Statistical Methods For The Social Sciences (Agresi&Finlay)

What are statistical methods? – Chapter 1

1.1 What is statistics and how can you learn it?

Statistics is used more and more often to study the behavior of people, not only by the social sciences but also by companies. Everyone can learn how to use statistics, even without much knowledge of mathematics and even with fear of statistics. Most important are logic thinking and perseverance.

To first step to using statistical methods is collecting data. Data are collected observations of characteristics of interest. For instance the opinion of 1000 people on whether marihuana should be allowed. Data can be obtained through questionnaires, experiments, observations or existing databases.

But statistics aren't only numbers obtained from data. A broader definition of statistics entails all methods to obtain and analyze data.

1.2 What is the difference between descriptive and inferential statistics?

Before being able to analyze data, a design is made on how to obtain the data. Next there are two sorts of statistical analyses; descriptive statistics and inferential statistics. Descriptive statistics summarizes the information obtained from a collection of data, so the data is easier to interpret. Inferential statistics makes predictions with the help of data. Which kind of statistics is used, depends on the goal of the research (summarize or predict).

To understand the differences better, a number of basic terms are important. The subjects are the entities that are observed in a research study, most often people but sometimes families, schools, cities etc. The population is the whole of subjects that you want to study (for instance foreign students). The sample is a limited number of selected subjects on which you will collect data (for instance 100 foreign students from several universities). The ultimate goal is to learn about the population, but because it's impossible to research the entire population, a sample is made.

Descriptive statistics can be used both in case data is available for the entire population and only for the sample. Inferential statistics is only applicable to samples, because predictions for a yet unknown future are made. Hence the definition of inferential statistics is making predictions about a population, based on data gathered from a sample.

The goal of statistics is to learn more about the parameter. The parameter is the numerical summary of the population, or the unknown value that can tell something about the ultimate conditions of the whole. So it's not about the sample but about the population. This is why an important part of inferential statistics is measuring and crediting how representative a sample is.

A population can be real (for instance foreign students) or conceptual (for instance the foreign students that will pass their statistics course this year).

1.3 What part does software play in statistics?

Software enables an easy application of complex methods. The most used software for statistics are SPSS, R, SAS and Stata.

Which kinds of samples and variables are possible? – Chapter 2

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 weight is continuous, because it's possible to weigh 70 kilo but also 70.52 kilo.

Categorical variables (nominal or ordinal) are always discrete because they have a limited number of categories. Quantitative variables can be both discrete or continuous. When quantitative variables happen to be able to have lots of possible values, they are considered continuous.

2.2 How does randomization work?

Randomization is the mechanism of obtaining a representative sample. In a simple random sample every subject of the population has an equal chance of becoming part of the sample. The randomness is important, because it needs to be guaranteed that the data isn't biased. Biased information would make inferential statistics useless, because then it's impossible to say anything about the population.

For a random sample a sampling frame is necessary; a list of all subjects within the population. Next all subjects are numbered and then at random numbers are drawn. Drawing random numbers can be done using software, for instance R. In R the following formula is used:

> sample(1:60, 4) #

[1] 22 47 38 44 #

The symbol > indicates that the program needs to execute a task. In this sample the goal is to select four random subjects from a list of 60 subjects in total. The program indicates which subjects are chosen: numbers 22, 47. 38 and 44.

Data can be collected using surveys, experiments and observational studies. All these methods can have a degree of randomization.

Different types of surveys are possible; online, offline etc. Every way to gather data has challenges in terms of representing the population accurately.

Experiments are used to measure and compare the reactions from subjects under different conditions. These conditions, so called treatments, are values of a variable that can influence the reaction. It is up to the researcher to decide which subjects will follow which treatments. This is where randomization plays a part; the researcher needs to divide the subjects into groups randomly. In this case an experimental design is used to constitute which subjects will follow which treatments.

In observational studies the researcher measures the values of variables without influencing or manipulating the situation. Who will be observed, is determined at random. The biggest