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.

Minimal property of standard deviation:

• Standard deviation is the minimum, never the maximum.
• The root mean square deviation about any arbitrary point $A$ is $S = \sqrt{\frac{1}{n} \sum (X_i - A)^2}$. By algebraic properties, $S^2 = \sigma^2 + (\bar{X} - A)^2$. Since $(\bar{X} - A)^2 \ge 0$, $S^2$ attains its unique absolute minimum when $A = \bar{X}$, meaning standard deviation is the least possible value of root mean square deviation.
• Standard deviation equals r.m.s.d. only when deviations are taken from the arithmetic mean, not universally.
• False proportional relationship.