Introduction to Statistics

Statistics involves working with data about individuals (people, animals, or objects) and the variables that describe them. There are two main types of variables you'll encounter:

Categorical variables place individuals into groups or categories (like state, gender, or preferred status). They tell us which group something belongs to rather than measuring an amount.

Quantitative variables take numerical values that represent amounts (like height, age, or arm span). These can be further divided into discrete variables (having gaps between possible values, usually whole numbers) and continuous variables (can take any value within a range, like time or height).

Quick Tip: When identifying variables, ask yourself: "Is this sorting things into categories, or is it measuring an amount?" If it's sorting, it's categorical. If it's measuring, it's quantitative.

When analyzing data, we use descriptive statistics to explore and summarize what we have, while inferential statistics helps us draw broader conclusions beyond our immediate data.

Working with Data Sets

Real-world data often comes in tables with multiple variables. For example, a survey might collect information about students' characteristics like height, gender, and communication preferences.

When examining a data set, first identify what each column represents. In the sample data shown, variables include Province (categorical), Gender (categorical), Number of languages spoken (quantitative), Handedness (categorical), Height (quantitative), Wrist circumference (quantitative), and Preferred communication (categorical).

The distribution of a variable tells us what values occur and how often they appear. Understanding these distributions is key to making sense of your data.

Remember: Each row in a data table represents one individual, while each column represents a different variable measured for all individuals.

Two-Way Tables and Relative Frequencies

Two-way tables display the relationship between two categorical variables. They organize counts of individuals who share specific combinations of characteristics.

When analyzing these tables, we use different types of relative frequencies:

• Marginal relative frequency tells you the proportion for one variable alone (like the percentage of all participants who are snowmobile owners)
• Joint relative frequency shows the proportion for a specific combination of two variables (like the percentage who both own snowmobiles and are in the environmental club)
• Conditional relative frequency shows proportions within a subset (like the percentage of snowmobile owners who are in the environmental club)

Think of it this way: Marginal frequencies look at one variable by itself, joint frequencies look at specific combinations, and conditional frequencies look at "if this, then what percent that?"

Analyzing Two-Way Tables

Two-way tables help us understand relationships between variables. For example, with the Titanic passenger data, we can calculate various proportions to understand survival patterns.

To find the overall survival rate, divide the total number of survivors by the total number of passengers. For class distribution, divide each class total by the overall passenger count.

When answering questions about specific combinations $like "third-class passengers who survived"$, you're calculating a joint relative frequency by dividing that specific count by the total number of passengers.

Real-world application: Statistical analysis of the Titanic disaster revealed that a passenger's social class strongly influenced their chances of survival, with first-class passengers having significantly higher survival rates than those in third class.

Interpreting Study Results

Studies often compare treatments or conditions using two-way tables. In medical research, for example, patients might receive different treatments (like St. John's wort, Zoloft, or a placebo), and their responses are recorded.

When analyzing such data, we calculate proportions to understand how treatments compare. If we want to know what percentage received a specific treatment, we divide the number in that treatment group by the total participants.

We can display these distributions using graphs. Side-by-side bar graphs show distributions separately, while segmented bar graphs stack percentages on top of each other. Mosaic plots modify segmented bar graphs by making the widths proportional to group sizes.

Key insight: When two variables have an association, knowing the value of one variable helps predict the value of the other. If knowing a patient's treatment helps predict their response, there's an association between treatment and response.

Visualizing Relationships in Data

When analyzing the Titanic data by class and survival, creating a segmented bar graph reveals important patterns. To make this graph, we calculate the proportion of survivors within each class of travel.

For first-class passengers, approximately 62% survived (197/319), while for third-class, only about 24% survived (151/627). These distributions clearly show that survival rates differed dramatically by class.

This is a perfect example of an association between variables - knowing a passenger's class helps predict their likelihood of survival. First-class passengers had the best chance of surviving, followed by second-class, with third-class passengers having the lowest survival rates.

Visual insight: Segmented bar graphs make relationships between variables immediately visible. When the proportions of segments differ noticeably across categories (as they do with Titanic class and survival), it signals an association between variables.

Analyzing Medical Study Results

In medical studies, researchers often want to know if treatments have different effects. The St. John's wort depression study compared three treatments using a two-way table.

To find what proportion of full responders took St. John's wort, divide the number of full responders on St. John's wort by the total full responders. To calculate the percent of St. John's wort patients showing no response, divide the number with no response by the total taking that treatment.

Creating a segmented bar graph for this data requires finding the distribution of response types within each treatment group. For each treatment, calculate what percentage had full, partial, or no response.

Critical thinking: When interpreting medical studies, look beyond just which treatment had the highest number of full responses. Consider the overall distribution of responses and whether differences between treatments are large enough to be meaningful rather than due to chance.