Properties of the Arithmetic Mean

The arithmetic mean has several algebraic properties that are frequently tested and that also explain why it behaves the way it does.

1. The sum of deviations of all observations from their arithmetic mean is always zero: $\Sigma(x - \bar{x}) = 0$. This is not a coincidence of any particular dataset — it follows directly from the definition of the mean, and it is the exact algebraic reason the assumed-mean method (§2) works at all. Worked Example 1 verifies this property numerically on its own data.
2. Adding (or subtracting) a constant to every observation shifts the mean by the same constant. If $y = x + k$ for every observation, then $\bar{y} = \bar{x} + k$.
3. Multiplying (or dividing) every observation by a constant scales the mean by the same constant. If $y = kx$, then $\bar{y} = k\bar{x}$. Properties 2 and 3 together are exactly what justify the step-deviation method — shifting by $A$ and scaling by $h$ change the mean of the deviations in a fully predictable, reversible way, which is why the final answer can always be recovered by undoing the shift and the scale.
4. The arithmetic mean is affected by every single value in the data, including extreme (very large or very small) values — unlike the median or the mode. This is exactly the property that makes the mean the most representative average when data is fairly symmetric, but also its biggest weakness when a few extreme values (outliers) are present (see §11 and the MCQ in Question 14). …