Describing, Exploring, and Comparing Data

Measures of Variation

Understanding the spread of data is essential in statistics. Measures of variation quantify how much the data values differ from each other.

• Range: The difference between the maximum and minimum values in a data set. Formula:

• Variance (Sample): The average of the squared differences from the mean. Formula:

• Standard Deviation (Sample): The square root of the variance, representing average distance from the mean. Formula:

The Empirical Rule (68-95-99.7 Rule)

The empirical rule applies to bell-shaped (normal) distributions and describes the spread of data:

• About 68% of data falls within 1 standard deviation of the mean.

• About 95% within 2 standard deviations.

• About 99.7% within 3 standard deviations.

Z-Scores and Significance

A z-score indicates how many standard deviations a value is from the mean.

• Formula:

• Values with z-scores less than -2 or greater than 2 are often considered significant (unusual).

Five Number Summary and Boxplots

The five number summary provides a quick overview of a data set:

• Minimum

• First Quartile (Q1)

• Median (Q2)

• Third Quartile (Q3)

• Maximum

A boxplot visually displays the five number summary and highlights outliers.

Identifying Outliers with IQR

The Interquartile Range (IQR) is the difference between Q3 and Q1. Outliers are values that fall below Q1 - 1.5*IQR or above Q3 + 1.5*IQR.

• Formula:

• Outlier Boundaries: ,

Probability

Probability Basics

Probability measures the likelihood of an event, ranging from 0 (impossible) to 1 (certain).

• Probability of an Event (E):

Sample Spaces and Simple Events

The sample space is the set of all possible outcomes. A simple event is an outcome with a single result.

• To construct a sample space, list all possible outcomes.

• The number of outcomes can be determined by counting or using the multiplication rule.

Classical Probability

Classical probability assumes all outcomes are equally likely.

• Formula:

Basic Probability Concepts

• Complement: The probability that event E does not occur:

• Mutually Exclusive Events: Events that cannot occur at the same time.

• Independent Events: The occurrence of one event does not affect the probability of the other.

Discrete Probability Distributions

Random Variables and Probability Distributions

A random variable assigns a numerical value to each outcome in a sample space. A probability distribution lists each value of the random variable with its probability.

• Probabilities must sum to 1.

• Each probability must be between 0 and 1.

Mean and Standard Deviation of a Probability Distribution

• Mean (Expected Value):

• Standard Deviation:

Constructing Probability Distributions

Probability distributions can be constructed from frequency distributions by dividing each frequency by the total number of outcomes.

Binomial Distributions

Identifying Binomial Distributions

A binomial distribution arises from a fixed number of independent trials, each with two possible outcomes (success or failure).

• Fixed number of trials (n)

• Each trial is independent

• Each trial has two outcomes

• Probability of success (p) is constant

Binomial Probability Formula

• Probability of x successes in n trials:

• Mean:

• Standard Deviation:

Normal Probability Distributions

Continuous Uniform Distribution

In a continuous uniform distribution, all intervals of the same length are equally probable.

• Probability for interval [a, b]: for

Standard Normal Distribution and Z-Scores

The standard normal distribution is a normal distribution with mean 0 and standard deviation 1. Z-scores are used to find probabilities and percentiles.

• Finding Probability for a Range of Z-Scores: Use standard normal tables (Table A2) to find the area under the curve between two z-scores.

• Finding Z-Scores for Given Areas: Use the table in reverse to find the z-score corresponding to a cumulative probability.

Example Table: Empirical Rule Coverage

Additional info: Students are allowed a handwritten cheat sheet and a graphing calculator for the test, but no phones or computers.