#### Chapter 11 Language of Descriptive Statistics

Section 11.3 Statistical Measures

# 11.3.1 Introduction

Suppose a sample of size $n$ is given for some quantitative property $X$. Let the original list be given by

$x\mathrm{ }=\mathrm{ }\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right) .$

##### Info 11.3.1

The arithmetic mean $\stackrel{‾}{x}$ (also called sample mean) of ${x}_{1},{x}_{2},\dots ,{x}_{n}$ is defined as

$\stackrel{‾}{x}\mathrm{ }=\mathrm{ }\frac{1}{n}·\sum _{k=1}^{n}{x}_{k}\mathrm{ }=\mathrm{ }\frac{{x}_{1}+{x}_{2}+\dots +{x}_{n}}{n} .$

In physical terms, $\stackrel{‾}{x}$ describes the centre of mass of a mass distribution given by equal masses at ${x}_{1},{x}_{2},\dots ,{x}_{n}$ on the massless number line.
##### Example 11.3.2
We have the following original list for a sample of size $n=20$:
 10 11 9 7 9 11 22 12 13 9 11 9 10 12 13 12 11 10 10 12

The investigated property could be, for example, the length of study (measured in semesters) of $20$ mathematics students at the TU Berlin. Summing up the values results in

$\sum _{k=1}^{20}{x}_{k}\mathrm{ }=\mathrm{ }223 ,$

so for the arithmetic mean we have

$\stackrel{‾}{x}\mathrm{ }=\mathrm{ }\frac{1}{20}·\sum _{k=1}^{20}{x}_{k}\mathrm{ }=\mathrm{ }\frac{223}{20}\mathrm{ }=\mathrm{ }11.15 .$

The arithmetic mean is rather sensitive to so-called statistical outliers: measurement values that vary strongly form the other data can significantly affect the arithmetic mean.
##### Example 11.3.3
Let us again consider the original list for the sample of size $n=20$ above. If we drop the value ${x}_{7}=22$, then for the arithmetic mean of the remaining $19$ data values we have

$\frac{1}{19}·\sum _{k=1,k\ne 7}^{n}{x}_{k}\mathrm{ }=\mathrm{ }\frac{201}{19}\mathrm{ }\approx \mathrm{ }10.58 .$

If a multiplicative or relative relation exists among the values in an original list (for example, for growth processes or continuous compounding interest), the arithmetic (additive) mean is not an appropriate measure. For such data values, the geometric mean is used:
##### Info 11.3.4

Let data of the form ${x}_{1}>0,\mathrm{ }{x}_{2}>0,\mathrm{ }\dots ,{x}_{n}>0$ be given. Then, the geometric mean ${\stackrel{‾}{x}}_{G}$ of ${x}_{1},{x}_{2},\dots ,{x}_{n}$ is given by

${\stackrel{‾}{x}}_{G}\mathrm{ }=\mathrm{ }\sqrt[n]{{x}_{1}·{x}_{2}·\dots ·{x}_{n}} .$

##### Example 11.3.5
Let us consider a population that consists of $50$ animals at time ${t}_{0}$. Every two years, the number of animals is counted again.
 Year Number of animals Growth rate ${t}_{0}$ 50 ${t}_{0}+2$ 100 doubled (${x}_{1}=2$) ${t}_{0}+4$ 400 quadrupled (${x}_{2}=4$) ${t}_{0}+6$ 1200 tripled (${x}_{3}=3$)

For the (geometric) mean growth rate, we have

${\stackrel{‾}{x}}_{G}\mathrm{ }=\mathrm{ }\sqrt[3]{2·4·3}\mathrm{ }=\mathrm{ }\sqrt[3]{24}\mathrm{ }\approx \mathrm{ }2.8845 .$

This example illustrates that applying the arithmetic mean to growth processes gives misleading results. We would get

$\stackrel{‾}{x}\mathrm{ }=\mathrm{ }\frac{1}{3}·\left(2+4+3\right)\mathrm{ }=\mathrm{ }\frac{9}{3}=\mathrm{ }3 .$

However, a theoretical tripling of the population size every two years would imply that the number of animals after six years would be $1,350$ which is obviously not the case. From an average growth rate of $2.8845$, we obtain the correct result: $50·\left(2.8845{\right)}^{3}\approx 1,200$.
##### Exercise 11.3.6
The growth rates per year of an investment are as follows:
 Year 2011 2012 2013 2014 2015 Growth rate $0.5%$ $1.1%$ $0.8%$ $1.2%$ $0.7%$
Calculate the mean growth rate over five years in percent: ${\stackrel{‾}{x}}_{G}$$=$
$%$, rounded mathematically to two fractional digits.
In this exercise you are allowed to use a calculator.