2.1 Which kinds of variables can be measured?

All characteristics of a subject that can be measured are variables. These characteristics can vary between different subjects within a sample or within a population (like income, sex, opinion). The use of variables is to indicate the variability of a value. As as example, the number of beers consumed per week by students. The values of a variable constitute the measurement scale. Several measurement scales, or ways to differ variables, are possible.

The most important divide is that between quantitative and categorical variables. Quantitative variables are measured in numerical values, such as age, numbers of brothers and sisters, income. Categorical variables (also called qualitative variables) are measured in categories, such as sex, marital status, religion. The measurement scales are tied to statistical analyses: for quantitative variables it is possible to calculate the mean (i.e. the average age), but for categorical variables this isn't possible (i.e. there is no average sex.

Also there are four measurement scales: nominal, ordinal, interval and ratio. Categorical variables have nominal or ordinal scales.

The nominal scale is purely descriptive. For instance with sex as a variable, the possible values are man and woman. There is no order or hierarchy, one value isn't higher than the other.

The ordinal scale on the other hand assumes a certain order. For instance happiness. If the possible values are unhappy, considerably unhappy, neutral, considerably happy and ecstatic, then there is a certain order. If a respondent indicates to be neutral, this is happier than considerably unhappy, which in turn is happier than unhappy. Important is that the distances between the values cannot be measured, this is the difference between ordinal and interval.

Quantitative variables have an interval or ratio scale. Interval means that there are measurable differences between the values. For instance temperate in Celcius. There is an order (30 degrees is more than 20) and the difference is clearly measurable and consistent.

The difference between interval and ratio is that for an interval scale the value can't be zero, but for a ratio scale it can be. So the ratio scale has numerical values, with a certain order, with measurable differences and with zero as a possible value. Examples are percentage or income.

Furthermore there are discrete and continuous variables. A variable is discrete when the possible values can only be limited, separate numbers. A variable is continuous when the values can be anything possible. For instance the number of brothers and sisters is discrete, because it's not possible to have 2.43 brother/sister. And for instance

4.1 What are the basic rules of probability?

Randomization is important for collecting data, the idea that the possible observations are known but it's yet unknown which possibility will prevail. What will happen, depends on probability. The probability is the proportion of the number of times that a certain observation is prevalent in a long sequence of similar observations. The fact that the sequence is long, is important, because the longer the sequence, the more accurate the probability. Then the sample proportion becomes more like the population proportion. Probabilities can also be measured in percentages (such as 70%) instead of proportions (such as 0.7). A specific branch within statistics deals with subjective probabilities, called Bayesian statistics. However, most of statistics is about regular probabilities.

A probability is written like P(A), where P is the probability and A is an outcome. If two outcomes are possible and they exclude each other, then the chance that B happens is 1- P(A).

Imagine research about people's favorite colors, whether this is mostly red and blue. Again the assumption is made that the possibilities exclude each other without overlapping. The chance that someone's favorite color is red (A) or blue (B), is P(A of B) = P (A) + P (B).

Next, imagine research that encompasses multiple questions. The research seeks to investigate how many married people have kids. Then you can multiply the chance that someone is married (A) with the chance that someone has kids (B).The formula for this is: P(A and B) = P(A) * P(B if also A). Because there is a connection between A and B, this is called a conditional probability.

Now, imagine researching multiple possibilities that are not connected. The chance that a random person likes to wear sweaters (A) and the chance that another random person likes to wear sweaters (B), is P (A and B) = P (A) x P (B). These are independent probabilities.

4.2 What is the difference in probability distributions for discrete and continuous variables?

A random variable means that the outcome differs for each observation, but mostly this is just referred to as a variable. While a discrete variable has set possible values,