Introduction

What is Statistics?

Statistics: Methods to collect, analyze, present, interpret data, and make decisions.

Two Types of Statistics:

1. Descriptive: Methods to organize, display, and describe data.
   • e.g., Tables, graphs, summary measures
2. Inferential: Methods that use sample results to help make decisions or predictions about a population.
   • e.g., T-test, confidence

Note: Probability lies in-between descriptive and inferential statistics.

Course Progression: Descriptive Statistics \to Probability \to Inferential Statistics

Terms in Statistics

Population: The entire group of individuals that we want information about.

• It’s usually impossible to observe the entire population.

Sample: The subset of the population we examine to gather information.

• Good Sample: Contains important features/traits of the population.

Random Sample: Sample where each element in the population has a chance of being selected.

Sample Size (n): The number of observations in a sample.

Variable: Characteristic of a person or thing.

• e.g., age

Observation (x_i): Value of a variable for an element.

• e.g., x_1 = 10 \text{years}

Data Set (X): Collection of observations on \ge 1 variables.

Example: Finding the average number of plushies students have in their bedrooms

• Population: All students at my college.
• Sample: The students walking past the entrance to the library who I polled as they passed.
• Sample Size: n = The number of students I asked.
• Variable: # of plushies in bedroom.
• Observation: “x_1 = 2, x_2 = 10, etc.”

Types of Variables

Two Types of Variables:

1. Numerical (Quantitative): Variables whose values are numeric.
   • e.g., age, weight, height
2. Categorical (Qualitative): Variables whose values are not numeric.
   • e.g., hair color, blood-type, gender

Important: Variable type is determined by how we use it.

• e.g.,
  • Numerical Age: 19, 42, 69, etc.
  • Categorical Age: Young, Average, Old

Note: Different types require different analyses.

Random Sampling

Simple R.S. Select n objects at random from the population.

• Every possible sample size n has the same chance of being selected.

Stratified R.S. Divide population into strata¹, then select simple random sample for each stratum.

• e.g., Splitting population into male/female, old/young, etc.

Cluster R.S. Divide population into clusters², select clusters at random, then select simple random sample for each cluster.

• e.g., A study on high school curriculum may select a few states (clusters)
  • Where is each state is a cluster, and each school district is a cluster.

Note: Clusters v.s. Strata (some v.s. all)

• Clusters are representative of their population.
  • e.g., You can describe the population using any cluster.
• Strata must be used together to describe their population.
  • e.g., If you have male and female strata, you can’t just use the male stratum to describe the population.

Systematic R.S. Select every kth element in the population.

• e.g., “I will measure every tenth student who enters the cafeteria”, if a basketball team walks past, they won’t skew the measurements heavily.

Multi-Stage R.S.: A mixture or combination of at least two methods above (except simple R.S.)

Note: In this course, we’ll assume the samples we’re given are good samples and gotten through simple R.S., unless told otherwise.