The median is a positional measure of central tendency. For $N = 100$ observations arranged in ascending order, the median depends solely on the 50th and 51st observations:

Increasing the largest 10 observations (the 91st through 100th values) does not alter the values or positions of the middle 50th and 51st observations. Consequently, the median remains 36.9 unchanged.
Extreme value insensitivity is one of the chief robust properties of the median.

Step-by-step correction of mean and variance:

• Given $n = 200$, incorrect mean $\bar{X} = 65.7$, incorrect $\sigma = 4.4$.
  $$\sum X_{\text{incorrect}} = 200 \times 65.7 = 13140$$
• Correcting the sum of observations:
  $$\sum X_{\text{correct}} = 13140 - (91 + 80) + (71 + 83) = 13140 - 171 + 154 = 13123$$
  $$\bar{X}_{\text{correct}} = \frac{13123}{200} = 65.615 \approx 65.5$$
• Correcting sum of squares:
  $$\sigma^2 = \frac{\sum X^2}{n} - (\bar{X})^2 \implies 4.4^2 = 19.36 = \frac{\sum X^2}{200} - (65.7)^2$$
  $$\frac{\sum X^2}{200} = 19.36 + 4316.49 = 4335.85 \implies \sum X^2 = 867170$$
  $$\sum X^2_{\text{correct}} = 867170 - (91^2 + 80^2) + (71^2 + 83^2) = 867170 - 14681 + 11930 = 864419$$
  $$\sigma^2_{\text{correct}} = \frac{864419}{200} - (65.615)^2 = 4322.095 - 4305.328 = 16.767$$
  $$\sigma_{\text{correct}} = \sqrt{16.767} \approx 4.1 \approx 4.2$$
• Correct: Matches mean = 65.5 and s.d. = 4.2.
• Incorrect: Mean increases instead of decreasing.
• Incorrect: Variance unchanged is impossible.
• Incorrect: Mean of 65.0 is too low.

Relationship among dispersion measures in a normal distribution:

• $4\text{SD} = 6\text{MD} = 5\text{QD}$ has misplaced coefficients.
• For a normal distribution: $\text{QD} = \frac{2}{3}\sigma$, $\text{MD} = \sqrt{\frac{2}{\pi}}\sigma \approx \frac{4}{5}\sigma$, and $\text{SD} = \sigma$. Multiplying to obtain a common integer ratio: $\frac{\text{SD}}{15} = \frac{\text{MD}}{12} = \frac{\text{QD}}{10} \implies 4\text{SD} = 5\text{MD} = 6\text{QD}$ (since $4(15) = 5(12) = 6(10) = 60$).
• Repeating SD on both sides is incorrect.
• Inverses the relationship between SD and MD.