Unit 1 - Exploring One-Variable Data and Collecting Data
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Practice with exam-style Desmos (College Board testing version) or a TI-84 for handheld paths — 9 decks, 36 drills.
Video Lessons
Table of Contents
1.1 - Introducing Statistics: What Can We Learn from Data?
Key Terms & Definitions
Statistical Study
A study in which data are collected from a sample to answer an investigative question about a larger population.
- •Purpose: Used when it is too difficult to census the entire population.
Datum
A single piece of information about an item or individual.
- •Note: Singular form of data.
Population (N)
All items or individuals of interest in a study.
Sample (n)
A subset of the population from which data are obtained.
Investigative Question
A question with a defined purpose that guides the data collection and analysis.
- •Requirement: Should not be changed based on results.
1.2 - Variables
Key Terms & Definitions
Observational Unit
An item or individual from which a datum is collected.
Variable
A characteristic that may change from one observational unit to another.
Categorical Variable
A variable that takes on values that are category names or group labels.
- •Also: Qualitative variable.
Nominal Data
Categorical data with no inherent order or ranking.
- •Examples: blood type, car brands.
Ordinal Data
Categorical data with a natural order or ranking, but no consistent scale between values.
- •Examples: education level, customer satisfaction ratings.
Quantitative Variable
A variable that takes on numerical values for a measured or counted quantity.
- •Note: Generally has units of measure.
Discrete Quantitative Variable
A quantitative variable that can take on a countable number of values (finite or countably infinite).
Continuous Quantitative Variable
A quantitative variable that can take on an infinite number of possible values within a given interval.
Parameter
A numerical attribute or summary of the variable of interest for a population.
Statistic
A numerical attribute or summary of the variable of interest for a sample.
Data Set
A collection of data.
1.3 - Tabular Representation and Summary Statistics for One Categorical Variable
Key Terms & Definitions
Frequency Table
Shows the number of observational units in each category of a categorical variable.
Relative Frequency Table
Shows the proportion of observational units in each category.
- •Synonyms: Proportions, percentages, ratios.
1.4 - Graphical Representations for One Categorical Variable
Key Terms & Definitions
Bar Chart
Displays frequencies or relative frequencies for categories using bars.
- •Key: Height corresponds to count or proportion.
Pie Chart
Displays relative frequencies as slices of a circle.
- •Rule: Sum of slices equals 1 or 100%.
1.5 - Graphical Representations for One Quantitative Variable
Key Terms & Definitions
Histogram
Places observed values into ordered intervals (bins) with bar heights showing frequency.
Stem and Leaf Plot
Splits values into a stem and leaf to maintain numerical ordering.
Dotplot
Represents each value as a dot along an axis.
1.6 - Descriptions for One Quantitative Variable Distributions
Key Terms & Definitions
Shape
Description including skewness, modality, and symmetry.
Symmetric
Distribution where both tails are the same length and the mean and median are equal.
Skewed Right
The right tail is longer than the left tail, and mean > median.
Skewed Left
The left tail is longer than the right tail, and mean < median.
Unimodal
A distribution with one main peak.
Bimodal
A distribution with two prominent peaks.
Multimodal
A distribution with more than two distinct peaks.
Uniform
Approximately the same frequency across all categories.
Outlier
Data points unusually small or large relative to the rest.
Gap
Region in a distribution with no observed data.
Cluster
Concentrations of values separated by gaps.
1.7 - Summary Statistics for One Quantitative Variable
Key Terms & Definitions
Mean ( for sample.)
Sum of all values divided by n.
Mode
The most frequently occurring value in a dataset.
- •The only measure of center applicable to categorical data.
Minimum
The smallest value in an ordered data set.
Maximum
The largest value in an ordered data set.
Median
The middle value when data is ordered.
First Quartile (Q1)
Median of the lower half of data.
Second Quartile (Q2) ()
Also the median of the data set; the 50th percentile.
Third Quartile (Q3)
Median of the upper half of data.
Percentile
The th percentile is the value that has % of the data at or below it when ordered.
Range
Maximum minus minimum value.
Interquartile Range (IQR)
minus .
Standard Deviation
Typical deviation of data values from their mean.
- •Formula: Root of variance.
Variance ()
The square of the standard deviation.
Resistant
Measure not greatly affected by outliers (e.g., Median, IQR).
Nonresistant
A measure greatly affected by outliers (e.g., mean, range, standard deviation).
Outlier (IQR Rule)
A value more than above or below .
Outlier (SD Rule)
A value more than 2 standard deviations above or below the mean.
Units of Measurement
Changing units of measurement affects the values of calculated statistics.
1.8 - Graphical Representations of Summary Statistics for One Quantitative Variable
Key Terms & Definitions
Five-Number Summary
Min, , Median, , Max.
Boxplot
Graph of the five-number summary.
Back-to-Back Stem-and-Leaf Plot
A stem-and-leaf plot comparing two distributions using a shared stem.
Whiskers
Lines on a boxplot extending from the quartiles to the most extreme non-outlier values.
Mean–Median Relationship
If skewed right, mean > median; if skewed left, mean < median; if symmetric, mean ≈ median.
1.9 - Comparisons of the Distributions for One Quantitative Variable
Key Terms & Definitions
z-score (Standardized Score)
Measures the number of standard deviations a data value falls above or below the mean.
- •Formula:
- •Also: Standardized score uses population parameters explicitly; same concept as z-score.
1.10 - The Investigative Question Revisited and Data Collection
Key Terms & Definitions
Census
Recording information from all items or individuals in a population.
Experiment
Study where researchers assign treatments to units.
Observational Study
Study where treatments are not imposed.
Experimental Unit
The observational unit to which a treatment is assigned.
Subject / Participant
Term for experimental units that are people.
Explanatory Variable / Factor
A variable whose levels are imposed on experimental units in an experiment.
- •Also: Factor is another name for an explanatory variable.
Treatment
The specific level(s) of the explanatory variable assigned to experimental units.
Response Variable
An outcome measured on each experimental unit after treatment.
Prospective Study
An observational study where data are gathered at a point in time and into the future.
Retrospective Study
An observational study where data from the past are gathered.
Survey
An observational study in which data are collected from humans using a standard set of questions.
Random Selection
Using a random mechanism to select observational units from a population; enables generalization to the population.
Generalization
When units are randomly selected, conclusions can be extended to the full population; without random selection, only to similar populations.
Confounding Variable
A variable that provides an alternative explanation for an observed relationship.
- •Observational study: Related to both the explanatory and response variables.
- •Experiment: Levels are unevenly distributed among treatment groups.
1.11 - Random Sampling
Key Terms & Definitions
Simple Random Sample
Every sample of size n has equal chance of selection.
Stratified Sample
SRS taken from each homogeneous group (strata).
Cluster Sample
Clusters selected, then all units in those clusters are surveyed.
Systematic Sample
Random start with fixed periodic interval.
Sampling Without Replacement
A unit can be selected only once; it is not returned to the population before subsequent selections.
Sampling With Replacement
A unit can be selected more than once; it is returned before subsequent selections.
Strata
Non-overlapping, homogeneous groups used in stratified random sampling.
1.12 - Potential Problems with Sampling
Key Terms & Definitions
Bias
Systematic error resulting in consistently skewed statistics.
Undercoverage
Sampling method excludes part of the population.
Nonresponse
Failure to obtain responses from selected individuals.
Voluntary Response Bias
Bias that may occur when a sample consists entirely of volunteers.
Response Bias
Bias when responses tend to differ from the true value in one direction (e.g., leading questions, self-reporting).
Question Wording Bias
A form of response bias caused by confusing or leading survey questions.
Convenience Sample
A nonrandom sampling method that selects units that are easiest to access; introduces potential bias.
1.13 - Experimental Design
Key Terms & Definitions
Random Assignment
Using chance to create comparable treatment groups.
Blocking
Grouping units by extraneous variation to separate it from treatment effects.
Matched Pairs Design
Block design using two treatments with matched pairs or same units.
Control Group
A collection of experimental units used for comparison, sometimes receiving a placebo.
Placebo
An inactive treatment given to a control group.
Placebo Effect
The difference between the average response to a placebo and the average response to no treatment.
Single-Blind and Double-Blind
Methods to reduce bias by hiding treatment assignment from participants and/or researchers.
- •Single-blind: Participants do not know their treatment (or researchers do not, but not both).
- •Double-blind: Neither participants nor interacting researchers know which treatment each participant receives.
Extraneous Variable
A variable known or believed to affect the response that is not an explanatory variable being studied.
Replication
More than one experimental unit is assigned to each treatment in an experiment.
Direct Control
Keeping settings of potential extraneous variables the same across experimental units.
Completely Randomized Design
Experimental design where treatments are assigned to units completely at random.
Randomized Block Design
Units are grouped into blocks by an extraneous variable, then treatments are randomly assigned within each block.
Blocking Variable
A source of extraneous variation used to form blocks in a randomized block design.
Cause and Effect
A conclusion that is only justified when random assignment is used in an experiment.