1. **Stating the problem:** We want to construct a frequency marginal distribution from a joint frequency distribution table. 2. **Understanding the concept:** A frequency marginal distribution summarizes the total frequencies for each category of one variable, ignoring the other variable. 3. **Formula:** If $f_{ij}$ is the joint frequency for category $i$ of variable $X$ and category $j$ of variable $Y$, then the marginal frequency for category $i$ of $X$ is: $$f_{i\cdot} = \sum_j f_{ij}$$ Similarly, for category $j$ of $Y$: $$f_{\cdot j} = \sum_i f_{ij}$$ 4. **Steps to construct:** - Sum the frequencies across each row to get the marginal distribution for $X$. - Sum the frequencies down each column to get the marginal distribution for $Y$. 5. **Example:** Suppose a joint frequency table: | | Y1 | Y2 | Y3 | |---|----|----|----| | X1| 3 | 5 | 2 | | X2| 4 | 1 | 6 | - Marginal for $X$: $$f_{X1} = 3 + 5 + 2 = 10$$ $$f_{X2} = 4 + 1 + 6 = 11$$ - Marginal for $Y$: $$f_{Y1} = 3 + 4 = 7$$ $$f_{Y2} = 5 + 1 = 6$$ $$f_{Y3} = 2 + 6 = 8$$ 6. **Interpretation:** These marginal frequencies tell us the total counts for each category of one variable regardless of the other variable. This method helps in understanding the distribution of each variable independently from the joint distribution.