Question 1166274: Test Scores. The table below shows eighth-grade mathematics test scores in Nebraska and New Jersey. The scores are sepa rated according to the race of the students. Also shown are the state average for all races.
White
281
283
Nonwhite
250
252
Average for all races
277
272
Nebraska
New Jersey
Source Based on National Assessment of Educational Progress scores for 1992, from Chance, Spring 1999.
a. Which state had the higher scores in both racial catego ries? Which state had the higher overall average across both
racial categories?
b. Explain how a state could score lower in both categories and still have a higher overall average. c. The following table gives the actual percentage of whites and non whites in each state. Use these percentages to verify that the overall average test score in Nebraska is 277, as
claimed in the first table.
Nebraska New Jersey
White
87% 66%
Nonwhite
13% 34%
d. Use the racial percentages to verify that the overall average test score in New Jersey is 272, as claimed in the first table. e Briefly explain why the results might seem surprising or paradoxical
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website! This problem is an example of **Simpson's Paradox**, where a trend appears in different groups of data but disappears or reverses when the groups are combined.
Here is the breakdown of the data and calculations:
## 📊 Test Score Analysis
### a. Comparison of Scores
* **Higher scores in both racial categories:**
* **White:** New Jersey (283) vs. Nebraska (281). **New Jersey is higher.**
* **Nonwhite:** New Jersey (252) vs. Nebraska (250). **New Jersey is higher.**
* **New Jersey** had the higher scores in both the White and Nonwhite categories.
* **Higher overall average:**
* Nebraska (277) vs. New Jersey (272). **Nebraska is higher.**
---
### b. Explaining the Paradox
Nebraska scored higher overall (277) despite scoring lower in both the White (281) and Nonwhite (250) categories compared to New Jersey (283 and 252, respectively).
This happens because the **overall average is a weighted average**, and the two states have very different **racial compositions**. Nebraska has a much higher percentage of students in the **higher-scoring White category** (87%), while New Jersey has a much higher percentage of students in the **lower-scoring Nonwhite category** (34%). Nebraska's high concentration of high-scoring students pulls its state average up, even though the performance within each group is slightly lower.
---
### c. Verification of Nebraska's Overall Average
To verify Nebraska's overall average, we calculate the weighted average using the scores and racial percentages:
* **White Score:** 281
* **Nonwhite Score:** 250
* **White Percentage:** 87% (or 0.87)
* **Nonwhite Percentage:** 13% (or 0.13)
$$\text{Nebraska Average} = (\text{White Score} \times \text{White \%}) + (\text{Nonwhite Score} \times \text{Nonwhite \%})$$
$$\text{Nebraska Average} = (281 \times 0.87) + (250 \times 0.13)$$
$$\text{Nebraska Average} = 244.47 + 32.5$$
$$\text{Nebraska Average} = \mathbf{276.97}$$
Since the table value is 277, the calculation **verifies** the claimed overall average for Nebraska (276.97 is rounded to 277).
---
### d. Verification of New Jersey's Overall Average
We perform the same weighted average calculation for New Jersey:
* **White Score:** 283
* **Nonwhite Score:** 252
* **White Percentage:** 66% (or 0.66)
* **Nonwhite Percentage:** 34% (or 0.34)
$$\text{New Jersey Average} = (\text{White Score} \times \text{White \%}) + (\text{Nonwhite Score} \times \text{Nonwhite \%})$$
$$\text{New Jersey Average} = (283 \times 0.66) + (252 \times 0.34)$$
$$\text{New Jersey Average} = 186.78 + 85.68$$
$$\text{New Jersey Average} = \mathbf{272.46}$$
Since the table value is 272, the calculation **verifies** the claimed overall average for New Jersey (272.46 is rounded to 272).
---
### e. Explaining the Surprising Results
The results are surprising because they illustrate **Simpson's Paradox**.
* If you look at the data by **race**, New Jersey is clearly better: its students outperform Nebraska's students in both racial groups.
* However, if you look at the data in **total**, Nebraska appears better.
This paradox occurs because the **group sizes are not equal** and the distribution of the population (the weighting factor) is heavily skewed. Nebraska benefits from having 87% of its students in the group that scores higher across both states (White), while New Jersey has a higher proportion (34%) of its students in the group that scores lower across both states (Nonwhite). The difference in demographic composition outweighs the slight difference in performance within each racial group.
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