Introduction to Statistics: Data Collection and Variables

Lecture 1: Introduction and Variables

Statistics: Definition and Purpose

Statistics is the science of collecting, organizing, summarizing, and analyzing information to answer questions. It also provides a measure of confidence in any conclusions drawn from data.

• Data are facts or propositions used to draw conclusions or make decisions. Data describe characteristics of individuals.

• Statistics helps us make informed decisions and quantify uncertainty.

Population, Individual, and Sample

Understanding the scope of a statistical study requires defining the population, individuals, and sample.

• Population: The entire group of individuals to be studied.

• Individual: A single person or object that is a member of the population.

• Sample: A subset of the population selected for study.

The Process of Statistics

The statistical process involves several key steps:

1. Identify the research objective: Determine the questions to be answered.

2. Collect the data needed: Gather data from a sample of the population.

3. Describe the data: Use descriptive statistics to summarize and understand the data. This helps determine which analytical methods to use. (See Chapters 2-4)

4. Perform inference: Apply inferential techniques to extend results from the sample to the population, reporting a level of reliability. (See Chapters 5-8)

Example: Coffee Consumption Survey

This example illustrates the statistical process:

• Research Objective: Determine the average number of cups of coffee per day that BU undergraduates drink.

• Population: All BU undergraduates.

• Sample: 200 students randomly selected from different majors.

• Describe the Data: The sample mean was 2.1 cups per day, with a range from 0 to 7 cups. This is descriptive statistics.

• Perform Inference: Infer that the true average for all undergraduates is about 2.1 cups per day. Using confidence intervals, we might say: "We are 95% confident that the true average lies between 1.9 and 2.3 cups per day."

Formula for Confidence Interval (Mean):

$\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$

Additional info: $\bar{x}$ is the sample mean, $z^*$ is the critical value for the desired confidence level, $\sigma$ is the population standard deviation, and $n$ is the sample size.

Variables

Definition and Importance

Variables are characteristics of individuals within the population that can vary from one individual to another.

• Variables are central to statistical analysis because they capture the diversity and variability in data.

• Example: In growing tomatoes, the variable of interest might be the weight of each tomato.

• If all tomatoes weighed the same, only one measurement would be needed. Variability motivates research.

Qualitative vs Quantitative Variables

Variables can be classified as qualitative or quantitative:

• Qualitative (Categorical) Variables: Classify individuals based on descriptive attributes (e.g., major, payment method).

• Quantitative Variables: Provide numerical measures of individuals, allowing for measurement and comparison (e.g., temperature, number of days studied).

Examples:

• Major chosen by student: Qualitative

• Temperature: Quantitative

• Number of days studied: Quantitative

• Zip code: Qualitative (though numeric, it is a label)

Discrete vs Continuous Variables

Quantitative variables are further classified as discrete or continuous:

• Discrete Variable: Has a finite or countable number of possible values (e.g., number of hospital visits).

• Continuous Variable: Has an infinite number of possible values that are not countable (e.g., body temperature).

Examples:

• Number of cavities: Discrete

• Body temperature: Continuous

• Number of hospital visits: Discrete

• Cholesterol level: Continuous

• Number of cells: Discrete

• Blood pressure: Continuous

• Heart rate: Continuous

• Number of doses: Discrete

Variables vs Data

The list of observations a variable assumes is called data.

• Qualitative data: Observations corresponding to a qualitative variable.

• Quantitative data: Observations corresponding to a quantitative variable.

• Discrete data: Observations corresponding to a discrete variable.

• Continuous data: Observations corresponding to a continuous variable.

Example: Bubble Tea Shop Survey

Survey Table

The following table summarizes a survey of customers at a bubble tea shop:

Customer	Drink Type	Topping	Payment Method	Amount Paid ($)	Drink Size (ml)
1	Milk Tea	Boba	Card	5.50	500
2	Fruit Tea	Jelly	Cash	6.00	600
3	Slushie	None	Mobile Pay	4.75	450
4	Milk Tea	Boba	Card	5.95	500
5	Fruit Tea	None	Mobile Pay	5.00	600
6	Milk Tea	Jelly	Cash	6.10	700
7	Slushie	Jelly	Card	4.90	450

Survey Analysis

• Individuals: The customers (Customer 1, 2, ..., 10). Each row represents one individual.

• Variables:

  • Qualitative: Drink Type, Topping, Payment Method

  • Quantitative: Amount Paid ($) (discrete), Drink Size (ml) (continuous)

• Example Observation (Customer 3): Slushie, None, Mobile Pay, $4.75, 450 ml