Calculations:

• Total sum of all 21 numbers $= 21 \times 9 = 189$.
• If the number 19 is removed, the sum of the remaining 20 numbers is:
  $$\text{New Sum} = 189 - 19 = 170$$
• The mean of the remaining 20 numbers is:
  $$\text{New Mean} = \frac{170}{20} = 8.5$$

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:
  $$\frac{1}{x_1} + \frac{1}{x_2} = \frac{(1-ab) + (1+ab)}{a} = \frac{2}{a}$$

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.

Step-by-step arithmetic mean correction:

• Original total sum $= n \times \bar{X} = 25 \times 56 = 1400\text{ cm}$.
• When the erroneous observation $64$ is omitted, the new sum becomes: $\text{New Sum} = 1400 - 64 = 1336\text{ cm}$.
• The new number of observations becomes $n' = 25 - 1 = 24$.
• The corrected mean is $\bar{X}' = \frac{1336}{24} = 55.666... \approx 55.67\text{ cm}$.