100 | 200 | 300 | |---|---|---|---| | 1 | 5 | 8 | 2 | | 2 | 3 | 1 | 7 | Identify the value of the…
Observe the following table of joint frequencies:
| $X \setminus Y$ | 100 | 200 | 300 |
|---|---|---|---|
| 1 | 5 | 8 | 2 |
| 2 | 3 | 1 | 7 |
Identify the value of the marginal frequency $n_{2 \bullet}$ (the total number of individuals with $X=2$) and the value of the marginal frequency $n_{\bullet 3}$ (the total number of individuals with $Y=300$).
The mean temperature $X$ (in °F) and the monthly rainy days $Y$ are recorded in several cities: | $X \setminus Y$ | 2 | 4 | 6 | |---|---|---|---| | 15 | 3 | 5 | 2 | | 20…
The mean temperature $X$ (in °F) and the monthly rainy days $Y$ are recorded in several cities:
| $X \setminus Y$ | 2 | 4 | 6 |
|---|---|---|---|
| 15 | 3 | 5 | 2 |
| 20 | 4 | 7 | 3 |
| 25 | 1 | 4 | 6 |
Extract the marginal distributions of $X$ and $Y$. Check numerically that the sum of the marginal frequencies of $X$ coincides with the sum of the marginal frequencies of $Y$, and that both are equivalent to the total sample size $N$.
From a list of pairs to a marginal distribution
Sometimes the data arrive as a list of pairs $(x, y)$, one pair per individual. You have two options.
-
Build the two-way table first, tallying each pair in its cell, then add rows and columns.
-
Shortcut: to get the marginal of $Y$, ignore the $x$-values and simply count how many times each $y$ appears (and the same for $X$).
The following pairs of values $(x_i, y_i)$ have been collected by observing 5 individuals: $(1, 5), (2, 5), (1, 6), (3, 5), (2, 6)$ From this data, construct the table…
The following pairs of values $(x_i, y_i)$ have been collected by observing 5 individuals:
$(1, 5), (2, 5), (1, 6), (3, 5), (2, 6)$
From this data, construct the table of the marginal distribution of the variable $Y$.
In a group of 10 friends, they were asked about the number of siblings ($X$) and the number of pets ($Y$). The results $(X, Y)$ are:…
In a group of 10 friends, they were asked about the number of siblings ($X$) and the number of pets ($Y$). The results $(X, Y)$ are:
$(1, 1), (2, 0), (1, 1), (0, 2), (2, 1), (1, 0), (3, 0), (1, 1), (0, 1), (2, 0)$
Write down the table of the marginal distribution of the number of pets ($Y$).
The number of weekly study hours ($X$) and the score obtained on an exam ($Y$) have been recorded for 10 students:…
The number of weekly study hours ($X$) and the score obtained on an exam ($Y$) have been recorded for 10 students:
$(5, 6), (7, 8), (4, 5), (8, 9), (5, 5), (6, 7), (7, 7), (9, 10), (4, 6), (6, 8)$
Construct the two-way table of absolute frequencies. From it, obtain the marginal distribution of the scores ($Y$) and determine what percentage of students obtained a marginal score higher than 6.
Marginal mean
Once you have a marginal distribution, treat it like any one-variable data set. Multiply each value by how often it occurs, add, and divide by $N$.
The marginal distribution of a two-dimensional variable for its component $X$ is given by the following table: | $x_i$ | 2 | 4 | 6 | |---|---|---|---| | $n_i$ | 3 | 5 |…
The marginal distribution of a two-dimensional variable for its component $X$ is given by the following table:
| $x_i$ | 2 | 4 | 6 |
|---|---|---|---|
| $n_i$ | 3 | 5 | 2 |
Calculate the marginal mean $\bar{x}$.
A biologist studies a species of amphibian, recording its mass $X$ (in pounds) and its length $Y$ (in inches). The results are grouped in the following two-way table: |…
A biologist studies a species of amphibian, recording its mass $X$ (in pounds) and its length $Y$ (in inches). The results are grouped in the following two-way table:
| $X \setminus Y$ | 15 | 20 | 25 |
|---|---|---|---|
| 0.5 | 4 | 2 | 0 |
| 1.0 | 1 | 5 | 3 |
| 1.5 | 0 | 2 | 3 |
Calculate the marginal means $\bar{x}$ and $\bar{y}$ of both variables.
An ecologist measures the amount of sunlight $X$ and the growth of a plant $Y$. The data are summarized in the following table: | $X \setminus Y$ | 10 | 20 |…
An ecologist measures the amount of sunlight $X$ and the growth of a plant $Y$. The data are summarized in the following table:
| $X \setminus Y$ | 10 | 20 |
|---|---|---|
| 5 | 2 | 3 |
| 8 | 4 | 1 |
Show numerically that calculating the marginal mean of $X$ using the full two-dimensional formula $\bar{x} = \frac{1}{N} \sum_{i} \sum_{j} x_i \cdot n_{ij}$ gives exactly the same result as if the marginal distribution of $X$ is extracted first and the one-dimensional formula $\bar{x} = \frac{1}{N} \sum_{i} x_i \cdot n_{i \bullet}$ is applied. Show the step-by-step working for both methods.
Marginal variance and standard deviation
The mean tells you the center. The variance and standard deviation tell you how spread out the values are. Because the table lists the whole group, we divide by $N$ (the population formulas).
The marginal distribution of the variable $Y$ in a statistical study is as follows: | $y_j$ | 0 | 1 | 2 | |---|---|---|---| | $n_j$ | 4 | 4 | 2 | Knowing that its…
The marginal distribution of the variable $Y$ in a statistical study is as follows:
| $y_j$ | 0 | 1 | 2 |
|---|---|---|---|
| $n_j$ | 4 | 4 | 2 |
Knowing that its marginal mean is $\bar{y} = 0,8$, calculate the marginal variance $\sigma_Y^2$.
The battery life $X$ (in hours) and the screen size $Y$ (in inches) of several tablet models are analyzed: | $X \setminus Y$ | 7 | 8 | 10 | |---|---|---|---| | 6 | 2 | 5…
The battery life $X$ (in hours) and the screen size $Y$ (in inches) of several tablet models are analyzed:
| $X \setminus Y$ | 7 | 8 | 10 |
|---|---|---|---|
| 6 | 2 | 5 | 1 |
| 8 | 4 | 8 | 3 |
| 10 | 1 | 4 | 6 |
| 12 | 0 | 2 | 4 |
Construct the marginal distribution of the battery life ($X$) and calculate its marginal standard deviation.
Grouped data: class midpoints
When $X$ or $Y$ is given in intervals, use the midpoint of each interval as its value $x_i$. The results are estimates, just like for grouped data in one variable.
$$\text{midpoint of } [a, b) = \frac{a+b}{2}$$In a market study on the price of a product $X$ (grouped into intervals of dollars) and the number of daily sales $Y$, the following is obtained: | $X \setminus Y$ | 10…
In a market study on the price of a product $X$ (grouped into intervals of dollars) and the number of daily sales $Y$, the following is obtained:
| $X \setminus Y$ | 10 | 20 | 30 |
|---|---|---|---|
| $[10, 20)$ | 5 | 12 | 3 |
| $[20, 30)$ | 8 | 15 | 7 |
| $[30, 40)$ | 2 | 6 | 12 |
Obtain the marginal distribution of $X$ using the class midpoints and calculate its marginal mean.
Comparing the spread of two variables
If $X$ and $Y$ have the same units, you can compare $\sigma_X$ and $\sigma_Y$ directly. If they have different units or very different sizes, a bigger $\sigma$ does not mean "more spread out relative to its size." Use the coefficient of variation.
In a clinical trial, the dose of a medication $X$ (in mg) is related to the days of recovery $Y$: | $X \setminus Y$ | 3 | 5 | 7 | |---|---|---|---| | 100 | 6 | 4 | 0 | |…
In a clinical trial, the dose of a medication $X$ (in mg) is related to the days of recovery $Y$:
| $X \setminus Y$ | 3 | 5 | 7 |
|---|---|---|---|
| 100 | 6 | 4 | 0 |
| 200 | 2 | 8 | 2 |
| 300 | 0 | 3 | 5 |
Calculate the marginal coefficient of variation of the variable $Y$ to analyze its relative dispersion.
The amount of fertilizer $X$ (in lb/acre) and the crop yield $Y$ (in tons/acre) in different plots are studied: | $X \setminus Y$ | 2 | 3 | 4 | |---|---|---|---| | 50 |…
The amount of fertilizer $X$ (in lb/acre) and the crop yield $Y$ (in tons/acre) in different plots are studied:
| $X \setminus Y$ | 2 | 3 | 4 |
|---|---|---|---|
| 50 | 5 | 2 | 1 |
| 100 | 3 | 6 | 2 |
| 150 | 0 | 4 | 7 |
Calculate the marginal variance of $X$ and the marginal variance of $Y$. Which of the two variables shows greater absolute dispersion?
Relative frequencies and bubble charts
Sometimes the table gives joint relative frequencies $h_{ij} = \dfrac{n_{ij}}{N}$ instead of counts. Marginals work the same way: add rows or columns. The grand total is $1$.
A bubble chart shows a two-way table as a picture: each bubble sits at a pair $(x, y)$, and the number inside it is the joint frequency $n_{ij}$. Read the numbers off the bubbles, then add by column to get the marginal of $X$, or by row to get the marginal of $Y$.
From the chart, the marginal of $X$ is $1 \to 5$, $2 \to 8$, $3 \to 2$ (so $N = 15$), and the marginal of $Y$ is $5 \to 7$, $10 \to 8$.
A study relates the income level $X$ (in thousands of dollars) and the satisfaction index $Y$ (from 1 to 3). Instead of absolute frequencies, the joint relative…
A study relates the income level $X$ (in thousands of dollars) and the satisfaction index $Y$ (from 1 to 3). Instead of absolute frequencies, the joint relative frequencies $h_{ij}$ are provided:
| $X \setminus Y$ | 1 | 2 | 3 |
|---|---|---|---|
| 15 | 0.10 | 0.15 | 0.05 |
| 25 | 0.05 | 0.25 | 0.10 |
| 35 | 0.00 | 0.10 | 0.20 |
Calculate the relative marginal distributions and, from them, obtain the marginal means $\bar{x}$ and $\bar{y}$.
The following bubble chart represents the two-dimensional distribution of the Risk Level ($X$) and the Expected Return ($Y$) of a set of investment portfolios. The…
The following bubble chart represents the two-dimensional distribution of the Risk Level ($X$) and the Expected Return ($Y$) of a set of investment portfolios. The number inside each bubble indicates the joint frequency.
From the graph, extract the marginal distributions of both variables and calculate the marginal mean of the Return ($\bar{y}$).
Finding missing entries
Marginal totals are powerful clues. Each total gives an equation, and a given mean or $N$ gives another. You need as many equations as unknowns.
In a basketball team, the number of fouls committed ($X$) and the points scored ($Y$) per game are analyzed. The joint table has an unknown frequency $f$: |…
In a basketball team, the number of fouls committed ($X$) and the points scored ($Y$) per game are analyzed. The joint table has an unknown frequency $f$:
| $X \setminus Y$ | 5 | 10 | 15 |
|---|---|---|---|
| 1 | 2 | $f$ | 1 |
| 2 | 3 | 4 | 2 |
Knowing that the marginal mean of the points scored is $\bar{y} = 10$, find the value of the unknown frequency $f$.
In a factory, the number of defects $X$ and the inspection time $Y$ (in minutes) of a batch of parts are analyzed. The joint frequency table contains two parameters $a$…
In a factory, the number of defects $X$ and the inspection time $Y$ (in minutes) of a batch of parts are analyzed. The joint frequency table contains two parameters $a$ and $b$:
| $X \setminus Y$ | 2 | 4 | 6 |
|---|---|---|---|
| 0 | 10 | 5 | 2 |
| 1 | 4 | $a$ | 3 |
| 2 | 1 | 2 | $b$ |
It is known that the total sample size is $N = 40$ and that the marginal mean of the number of defects is $\bar{x} = 0,8$. Determine the values of the frequencies $a$ and $b$.
The tensile strength $X$ (in MPa) and the carbon percentage $Y$ in steel alloys are evaluated. The table is incomplete, but the marginal totals are known: |…
The tensile strength $X$ (in MPa) and the carbon percentage $Y$ in steel alloys are evaluated. The table is incomplete, but the marginal totals are known:
| $X \setminus Y$ | 0.2 | 0.4 | 0.6 | Total $X$ |
|---|---|---|---|---|
| 400 | 8 | ? | 2 | 15 |
| 500 | ? | 12 | ? | 25 |
| 600 | 1 | ? | 5 | 10 |
| Total $Y$ | 12 | 22 | 16 | $N=50$ |
Reconstruct the complete two-way table. Then, calculate the marginal variance of $Y$.
Changing the origin and the scale
Prices rise, ages are recorded a year late, units get converted. Instead of transforming every value, you can transform the mean and variance directly. Everything below also holds for a marginal, because a marginal is just a one-variable distribution.
The age $X$ (in intervals) and the number of children $Y$ of a group of families are analyzed: | $X \setminus Y$ | 0 | 1 | 2 | 3 | |---|---|---|---|---| | $[20, 30)$ |…
The age $X$ (in intervals) and the number of children $Y$ of a group of families are analyzed:
| $X \setminus Y$ | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| $[20, 30)$ | 15 | 10 | 2 | 0 |
| $[30, 40)$ | 8 | 18 | 12 | 4 |
| $[40, 50)$ | 3 | 7 | 15 | 6 |
Calculate the marginal mean and variance of the Age ($X$) using the class midpoints. If, due to a recording error, a correction factor must be applied by adding 2 years to each class midpoint, what will the new marginal mean and variance of $X$ be without needing to recalculate the whole table? Justify your answer based on the properties of the mean and variance.
A store records the base price $X$ (in dollars) and the number of units sold $Y$ of different items. The marginal distribution of the base prices $X$ obtained from the…
A store records the base price $X$ (in dollars) and the number of units sold $Y$ of different items. The marginal distribution of the base prices $X$ obtained from the two-dimensional table is:
| $x_i$ | 10 | 20 | 30 | 40 |
|---|---|---|---|---|
| $n_i$ | 25 | 40 | 20 | 15 |
Knowing that the final retail price ($X'$) is calculated by applying a fixed tax of $\$5$ plus a 20% increase on the base price (that is, $X' = 1,20X + 5$), calculate the marginal mean and standard deviation of the final prices $X'$ without having to transform each of the values individually.
Back to the start: Ms. Patel's question is answered by the row for $X = 6$ in the Algebra/Science table. Adding $1+5+3$ gives 9 students, whatever their Science score.