1.1 Introduction to the Practice of Statistics

Definition and Purpose of Statistics

Statistics is the science of collecting, organizing, analyzing, and interpreting data to make decisions. It is essential for understanding variability in data and making informed conclusions in the presence of uncertainty.

• Data: Information collected from observations, counts, measurements, or responses.

• Statistics helps counter anecdotal claims and supports evidence-based conclusions (e.g., effectiveness of drugs, trends in house prices).

• Variability is central to statistics—differences exist among individuals and within the same individual over time.

Statistics as a Process

1. Identify the research objective: Define the question or problem to be studied.

2. Collect data: Gather relevant information to address the research objective.

3. Describe the data: Summarize and explore the data using tables, graphs, and numerical measures.

4. Perform inference: Draw conclusions about the population based on sample data.

Key Statistical Terms

• Population: The entire group of individuals or items of interest in a study.

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

• Individual: A single member of the population or sample.

• Parameter: A numerical summary describing a characteristic of the population.

• Statistic: A numerical summary describing a characteristic of a sample.

Example 1

Suppose the proportion of all students on your campus who have a job is 0.849 (parameter). A sample of 250 students is taken, and 0.864 of them have a job (statistic). Here, the population is all students on campus, the sample is the 250 students, individuals are each student, the parameter is 0.849, and the statistic is 0.864.

Example 2

In a medical study, a random sample of 165 US adults with high blood pressure is taken. After 6 months, the drug lowers blood pressure by an average of 14 points. The population is all US adults with high blood pressure, the sample is the 165 adults, individuals are each adult, the parameter is the average reduction for all adults, and the statistic is the average reduction in the sample.

Types of Variables

Definition of Variable

A variable is any characteristic, number, or quantity that can be measured or counted and varies among individuals in a population.

Qualitative (Categorical) vs. Quantitative Variables

• Qualitative (Categorical) Variables: Describe qualities or categories (e.g., education level, phone type).

• Quantitative Variables: Represent numerical values (e.g., age, temperature, income).

Examples

• Education level: Qualitative

• Today's high temperature: Quantitative

• Daily intake of whole grains: Quantitative

• Number of vending machines: Quantitative

• Student preparedness: Qualitative

Discrete vs. Continuous Variables

• Discrete Variable: Quantitative variable with countable values (e.g., number of students).

• Continuous Variable: Quantitative variable with infinite possible values within a range (e.g., time exercised, temperature).

Examples

• Internet provider: Qualitative

• Income: Quantitative, Continuous

• Grade in Algebra: Quantitative, Continuous

• Number of students: Quantitative, Discrete

Levels of Measurement

Levels of measurement classify data based on the nature of the information within the values assigned to variables.

• Nominal: Names, labels, or categories without order (e.g., phone type).

• Ordinal: Categories with a meaningful order but no consistent difference (e.g., education level: High School, Bachelor’s, etc.).

• Interval: Ordered categories with consistent differences, but no true zero (e.g., temperature in Celsius).

• Ratio: Like interval, but with a true zero (e.g., number of students, age in years).

Examples

• Internet provider: Nominal

• Age (Young, Middle, Old): Ordinal

• Age (in years): Ratio

• Temperature (Celsius): Interval

• Number of students: Ratio

1.2 Observational Studies Versus Designed Experiments

Types of Statistical Studies

• Explanatory Variable: The variable that explains or influences changes in a response variable.

• Response Variable: The outcome or variable being studied.

• Observational Study: Researchers observe and measure characteristics without influencing them. Can show association but not causation.

• Designed Experiment: Researchers apply a treatment and observe its effects. Can provide evidence for causation.

Examples

• Surveying new mothers about postpartum depression: Observational study.

• Giving children milk with and without xylitol to compare taste: Designed experiment.

Note: Observational studies may have lurking variables (unmeasured variables affecting the response), while designed experiments may have confounders (variables whose effects cannot be separated from the explanatory variable).

Other Types of Data Collection

• Census: Collecting data from every individual in the population.

• Web Scraping (Data Mining): Using automated tools to collect large amounts of data from the internet.

1.3 Simple Random Sampling

Random Sampling

Random sampling ensures every individual in the population has an equal chance of being selected, reducing bias and increasing representativeness.

• Simple Random Sample: Every possible sample of a given size has the same chance of being chosen.

Steps for Obtaining a Simple Random Sample

1. List all individuals in the population (create a frame).

2. Assign numbers to each individual.

3. Use a random method (table, calculator, software) to select the sample.

Example

To select 5 members from 435 in the House of Representatives, number each member 1–435 and randomly select 5 numbers.

1.5 Bias in Sampling

Types of Bias

If a sample does not accurately represent the population, it is biased. Main sources of bias include:

• Sampling Bias: Some members of the population are less likely to be included in the sample.

• Nonresponse Bias: Individuals selected for the sample do not respond.

• Response Bias: Responses are inaccurate due to question wording, interviewer influence, or respondent dishonesty.

Types of Error

• Nonsampling Error: Errors not related to the act of selecting a sample (e.g., data entry errors, poorly worded questions).

• Sampling Error: The natural variation that occurs because a sample, not the entire population, is surveyed.

Examples of Bias and Error

• Polling only magazine readers or car owners (as in the 1936 Literary Digest poll) leads to sampling bias.

• Low response rates or robocalls may cause nonresponse bias or nonsampling error.

• Survey questions with limited answer choices may introduce response bias.

Additional info: Understanding and minimizing bias is crucial for valid statistical inference. Proper sampling methods and careful survey design help ensure reliable results.