Wiki Page, Math 13 - Perez Sandbox : Module 3 - The Big Picture

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2026-01-17T21:15:48-08:00

The Big Picture of Statistics

The Big Picture of Statistics is a diagram of how all of the pieces of this course relate to each other. As we learn new material, we will always look at how these new ideas relate to the Big Picture.

The Big Picture Summarizes the Steps in a Statistical Investigation.

We will now walk through an overview of the course using the Big Picture. Do not worry if there are details on this page that you do not understand. We want you to get a general sense of what we will be learning in this course. The details will become clear as we continue to move through the course.

Start with a research question

We begin a statistical investigation with a research question. The research question is frequently something we want to know about a population. The population can be people or other things, such as animals or objects. For example, we might want to know the answer to questions such as:

What percentage of U.S. adults supports the death penalty? (Population: U.S. adults)

Do cell phones affect bees? (Population: bees)

Do cars get better gas mileage with a new gasoline additive? (Population: cars)

The population is the entire group that we want to know something about:

Produce data

In most cases, the population is a large group. Often, the population is so large that we cannot collect information from every individual in the population, so we select a sample from the population.

We collect data from this sample. Data is information or measurements that will help us answer the research question.

Of course, we need a sample that represents the population well. For example, if our goal is to determine the percentage of U.S. adults who favor the death penalty, we do not want our sample to contain only Democrats or only Republicans because these groups may have different opinions. Therefore, we collect a random sample so that everyone has the same chance to be in the sample.

Analyze the data

To make sense of the data we collect from the sample, we summarize it using graphs and different numerical measures, such as percentages or averages. We call this step exploratory data analysis.

Use probability to understand how samples vary

Remember, our goal is to answer a question about a population based on a sample. Of course, samples will vary due to chance, and we will need to answer our question in spite of this variability. For this reason, we need to understand how sample results will vary and how sample results relate to the population as a whole when chance is involved. This is where probability comes in.

Draw a conclusion

Probability is the “machinery” behind statistical inference. We infer something about a population based on a random sample and we use probability to describe the degree of uncertainty that is due to the sampling process.

EXAMPLE

At the end of April 2005, ABC News and the Washington Post conducted a poll to determine the percentage of U.S. adults who support the death penalty.

Research question: What percentage of U.S. adults support the death penalty?

Steps in the statistical investigation:

1. Produce Data: Write the survey question. Select the random sample.

The poll selected 1,082 U.S. adults at random. Each adult answered this question: “Do you favor or oppose the death penalty for a person convicted of murder?”

2. Analyze the Data: Summarize the opinions in the survey using a percentage.

In the sample, 65% favored the death penalty.

3. Draw a Conclusion: Use the data, probability, and statistical inference to draw a conclusion about the population.
Use probability to describe our uncertainty: Our goal is to determine the percentage of the U.S. adult population that supports the death penalty. We know that different samples will give different results. What are the chances that our sample reflects the opinions of the population within 3%? Probability describes the likelihood that our sample is this accurate. For example, we may find that there is a 95% chance that the opinions of random samples are within 3% of the opinions in the population.
Draw a conclusion: We can say with 95% confidence that between 62% and 68% of the population favor the death penalty.

Let’s Summarize:

A statistical investigation begins with a research question. Then the investigation proceeds with the following steps:

1. Produce data: Determine what to measure then collect the data

2. Explore the data: Analyze and summarize the data (also called exploratory data analysis),

3. Use probability to understand how samples vary

4. Draw a conclusion about the population.

In this course, we discuss these sequenced steps. In reality, a statistical investigation may not be this orderly. For example, an initial exploration of the data may reveal a more interesting question that we decide to investigate. This change in the research question may require a different plan for collecting data. So we restart the investigation before we draw a conclusion. Or, after exploring the data, we might realize that the data does not meet the requirements for a statistical inference procedure we want to use. So we go back and collect a new set of data.

Unit 1 consists of Modules 1, 2, and 3. This is the end of Unit 1.

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