Agricultural Statistics

Graphical determination of median:

• Arithmetic mean cannot be directly determined by the intersection of cumulative frequency curves.
• Geometric mean is not located via ogives.
• Harmonic mean is not located via ogives.
• The 'less than' ogive rises from $0$ to $N$, while the 'more than' ogive falls from $N$ to $0$. They intersect precisely at cumulative frequency $N/2$. Projecting this intersection point onto the horizontal X-axis gives the median.

Let the $n$ numbers be $x_1, x_2, \dots, x_n$.
The arithmetic mean is given by:

The total sum can be written as:

Given that the sum of the last $n-1$ numbers is $k$ (i.e. $\sum_{i=2}^n x_i = k$):

A fundamental algebraic property of the Arithmetic Mean states that the sum of squares of deviations (and hence the mean square deviation, $\frac{1}{N}\sum (x_i - A)^2$) is minimum when taken about the arithmetic mean (i.e. when $A = \bar{x}$).
Proof: Let $S = \sum (x_i - A)^2$. Differentiating with respect to $A$ and setting to zero yields $-2\sum (x_i - A) = 0 \implies A = \frac{\sum x_i}{N} = \bar{x}$.
Note: The sum of absolute deviations $\sum |x_i - A|$ is minimum when taken about the Median.