STAT311 E1

Observation

An individual, object, or unit from which data is collected.

Variable

A characteristic measured or recorded for each observation.

Categorical Variable

A variable that describes groups or categories rather than numerical quantities.

Numerical Variable

A variable that represents a measurable or countable quantity.

Discrete Variable

A numerical variable that takes countable values, typically whole numbers.

Continuous Variable

A numerical variable that can take any value within a range, including decimals.

Population

The entire group of individuals we want to study.

Sample

A subset of the population from which we collect data.

Census

An attempt to collect data from every individual in a population.

Sampling Frame

The list or source from which individuals are selected for a sample.

Population Parameter

A numerical value describing a characteristic of an entire population, typically unknown.

Sample Statistic

A numerical value calculated from a sample and often used to estimate a population parameter.

Sampling Variability

Natural differences in sample statistics that occur when different random samples are selected from the same population.

Simple Random Sample

A sampling method in which every possible group of a given size has an equal chance of being selected.

Stratified Sampling

Dividing a population into subgroups and randomly selecting individuals from each subgroup to ensure representation.

Cluster Sampling

Randomly selecting entire groups and including every individual within the selected groups.

Multistage Sampling

Randomly selecting groups and then randomly selecting individuals within those groups.

Systematic Sampling

Selecting every nth individual from a list, usually after choosing a random starting point.

Convenience Sampling

Selecting individuals who are easiest to access, which can introduce sampling bias.

Voluntary Response Sampling

Allowing individuals to choose whether to participate, often attracting people with strong opinions.

Snowball Sampling

Asking participants to recruit additional participants, often used to reach difficult-to-access populations.

Sampling Bias

A systematic problem in the selection process that causes certain groups to be overrepresented or underrepresented.

Nonresponse Bias

Bias that occurs when people who do not respond differ meaningfully from those who do.

Question Bias

Bias caused by wording or asking questions in ways that influence responses.

Incentive Bias

Bias that can occur when rewards influence who participates or how participants respond.

Response Bias

Bias caused by participants providing inaccurate or misleading answers.

Effect of Increasing Sample Size on Bias

A larger sample does not automatically eliminate bias because a flawed sampling method can remain biased regardless of sample size.

Observational Study

A study in which researchers observe variables without assigning treatments or manipulating conditions.

Experiment

A study in which researchers deliberately assign treatments to investigate their effects.

Correlation

An association between two variables that does not necessarily indicate causation.

Causation

A relationship in which changing one variable directly influences another variable.

Confounding Variable

A variable whose effects cannot be separated from those of the explanatory variable, making causal conclusions difficult.

Lurking Variable

A variable not included in the analysis that may influence the observed relationship between variables.

Random Sampling

Randomly selecting individuals from a population to help obtain a representative sample and support generalization.

Random Assignment

Randomly assigning participants to treatment groups to help balance confounding variables and support causal conclusions.

Random Sampling vs. Random Assignment

Random sampling helps generalize results to a population, while random assignment helps establish causation.

Treatment Group

The group receiving the treatment or intervention being studied.

Control Group

The comparison group that does not receive the treatment of interest or receives a standard treatment or placebo.

Placebo

An inactive treatment that resembles the real treatment and helps control for psychological expectations.

Placebo Effect

Changes in participants' responses resulting from their expectations about receiving treatment.

Blind Study

A study in which participants do not know which treatment they receive.

Double-Blind Study

A study in which neither participants nor the researchers interacting with or evaluating them know their treatment assignments.

Blocking

Grouping participants with similar characteristics before randomly assigning treatments within each group.

Purpose of Blocking

To reduce variability caused by known differences among participants, making treatment effects easier to identify.

Replication

Applying treatments to multiple experimental units and designing studies so results can be reproduced.

Noncompliance

When participants do not follow their assigned treatment, potentially complicating the interpretation of results.

Independent Observations

Observations whose outcomes do not influence one another.

Random Process

A process whose individual outcomes are uncertain but whose long-run patterns can be described using probability.

Probability

A measure of how likely an event is to occur, ranging from 0 to 1.

Event

A collection of possible outcomes from a random process.

Sample Space

The set of all possible outcomes of a random process.

Probability Distribution

A description of the possible values of a random variable and their associated probabilities.

Valid Probability Distribution

A distribution in which every probability is between 0 and 1 and the probabilities of all possible outcomes sum to 1.

Degenerate Probability

An event with a probability of exactly 0 or 1.

Complement

The event that the original event does not occur.

Complement Rule

The probability of an event not occurring equals 1 minus the probability of that event.

Union

The event that A occurs, B occurs, or both occur; represented by A ∪ B.

Intersection

The event that both A and B occur; represented by A ∩ B.

Addition Rule

The probability of A or B equals the probability of A plus the probability of B minus the probability of both, preventing double-counting.

Multiplication Rule

The probability of both A and B occurring equals the probability of A multiplied by the probability of B given A.

Conditional Probability

The probability of an event occurring given that another event has already occurred.

Effect of Conditioning

Knowing that one event occurred may change the probability of another event by restricting the possible outcomes.

Independent Events

Events in which knowing whether one occurs does not change the probability of the other.

Dependent Events

Events in which knowing whether one occurs changes the probability of the other.

Mutually Exclusive Events

Events that cannot occur simultaneously.

Independent vs. Mutually Exclusive Events

Independent events do not influence each other's probabilities, while mutually exclusive events cannot happen together.

Mutually Exclusive Events and Independence

Two mutually exclusive events with positive probabilities cannot be independent because the occurrence of one makes the other impossible.

Addition Rule for Mutually Exclusive Events

When events cannot happen together, their probabilities can be added without subtracting an intersection.

Multiplication Rule for Independent Events

When events are independent, the probability of both occurring equals the product of their individual probabilities.

Gambler's Fallacy

The incorrect belief that previous independent random outcomes influence the probability of future outcomes.

Law of Large Numbers

As the number of independent observations increases, the sample average tends to approach the expected value.

Expected Value

The theoretical long-run average outcome of a random variable, calculated by weighting each possible outcome by its probability.

Expected Value Interpretation

The expected value represents an average over many repetitions, not necessarily an outcome that can occur in a single trial.

Variance

A measure of how spread out values are around their mean, expressed in squared units.

Standard Deviation

A measure of the typical spread of observations around their mean, expressed in the original units.

Variance vs. Standard Deviation

Variance measures squared deviations, while standard deviation is the square root of variance.

Effect of Increasing Variance

Greater variance means outcomes are more spread out around the mean.

Binomial Distribution

A probability distribution describing the number of successes in a fixed number of independent trials with two possible outcomes and a constant probability of success.

Conditions for a Binomial Distribution

Fixed number of trials, two possible outcomes per trial, independence, and constant probability of success.

Binomial Success Probability

The probability of the outcome classified as a success on each trial.

Effect of Increasing Binomial Trials

Increasing the number of trials increases the expected number of successes when the probability of success remains constant and positive.

Mean of a Binomial Distribution

The number of trials multiplied by the probability of success, representing the expected number of successes.

Bar Chart

A graph used to compare frequencies or proportions across categories.

Pareto Chart

A bar chart that displays categories in order from highest to lowest frequency or quantity.

Histogram

A graph showing how numerical observations are distributed across intervals.

Bar Chart vs. Histogram

Bar charts represent categorical data, while histograms represent numerical data grouped into intervals.

Scatterplot

A graph used to examine the relationship between two numerical variables.

Line Graph

A graph commonly used to show changes in a variable over time.

Pie Chart

A graph showing how categories contribute to a whole.

Mean

The arithmetic average of observations, calculated by adding all values and dividing by the number of observations.

Median

The middle value of an ordered dataset, with the average of the two middle values used when there is an even number of observations.

Mean vs. Median

The mean is sensitive to extreme values, while the median is resistant to outliers.

Effect of a High Outlier on the Mean

An extremely high observation pulls the mean upward.

Effect of a Low Outlier on the Mean

An extremely low observation pulls the mean downward.

Effect of Outliers on the Median

Outliers generally have less influence on the median because it depends on the order of observations rather than their magnitude.

Quantile

A value that divides an ordered distribution so that a specified proportion of observations falls at or below it.

First Quartile (Q1)

The 25th percentile, with approximately 25% of observations at or below it.

Second Quartile (Q2)

The median or 50th percentile.

Third Quartile (Q3)

The 75th percentile, with approximately 75% of observations at or below it.

Interquartile Range (IQR)

The difference between Q3 and Q1, measuring the spread of the middle 50% of observations.