Chapter 11 Language of Descriptive Statistics

Section 11.3 Statistical Measures

11.3.3 Measures of Dispersion

Means and quantiles are measures of position, i.e. they give information on the absolute position of the qualitative values $x_j$. If we add a constant $c$ to every value $x_j$, then the position measures also increase by $c$. In contrast, measures of dispersion are measures that give information on the dispersion or relative distribution of the data values independent of their absolute position. Consider a sample of size $n\ge 2$ of a quantitative property $X$. Let the original list be given by $x=(x_1,x_2,\dots ,x_n)\in ℝ{}^n$.

The sample variance is a measure of dispersion that describes the variability of the observation sample. The smaller the variance the "closer" the data values lie to each other. A variance $s_x^2=0$ is only possible if all data values are equal. Typically, it strongly increases with increasing $n$. The standard deviation is a more appropriate measure for the "broadness" of the distribution of data values. The two formulas given above have a few pitfalls:

• Before the variance can be calculated the mean $\bar{x}$ must already be known.
  
• The fact that in the definition of $s_x^2$ is divided by $n-1$ and not by $n$ is for deeper mathematical reasons that can only be discussed in a statistics lecture.
  
• The notation $s_x=+\sqrt{s_x^2}$ is a little misleading. You must not cancel the square by the square root, since the sum $s_x^2$ must be calculated (and this value is not defined as a single square) to determine $s_x$.
  
• Be careful using a scientific calculator with statistical functions: the sample variance is available via the $s^2$ key. The ${\sigma }^2$ key, however, provides the sum with denominator $n$ instead of $n-1$. This is not the sample standard deviation.