Agricultural Statistics

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.

Mathematical derivation of harmonic mean:
Let $x_1 = \frac{a}{1-ab}$ and $x_2 = \frac{a}{1+ab}$.

• Incorrect: The calculation yields $a$, not $a/b$.
• Correct: The harmonic mean $H$ of two quantities $x_1$ and $x_2$ is given by:
  $$H = \frac{2}{\frac{1}{x_1} + \frac{1}{x_2}}$$
  Reciprocals are:
  $$\frac{1}{x_1} = \frac{1-ab}{a}, \quad \frac{1}{x_2} = \frac{1+ab}{a}$$
  Adding the reciprocals:
  $$\frac{1}{x_1} + \frac{1}{x_2} = \frac{(1-ab) + (1+ab)}{a} = \frac{2}{a}$$

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.

Fundamental algebraic property of arithmetic mean:

• By definition, $\sum_{i=1}^{n} (X_i - \bar{X}) = \sum X_i - n\bar{X} = n\bar{X} - n\bar{X} = 0$. The algebraic sum of deviations about the arithmetic mean is always identically zero.
• It can never be negative.
• It can never be positive.
• It is a finite real quantity ($0$), never infinity.