Descriptive Statistics

Measures of Central Tendency and Spread

Descriptive statistics summarize and describe the main features of a data set. The two primary categories are measures of central tendency and measures of spread.

• Measures of Central Tendency: These indicate the center or typical value of a data set.

  • Mean: The arithmetic average of the data.

  • Median: The middle value when data are ordered.

  • Mode: The most frequently occurring value.

• Measures of Spread: These describe the variability or dispersion in the data.

  • Range: Difference between the largest and smallest values.

  • Variance: Average squared deviation from the mean.

  • Standard Deviation: Square root of the variance.

Summation Notation: Data can be represented as lists . The capital-Sigma notation is used to sum elements from to .

Parameters vs. Estimators

Parameters are characteristics of a population, while estimators are calculated from a sample to estimate population parameters.

Formulas:

• If the data is a population:

• If the data is a sample:

Variance: Properties and Calculation

Variance measures the average squared deviation from the mean. For sample variance:

• Equivalent form for calculation:

• Important property: (variance is always nonnegative).

• Also,

Basic Probability Theory

Motivation and Definition

Probability quantifies the likelihood of uncertain events. It helps distinguish likely from unlikely outcomes and guides decision-making in uncertain situations.

• Examples: Rolling a 3 with a fair die, electricity demand exceeding capacity, predicting a recession.

Sample Space, Events, and Set Theory

Probability theory begins with defining the sample space and events using set theory.

• Sample Space (S): The set of all possible outcomes of an experiment.

• Event: A subset of the sample space.

• Set Theory Operations:

  • Empty set (): Contains no objects.

  • Subset (): All elements of are in .

  • Intersection (): Elements in both and .

  • Union (): Elements in , , or both.

  • Complement (): Elements in not in .

  • Difference (): Elements in but not in .

Example (Coin Toss): ; possible events: , , , .

Example (Die Roll): ; event (outcome three or less), event (even outcome).

Probability Axioms and Properties

A probability function assigns a number in to each event, satisfying:

• For any event ,

• For disjoint events and ,

If these properties are not met, is not a valid probability function.

Probability Calculations: Examples

• Uniform Probability (Die): For , for each .

• Event Probability: For ,

• Intersection:

• Union:

Probability Properties

• Complement Rule:

• Null Set:

• Addition Rule:

• If and are disjoint,

Conditional Probability

Conditional probability updates the likelihood of an event based on new information.

• Definition: For events and with ,

• Interpretation: Probability of given has occurred.

• Note:

Example (Medical Test):

Multiplication Rule

The multiplication rule relates joint probability to conditional probability:

Example: If (defective), (pass inspection given defective), then

Independence

Two events and are independent if the occurrence of one does not affect the probability of the other.

• Definition:

• If and , then and are independent.

Example (Workforce Table):

• (not equal to , so not independent)

Law of Total Probability

The law of total probability expresses the probability of an event as the sum over a partition of the sample space:

• , where partition

Example: If , , , :

Bayes' Rule

Bayes' rule allows us to update the probability of a cause given an observed effect:

Example (Entomology): Suppose 98% of rare beetles have a pattern, 5% of common beetles have it, and rare beetles are 0.1% of all beetles. What is the probability a beetle is rare given the pattern?

References

• Telhammer, R. C. (2013). Mathematical Statistics for Economics and Business. Springer.