# Chapter 2 Introduction to Statistics

## 2.1 What is Statistics?

Statistics is the process of describing variability and testing hypotheses.

## 2.2 Variability

Why is it that everyone is not the same on ____? Why are some employees more content than others? Why do some people get heart disease and others don’t? Why are some people taller or shorter than others? Why is it that only some people suffer from a disorder? Variability is the existence of differences in an attribute. Everyone is not the same; we say there is *variability* in people’s height. If everyone were the same height, there would be no variability in height. When there is no variability, we call the value a *constant*. For anything you can think to measure about people, there are few, if any constants. We live in a world of complex variability.

## 2.3 Descriptive versus Inferential Statistics

Statistics is a tool to clarify our observations of variability. This is the first function of statistics, *descriptive statistics*. Descriptive statistics offers a number of tools to describe variability. Descriptive statistics is useful because we can understand variability better with statistics better than we can intuitively. Is it likely that two people in this class have the same birthday? There are 365 possibilities, so common sense would say that the chances are pretty low. Computing the probability reveals that in a class of 23, the probability is 50%. In a group 57 people, the probability that at least two people will have the same birthday is 99%. This is commonly called the Birthday problem.

If you’re interested, look up the formula for the birthday problem and calculate the probability that someone in our class has the same birthday as yours.

Inferential statistics is the process of drawing conclusions about the variability of a group of interest (called a population) using a limited set of data (called a sample). Fundamentally, inferential statistics are clever uses of probability theory and logic that allow you to make conclusions about populations.

Example: I am interested in middle school students’ reading comprehension in the United States, and I want to see if it changes over time. To understand this population directly, I would have to measure the reading comprehension of every member. This is impossible. Instead, I take a random sample from the population by mailing surveys to 50 random middle school students with consent of their parents, I can use descriptive statistics to understand my sample data (50 scores) and inferential statistics to generalize the results to the population (thousands of scores).

You may think of statistics as old and established, and it is. At the same time, it is more relevant than ever. Performing the large number of computations fast enough and having access to data were big challenges decades ago. Computers and the digital revolution have solved the first problem. The widespread use of the Internet means that collecting and organizing large amounts of data has never been easier (although many challenges remain). As a result, we live an increasingly data-driven society where statistics is a critical tool for understanding data.

This course will prepare you to use statistics within and outside of your field. Concepts presented in this course will be useful to critical thinkers in an increasingly data-focused society, and the techniques covered will provide a foundation for conducting real-world research. This course will help you answer the question: what do our observations tell us about the world?

## 2.4 Populations and Samples

A population is the entire group of interest. Examples: people, nursing home residents, repeat customers, etc. The population is the group we want to study.

Populations can be any group you want to draw conclusions about. The researcher defines the population, and this frames the entire research project. The findings of a study intending to measure college students may not apply to older adults. The population is the group to which you will generalize your findings.

The descriptive statistics we will cover can be applied to populations. If we can measure everybody and calculate the average, then we have calculated a population parameter. A dataset drawn from every member of a population is called a census. Most of the time, our populations are very large, and it’s impossible to measure everybody. How could you give a survey to every single college student in the United States? The solution is to use a sample, which will represent our population.

A *sample* is a smaller set from the population. When descriptive statistics are computed for a sample, they are called sample statistics.

The best way to get a sample is to obtain a *random sample*. A random sample means that every member of the population has an equal chance of being selected. To do this properly, a researcher should generate a list of every member of the population and select from the list at random. To be a truly random sample, every individual selected would have to participate in your study. Ethically, you cannot compel people to participate, and you’ll never get a 100% response rate to your study. A bigger issue is how you would generate a list of every member of your population, as populations can be huge. Random sampling is not an all-or-nothing technique; the closer to a random sample of the population, the better the sample will represent the population.
The other end of this spectrum is a *convenience sample*. A researcher using a convenience sample asks whoever is available to participate in the study. The resulting sample is biased due to proximity, availability, and convenience. The further away from a true random sample, the less likely it is that the sample collected will represent the population.

## 2.5 Constructs versus Measures

There is an important distinction between constructs and measures. A construct is a “concept, model, or schematic idea” (Shadish, Cook, & Campbell, 2002, p. 506). Constructs are the big ideas that researchers are interested in measuring: depression, patient outcomes, prevalence of cumulative trauma disorders, or even sales. For constructs in the social sciences, there is often disagreement and debate about how to define a construct. To do science, we must be able to quantify our observations (collect data) on the constructs. To go from a construct (the idea) to a measure requires an operational definition. An *operational definition* describes how a construct is measured.

## 2.6 Classifying Measurement Scales

No matter your field, the foundation of statistics is measurement. We can classify measures in three ways: according to their level of measurement, whether or not they are continuous or discrete, and whether they represent qualitative or quantitative data.

### 2.6.1 Level of Measurement

A stair diagram is used because higher levels of measurement satisfy all the requirements of the levels below.

```
Ratio scale/ratio measurement. Examples: weight, length
Interval scale/interval measurement. Example: Fahrenheit temperature
Ordinal scale/ordinal measurement. Example: the order in which people finish a race
Nominal scale/Nominal measurement. Example: gender
```

Notice that these levels are stair steps. Each level has all the characteristics of the level below it. So interval scales meet all the requirements of ordinal and nominal scales as well (plus they meet the additional requirement for interval scales).

To determine the level of measurement, ask yourself these questions:

- Can you rank/order the numbers? (if no, nominal scale. if yes, keep going) example: kinds of fish. can you rank halibut and mullet? (no, nominal scale) example: Olympic medals, can you rank gold, silver, and bronze? (yes, keep going)
- If you add/subtract the numbers, does the result have meaning? (if no, ordinal scale. if yes, keep going) example: 30 degrees F plus 10 degrees equals 40 degrees (yes, keep going) example: 1st place plus 2 equals 3rd place? (no, this doesn’t make sense, ordinal scale)
- Does the score have a value of 0 that means ‘none’ or ‘nothing?’ (if no, interval scale. if yes, ratio scale) example: counting people; 0 people means no people (yes, ratio scale) example: 0 degrees F means no heat? (no, interval scale)

### 2.6.2 Continuous or Discrete

Separately, decide if your variable is continuous or discrete. If you can have an infinite number of fractions of a value, it’s continuous. If you cannot, the measure is discrete. example: 5 yards, 5.0005 yards, 5.5 years, and 5.500001 yards are all valid measurements (continuous) example: Olympic medals; the measurement between gold and silver does not exist (discrete) There may be instances where a grey area exists; at some level, all variables are discrete. For example, you could subdivide a measurement of length down to the molecule. At that point, you cannot have fractional values. Try to avoid over-thinking this issue. If you can reasonably talk about fractional values (half seconds; twenty-five cents are a fraction of a dollar) then the measure is continuous. If you cannot (there is no such thing as half a dog or an eighth of an employee), then the measure is discrete.

### 2.6.3 Qualitative or Quantitative

**Quan**titative data is associated with a numerical value. **Qual**itative data is associated with labels that have no numerical value. Nominal and ordinal data are qualitative. Interval and ratio data are quantitative.

## 2.7 Experimental, Quasi-Experimental, and Non-Experimental Studies

Research psychology is a process of identifying constructs and describing how they relate to other constructs. We can classify research designs as experiments, quasi-experiments, and non-experiments.

**Experiments** are the only kind of research that shows causal relationships (that is, that construct A causes a change in construct B). So an experiment could show if smoking causes lung cancer. To do this, experiments need two things (or they are not experiments)

All experiments have a manipulation. This means that the experimenter changes something within the environment of the experiment (called an independent variable) to see if it causes a change in the outcome (called a dependent variable). For our smoking example, a manipulation would be assigning one group of participants to a lifetime of smoking and another group of participants to a lifetime of no smoking.

Experiments require random assignment. The experimenter decides when to vary the levels of the manipulation (change the manipulation) based on random assignment. Random assignment means that every participant has the same chance as being in one condition as another. For our smoking example, random assignment means each participant has a 50% chance of being in the smoking group.

As may be clear from the smoking example, we cannot always do experiments because of ethical (it would be wrong to assign people to smoke) or practical reasons (you cannot randomly assign people to genders, for example). The solution is a quasi- or non-experimental study.

In summary: experiments are powerful because they uniquely demonstrate causality (causal relationships). However, experiments require a manipulation and random assignment, which are not always possible.

In a **quasi-experimental** study, there is a manipulation but no random assignment. Whenever participants are assigned to levels of a manipulation non-randomly, the research is quasi-experimental. In a quasi-experimental smoking study, we could ask people if they had smoked before and assign them to smoking or non-smoking groups based on that answer.

In summary: quasi-experiments do not require random assignment, but they do not show casual relationships.

In a **non-experimental** study, no manipulation is done. If you want to look at the effects of gender on lung cancer, you would simply observe (collect data on) the genders of patients. By only observing, you would not be manipulating gender.

The differences between quasi- and non-experimental studies are slight (Pedhauzer & Schmelkin, 1991); it is more important that you can distinguish experiments from quasi- and non-experiments.

In summary: non-experimental studies are observational. Like quasi-experimental studies, they do not show causal relationships.

It’s worth repeating that only experiments demonstrate causality. Quasi- and non-experiments can show that a relationship exists but do not say whether one variable causes the other. Any non-causal relationship has three possible explanations:

- A -> B one variable causes another; in an experiment, this is the only explanation
- B -> A the relationship is reversed; the first variable is actually the outcome
- C -> A; C -> B a third variable exists that was not measured in the study; the third variable causes a change in both A and B. There are many ‘C’ variables, potentially.

In a non-experimental smoking study, you could not say whether smoking causes lung cancer or people who are predisposed to lung cancer are more likely to smoke. A third possibility is that a separate, third variable causes both lung cancer and a desire to smoke.

## 2.8 The word “data” is Plural

Finally, a bit of trivia. Did you know that data is actually the plural form of datum? One piece of information is a datum. Multiple pieces of information are data.