Marginal distributions
In a two-way table you have two kinds of information mixed together: how $X$ and $Y$ are distributed separately, and how they are related to each other. If for a moment you ignore one of the two variables, what is left is an ordinary distribution of a single variable: the marginal distribution.
What a marginal distribution is
Given the following two-way table for variables $X$ and $Y$: | $X \setminus Y$ | 3 | 4 | |---|---|---| | 1 | 2 | 5 | | 2 | 4 | 1 | Write down the table of the marginal…
Given the following two-way table for 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 $X$.
In the following two-way table, the marginal totals (the row and column sums) are missing. | $X \setminus Y$ | 1 | 2 | 3 | Total $X$ | |---|---|---|---|---| | 5 | 2 | 0…
In the following two-way table, the marginal totals (the row and column sums) 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 fill in the missing marginal values.
Marginal mean and variance
Once you have the marginal distribution, it is treated like any other: with its own $\bar{x}$, $\sigma_x^2$ and $\sigma_x$, using the same formulas as for one variable.
The marginal distribution of the variable $Y$ in a statistical study is as follows: | $y_j$ | 0 | 1 | 2 | |---|---|---|---| | $n_j$ | 4 | 4 | 2 | Given that its marginal…
The marginal distribution of the variable $Y$ in a statistical study is as follows:
| $y_j$ | 0 | 1 | 2 |
|---|---|---|---|
| $n_j$ | 4 | 4 | 2 |
Given that its marginal mean is $\bar{y} = 0,8$, calculate the marginal variance $\sigma_Y^2$.
The battery life $X$ (in hours) and screen size $Y$ (in inches) of several tablet models are analysed: | $X \setminus Y$ | 7 | 8 | 10 | |---|---|---|---| | 6 | 2 | 5 | 1…
The battery life $X$ (in hours) and screen size $Y$ (in inches) of several tablet models are analysed:
| $X \setminus Y$ | 7 | 8 | 10 |
|---|---|---|---|
| 6 | 2 | 5 | 1 |
| 8 | 4 | 8 | 3 |
| 10 | 1 | 4 | 6 |
| 12 | 0 | 2 | 4 |
Construct the marginal distribution of battery life ($X$) and calculate its marginal standard deviation.
Properties under changes of origin and scale
The properties of the variance that you already know for one variable work in exactly the same way for a marginal: it depends only on that variable, not on the other.
A shop records the base price $X$ (in euros) and the number of units sold $Y$ of different items. The marginal distribution of base prices $X$ obtained from the…
A shop records the base price $X$ (in euros) and the number of units sold $Y$ of different items. The marginal distribution of base prices $X$ obtained from the two-dimensional table is:
| $x_i$ | 10 | 20 | 30 | 40 |
|---|---|---|---|---|
| $n_i$ | 25 | 40 | 20 | 15 |
Given that the final retail price ($X'$) is calculated by applying a fixed tax of 5 euros 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 value individually.
Comparing the spread of the two variables
When $X$ and $Y$ have different units (or very different means), to find out which one is more spread out relative to its size you use the marginal coefficient of variation of each one, just as with a single variable.
In a clinical trial, the dosage of a medication $X$ (in mg) is related to recovery days $Y$: | $X \setminus Y$ | 3 | 5 | 7 | |---|---|---|---| | 100 | 6 | 4 | 0 | | 200…
In a clinical trial, the dosage of a medication $X$ (in mg) is related to recovery days $Y$:
| $X \setminus Y$ | 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 analyse its relative dispersion.
The amount of fertiliser $X$ (in kg/ha) and crop yield $Y$ (in tonnes/ha) are studied in different plots: | $X \setminus Y$ | 2 | 3 | 4 | |---|---|---|---| | 50 | 5 | 2…
The amount of fertiliser $X$ (in kg/ha) and crop yield $Y$ (in tonnes/ha) are studied in different plots:
| $X \setminus Y$ | 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?