Statistical Paradoxes and Data Interpretation Notes

Objective: To investigate common paradoxes within the field of statistics.
Key Concept: Understanding how statistical results can be counterintuitive, such as why a "90% accurate" test can result in a majority of people who fail being innocent or truthful.

The Phenomenon: It is possible for a specific data set to appear more favorable in individual group comparisons but yield a worse result when the data are aggregated into an overall total.
Cause: This paradox occurs because of how overall results are divided into unequally sized groups. The weighting of the groups impacts the aggregate mean or percentage differently than the individual parts.

Scenario: Comparing the shooting performance of two players, Sheryl and Candace, across two halves of a game.
First Half Data:
- Sheryl: 4 baskets on 10 attempts = $40\%$
- Candace: 1 basket on 4 attempts = $25\%$
- Observation: Sheryl performed better ( $40\%$ vs. $25\%$ ).
Second Half Data:
- Sheryl: 3 baskets on 4 attempts = $75\%$
- Candace: 7 baskets on 10 attempts = $70\%$
- Observation: Sheryl performed better ( $75\%$ vs. $70\%$ ).
Overall Game Statistics:
- Sheryl's total: $4 + 3 = 7$ baskets on $10 + 4 = 14$ shots.
- Sheryl's overall percentage: $\frac{7}{14} = 50\%$
- Candace's total: $1 + 7 = 8$ baskets on $4 + 10 = 14$ shots.
- Candace's overall percentage: $\frac{8}{14} \approx 57.1\%$
Conclusion: Even though Sheryl had a higher shooting percentage in both individual halves, Candace had the better overall game percentage. This illustrates the fundamental nature of the paradox where group sizes (10 vs 4 and 4 vs 10 shots) shift the weight of the percentages.

Tumor: Medically defined as any kind of abnormal swelling or tissue growth.
Malignant: A tumor caused by cancer; also referred to as cancerous.
Benign: Any tumor that is not cancerous.
Prevalence: Approximately 1 in 100 breast tumors (or $1\%$ ) are found to be malignant.

Assumed Accuracy: Let's assume a mammogram is $85\%$ accurate. This means:
- It identifies $85\%$ of malignant tumors as malignant (True Positives).
- It identifies $85\%$ of benign tumors as benign (True Negatives).
Hypothetical Study (10,000 Women with Breast Tumors):
- Actual Cancer Cases ( $1\%$ ): $0.01 \times 10,000 = 100$ women.
- Actual Benign Cases: $10,000 - 100 = 9,900$ women.

Result Type	Tumor is Malignant	Tumor is Benign	Total
Positive Mammogram	85 (True Positives)	1,485 (False Positives)	1,570
Negative Mammogram	15 (False Negatives)	8,415 (True Negatives)	8,430
Total	100	9,900	10,000

Analysis of Positive Results:
- The screening yields a total of 1,570 positive results.
- Only 85 of these are actually from women with cancer.
- The probability that a positive result actually means cancer is: $\frac{85}{1,570} \approx 5.4\%$
- Paradoxical Finding: Despite the test being $85\%$ accurate, a positive result still only carries a $5.4\%$ chance of the patient having cancer.
Analysis of Negative Results (False Negatives):
- Total negative results = 8,430.
- Total women with cancer who received a negative result = 15.
- The probability that a woman with a negative mammogram actually has cancer is: $\frac{15}{8,430} \approx 0.00178$ (or slightly less than 2 in 1,000).

Better Technology: Newer methods like digital mammograms and ultrasounds are reaching accuracies near $98\%$ .
Definitive Testing: The biopsy is the most definitive test for cancer, although even a biopsy can miss cancer cells if not performed with sufficient care.
Recommendation: Patients with negative results who still suspect abnormalities should seek a second opinion.

Scenario: The government administers a polygraph to 1,000 applicants for sensitive security jobs.
Prevalence of Lying: Suppose 10 people lie and 990 tell the truth.
Test Accuracy: $90\%$
Results for Liars:
- Correctly identified: $90\% \times 10 = 9$ people fail.
- Incorrectly identified: 1 person passes (False Negative).
Results for Truth-Tellers:
- Correctly identified: $90\% \times 990 = 891$ people pass.
- Incorrectly identified: $10\% \times 990 = 99$ people fail (False Positives).
Overall Impact:
- Total failures: $9 (liars) + 99 (honest people) = 108$ .
- The fraction of failures who were actually truthful: $\frac{99}{108} \approx 91.7\%$
- Implication: Almost $92\%$ of applicants rejected for failing the polygraph were actually telling the truth.

Scenario: 1,000 athletes in a track championship undergo a drug test with $95\%$ accuracy.
Assumed Drug Use Rate: $4\%$ .
Population Breakdown:
- Drug users: $0.04 \times 1,000 = 40$ athletes.
- Non-users: 960 athletes.
Test Results:
- User Group: $95\%$ of 40 users fail = 38 athletes. (2 pass incorrectly).
- Non-User Group: $5\%$ of 960 non-users fail = $0.05 \times 960 = 48$ athletes (False Positives).
Total Failures: $38 + 48 = 86$ .
Conclusion: The fraction of athletes suspended despite being innocent is $\frac{48}{86} \approx 55.8\%$ . More than half of all students who fail the test are innocent, despite the test's high accuracy rating.

Question: While the chance of cancer with a negative mammogram is small (2 in 1,000), it is not zero. Should biopsies be routine for all tumors, or just for cases of positive mammograms?
Considerations: Biopsies are invasive, involve surgery, can be painful, and are expensive.

Question: If you were falsely accused of a crime and police suggested a polygraph to prove your innocence, would you take it?
Statistical Context: Based on the data showing a high rate of false positives ( $91.7\%$ of failures are truth-tellers in specific low-prevalence scenarios), taking the test might carry a significant risk of a "fail" result despite innocence.