Introduction to Statistics

Types of Statistical Applications

Statistics is the science of collecting, analyzing, interpreting, and presenting data. It is broadly divided into two main branches:

• Descriptive Statistics: Methods for summarizing and presenting data to identify patterns and make the data easier to understand. Examples include calculating averages, creating tables, and drawing graphs.

• Inferential Statistics: Techniques that use sample data to make generalizations, estimates, predictions, or decisions about a population. This often involves probability theory and hypothesis testing.

Populations vs. Samples

Understanding the difference between populations and samples is fundamental in statistics:

• Population: The entire group of individuals or items that are of interest in a study. For example, all UCF students.

• Sample: A subset of the population, selected for the purpose of analysis. For example, 30 UCF students surveyed about scholarships.

Parameters vs. Statistics

Statistical analysis distinguishes between values describing populations and those describing samples:

• Parameter: A numerical value that describes a characteristic of a population (e.g., the proportion of all UCF students with scholarships).

• Statistic: A numerical value that describes a characteristic of a sample (e.g., the proportion of surveyed UCF students with scholarships).

Example

• Research Question: What proportion of UCF students have scholarships?

• Sample Data: 30 UCF students surveyed; 60% have scholarships.

• Population: All UCF students.

• Sample: The 30 students surveyed.

• Descriptive Statistic: 60% of the sample have scholarships.

• Inferential Statistic Example: If 50% of UF students have scholarships, inferential statistics might be used to compare UCF and UF students or to estimate the proportion of all Florida students with scholarships.

Types of Data

Quantitative vs. Qualitative Data

Data can be classified as quantitative or qualitative:

• Quantitative Data: Numerical values that represent counts or measurements (e.g., number of Instagram checks, room temperature).

• Qualitative Data: Non-numerical values or numbers that do not represent counts or measurements; these are classified into categories (e.g., political party affiliation, eye color, zip code).

Examples

• Political party affiliation: Qualitative

• How many times you’ve checked Instagram today: Quantitative

• Temperature of the room: Quantitative

• Eye color: Qualitative

• Zip code: Qualitative (numbers, but not a measurement or count)

Levels of Measurement

Measurement Scales

Data can be measured at different levels, which determine the types of statistical analyses that can be performed:

• Nominal: Qualitative only. Data cannot be ordered or ranked. (e.g., movie genres, eye color)

• Ordinal: Qualitative or quantitative. Data can be ordered or ranked, but differences between values are not meaningful. (e.g., pain level 1-10, top 5 occupations)

• Interval: Quantitative only. Data can be ordered, and meaningful differences exist, but there is no true zero. (e.g., body temperature in Fahrenheit, years of World Series wins)

• Ratio: Quantitative only. Data can be ordered, meaningful differences exist, and there is a true zero representing "none". (e.g., number of people at Disney World, heart rate)

Examples of Levels of Measurement

Statistical Studies

Types of Studies

Statistical studies are designed to collect data in different ways:

• Observational Study: The researcher observes and measures characteristics without influencing the subjects or changing existing conditions.

• Experiment: The researcher applies a treatment to subjects and observes the effect. Includes treatment and control groups, and may use placebos.

• Treatment: The intervention applied to the treatment group.

• Control Group: Subjects who do not receive the treatment; may receive a placebo.

• Experimental Units: The subjects in both treatment and control groups.

Example: Detecting Lung Cancer

• Study: 50,000 smokers randomly assigned to CT or X-ray screening.

• Type: Experiment (treatment applied: CT or X-ray)

• Experimental Units: The smokers participating in the study.

• Variable Measured: Age at which a tumor is detected (Quantitative)

• Population: All smokers eligible for screening.

• Sample: The 50,000 smokers in the trial.

• Inference: Which screening method is more effective for early detection of lung cancer.

Sampling Methods

Types of Sampling

Sampling is the process of selecting a subset of the population for study. Different methods are used to ensure representativeness and minimize bias:

• Simple Random Sample: Every member of the population has an equal chance of being selected.

• Stratified Sample: The population is divided into strata (subgroups) based on a characteristic, and random samples are taken from each stratum.

• Cluster Sample: The population is divided into clusters, and all members of selected clusters are included in the sample.

• Systematic Sample: Members are selected at regular intervals from a randomly chosen starting point.

• Census: Selecting all members of the population.

Sampling Examples

Additional info: Stratified sampling ensures representation from each subgroup, while cluster sampling is efficient when clusters are naturally occurring and similar. Systematic sampling is easy to implement but may introduce bias if there is a pattern in the population.