Lesson 11-5: Associations in Two-Way Tables

Property

Raw counts are hard to compare, so we often convert them into percentages called relative frequencies.

To find a Joint or Marginal Relative Frequency, you ALWAYS divide by the Grand Total (the total number of all people surveyed, located in the bottom-right corner).

• Joint Relative Frequency = $\frac{\text{Joint Frequency}}{\text{Grand Total}}$
• Marginal Relative Frequency = $\frac{\text{Marginal Frequency}}{\text{Grand Total}}$

Property

Relative frequencies can reveal associations between categorical variables. When the relative frequencies for one variable differ significantly across categories of another, an association exists.

• Direction: Identifies which category is more likely based on the larger relative frequency.
• Strength: Determined by the magnitude of the difference ($\Delta$) between the relative frequencies. A larger difference indicates a stronger association.

Examples

• Strong Association: The relative frequency of adults who prefer coffee is 0.78, while for teens it is 0.21. The large difference ($\Delta = |0.78 - 0.21| = 0.57$) indicates a strong association, showing adults are much more likely to prefer coffee.
• Weak Association: The relative frequency of left-handed students who play sports is 0.62, and for right-handed students is 0.59. The small difference ($\Delta = |0.62 - 0.59| = 0.03$) indicates a weak association, meaning handedness has little meaningful effect.
• No Association: If 45% of 7th graders and 47% of 8th graders prefer math, there is likely no association because the percentages are very similar.

Explanation

To identify an association, compare the relative frequencies for one variable across the different categories of the second variable. The direction tells you which specific categories are connected, while the strength depends on the gap between the values. A large gap means the variables are strongly linked, whereas a very small gap suggests the relationship is weak and might just be due to chance.

Property

When analyzing two-way tables, avoid these two common errors to ensure accurate conclusions:

1. Comparing raw counts: Always compare relative frequencies when row or column totals differ.
2. Dividing by the wrong total: Match the denominator to the specific group being analyzed (the condition).

Examples

• Error 1 (Raw Counts): 30 out of 50 adults prefer coffee, while 40 out of 100 teens prefer coffee. Comparing the raw counts ($30 < 40$) incorrectly suggests teens prefer coffee more. Comparing relative frequencies (60% for adults vs. 40% for teens) correctly shows adults have a higher preference.
• Error 2 (Wrong Total): A table shows 15 left-handed boys out of 100 total boys, and 200 total students. To find the frequency of being left-handed given that a student is a boy, you must divide by the row total for boys (100), not the grand total (200). The correct calculation is 15 / 100 = 0.15.

Explanation

Comparing raw counts can be highly misleading if the total number of observations in each group is different. To accurately detect associations, you must compare relative frequencies to account for these differing group sizes. Additionally, pay close attention to the wording of the question to ensure you divide by the correct row or column total, as misidentifying the condition will result in false conclusions.