Introduction to Statistics

Parameters vs Statistics

Statistics is the science of collecting, organizing, analyzing, and interpreting data to make informed decisions. Understanding the distinction between populations and samples, as well as parameters and statistics, is fundamental in statistical analysis.

• Population: The entire set of individuals or items of interest. For example, all employees at a marketing firm.

• Sample: A subset of the population, selected for study. For example, 12 out of 100 employees at a marketing firm.

• Parameter: A numerical value summarizing a characteristic for the entire population (e.g., the average salary of all employees).

• Statistic: A numerical value summarizing a characteristic for a sample (e.g., the average salary of 12 employees).

• Data: Information gathered from counting, measuring, or collecting responses.

Example: If you collect the test scores of every other student in a class, this is a sample. If you know 46.5% of all registered voters are registered democrats, this is a parameter.

Types of Data

Qualitative vs Quantitative Data

Data can be categorized as qualitative or quantitative, and quantitative data can be further classified as discrete or continuous.

• Qualitative Data: Describes qualities or categories (e.g., favorite color, eye color).

• Quantitative Data: Describes quantities or numerical values.

  • Discrete: Countable values that cannot be broken down further (e.g., dice roll, number of students).

  • Continuous: Measurable values that can be broken down further (e.g., time, temperature).

Example: Surveying the nationalities of people is qualitative data. Measuring the distance people walk is quantitative and continuous.

Levels of Measurement

Classification of Data by Measurement Level

Levels of measurement describe the nature of data and what mathematical operations are meaningful.

• Nominal: Categories, names, or labels with no order or calculations (e.g., hair color, favorite music genre).

• Ordinal: Ordered categories, but differences between values are not meaningful (e.g., letter grades, satisfaction ratings).

• Interval: Ordered, differences are meaningful, but no true zero (e.g., temperature in Celsius or Fahrenheit).

• Ratio: Ordered, differences and ratios are meaningful, with a true zero (e.g., heights, distances, working hours).

Example: Birth years are interval data; satisfaction ratings are ordinal; working hours are ratio; favorite music genre is nominal.

Collecting Data: Observational Studies vs Experiments

Methods of Data Collection

There are two main ways to collect data: observational studies and experiments.

• Experiment: Apply a treatment and measure its effects. You can assume causation if the study is well-designed.

• Observational Study: Do not change anything; just measure characteristics. You cannot assume causation.

Example: Testing a medication by giving subjects a placebo or the actual medication is an experiment. Surveying students about their sleep habits is an observational study.

Sampling Methods

Simple Random Sampling and Representative Samples

Sampling is the process of selecting a subset of subjects from a population. A representative sample reflects the characteristics of the population. In simple random sampling (SRS), each subject and group is equally likely to be chosen.

• Simple Random Sampling (SRS): Every subject and group has an equal chance of being selected.

• Representative Sample: The sample mirrors the population's characteristics.

Example: Randomly selecting marbles from a bag or students from a class.

Other Sampling Methods

When SRS is not practical, other sampling methods can be used:

• Systematic Sampling: Select every nth subject (e.g., every 12th cookie).

• Cluster Sampling: Divide the population into groups (clusters), then randomly select one or more clusters and survey all subjects within them.

• Stratified Sampling: Divide the population into groups (strata) based on shared characteristics, then randomly select subjects from each stratum.

Example: A university surveys random undergrads and grad students (stratified). A bakery tests every 12th cookie (systematic).

Summary Table: Levels of Measurement

Key Formulas

• Sample Mean:

• Population Mean: