ECON6037 Experimental Economics

Stats refresh: Random variables

Gerhard Riener

g.riener@soton.ac.uk

February 19, 2024

Data Analysis and Experiments

Good experimental design makes for clean data analysis.
Knowing which statistical techniques you analyse helps to plan your design.
Choose the statistical approach that best fits your needs (graphs, tests, confidence intervals, regressions).
Think of what kind of data you can collect, to get the cleanest possible test of your hypothesis.
Compute the sample size necessary to meaningfully test your hypotheses.

Random Variable Definition

A random variable \(X\) is a measurable function \(X: \Omega \rightarrow E\) from a sample space \(\Omega\) (a set of possible outcomes) to a measurable space \(E\).

Technical Axiomatic Definition

Requires the sample space \(\Omega\) to be part of a probability triple \((\Omega, \mathcal{F}, P)\), as per the measure-theoretic definition.
A random variable is often denoted by capital Roman letters such as \(X, Y, Z, T\).

Probability of a Random Variable

The probability that \(X\) takes on a value in a measurable set \(S \subseteq E\) is denoted as:

\[ P(X \in S) = P(\{\omega \in \Omega \mid X(\omega) \in S\}) \]

Examples of Random Variables

Experiment	Random variable
Toss two dice	\(Y =\) sum of the numbers
Toss a coin 25 times	\(Y =\) number of heads in 25 tosses
Apply different amounts of fertilizer	\(Y =\) yield per acre
Can communicate / can not communicate	\(Y =\) contribution to public good

Variable types

What types of variables exist

Quantitative

Reflects a notion of magnitude, that is, if the values it can take are numbers. A quantitative variable represents thus a measure and is numerical.

Discrete countable finite number.
- Number of children per family
- Number of students in a class
- Number of citizens of a country
Continuous not countable
- Age
- Weight
- Height

Note: For all measurements, we usually stop at a standard level of granularity

Qualitative

also referred as categorical variables or factors are variables that are not numerical and which values fits into categories.

In other words, a qualitative variable is a variable which takes as its values modalities, categories or even levels, in contrast to quantitative variables which measure a quantity on each individual.

Nominal: no ordering
- Gender: female/male/ungendered/others
- Eye color: blue/brown/green

Note: that a qualitative variable with exactly 2 levels is also referred as a binary or dichotomous variable.

Qualitative: Ordinal

On the other hand, a qualitative ordinal variable is a qualitative variable with an order implied in the levels. For instance, if the severity of road accidents has been measured on a scale such as light, moderate and fatal accidents, this variable is a qualitative ordinal variable because there is a clear order in the levels.

Another good example is health, which can take values such as poor, reasonable, good, or excellent. Again, there is a clear order in these levels so health is in this case a qualitative ordinal variable.