Home » Mathematics » Statistics » Correlation: r vs. r-squared

Dr. Ji Son

Dr. Ji Son

Correlation: r vs. r-squared

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Table of Contents

Section 1: Introduction
Descriptive Statistics vs. Inferential Statistics

25m 31s

Intro
0:00
Roadmap
0:10
Roadmap
0:11
Statistics
0:35
Statistics
0:36
Let's Think About High School Science
1:12
Measurement and Find Patterns (Mathematical Formula)
1:13
Statistics = Math of Distributions
4:58
Distributions
4:59
Problematic… but also GREAT
5:58
Statistics
7:33
How is It Different from Other Specializations in Mathematics?
7:34
Statistics is Fundamental in Natural and Social Sciences
7:53
Two Skills of Statistics
8:20
Description (Exploration)
8:21
Inference
9:13
Descriptive Statistics vs. Inferential Statistics: Apply to Distributions
9:58
Descriptive Statistics
9:59
Inferential Statistics
11:05
Populations vs. Samples
12:19
Populations vs. Samples: Is it the Truth?
12:20
Populations vs. Samples: Pros & Cons
13:36
Populations vs. Samples: Descriptive Values
16:12
Putting Together Descriptive/Inferential Stats & Populations/Samples
17:10
Putting Together Descriptive/Inferential Stats & Populations/Samples
17:11
Example 1: Descriptive Statistics vs. Inferential Statistics
19:09
Example 2: Descriptive Statistics vs. Inferential Statistics
20:47
Example 3: Sample, Parameter, Population, and Statistic
21:40
Example 4: Sample, Parameter, Population, and Statistic
23:28
Section 2: About Samples: Cases, Variables, Measurements
About Samples: Cases, Variables, Measurements

32m 14s

Intro
0:00
Data
0:09
Data, Cases, Variables, and Values
0:10
Rows, Columns, and Cells
2:03
Example: Aircrafts
3:52
How Do We Get Data?
5:38
Research: Question and Hypothesis
5:39
Research Design
7:11
Measurement
7:29
Research Analysis
8:33
Research Conclusion
9:30
Types of Variables
10:03
Discrete Variables
10:04
Continuous Variables
12:07
Types of Measurements
14:17
Types of Measurements
14:18
Types of Measurements (Scales)
17:22
Nominal
17:23
Ordinal
19:11
Interval
21:33
Ratio
24:24
Example 1: Cases, Variables, Measurements
25:20
Example 2: Which Scale of Measurement is Used?
26:55
Example 3: What Kind of a Scale of Measurement is This?
27:26
Example 4: Discrete vs. Continuous Variables.
30:31
Section 3: Visualizing Distributions
Introduction to Excel

8m 9s

Intro
0:00
Before Visualizing Distribution
0:10
Excel
0:11
Excel: Organization
0:45
Workbook
0:46
Column x Rows
1:50
Tools: Menu Bar, Standard Toolbar, and Formula Bar
3:00
Excel + Data
6:07
Exce and Data
6:08
Frequency Distributions in Excel

39m 10s

Intro
0:00
Roadmap
0:08
Data in Excel and Frequency Distributions
0:09
Raw Data to Frequency Tables
0:42
Raw Data to Frequency Tables
0:43
Frequency Tables: Using Formulas and Pivot Tables
1:28
Example 1: Number of Births
7:17
Example 2: Age Distribution
20:41
Example 3: Height Distribution
27:45
Example 4: Height Distribution of Males
32:19
Frequency Distributions and Features

25m 29s

Intro
0:00
Roadmap
0:10
Data in Excel, Frequency Distributions, and Features of Frequency Distributions
0:11
Example #1
1:35
Uniform
1:36
Example #2
2:58
Unimodal, Skewed Right, and Asymmetric
2:59
Example #3
6:29
Bimodal
6:30
Example #4a
8:29
Symmetric, Unimodal, and Normal
8:30
Point of Inflection and Standard Deviation
11:13
Example #4b
12:43
Normal Distribution
12:44
Summary
13:56
Uniform, Skewed, Bimodal, and Normal
13:57
Sketch Problem 1: Driver's License
17:34
Sketch Problem 2: Life Expectancy
20:01
Sketch Problem 3: Telephone Numbers
22:01
Sketch Problem 4: Length of Time Used to Complete a Final Exam
23:43
Dotplots and Histograms in Excel

42m 42s

Intro
0:00
Roadmap
0:06
Roadmap
0:07
Previously
1:02
Data, Frequency Table, and visualization
1:03
Dotplots
1:22
Dotplots Excel Example
1:23
Dotplots: Pros and Cons
7:22
Pros and Cons of Dotplots
7:23
Dotplots Excel Example Cont.
9:07
Histograms
12:47
Histograms Overview
12:48
Example of Histograms
15:29
Histograms: Pros and Cons
31:39
Pros
31:40
Cons
32:31
Frequency vs. Relative Frequency
32:53
Frequency
32:54
Relative Frequency
33:36
Example 1: Dotplots vs. Histograms
34:36
Example 2: Age of Pennies Dotplot
36:21
Example 3: Histogram of Mammal Speeds
38:27
Example 4: Histogram of Life Expectancy
40:30
Stemplots

12m 23s

Intro
0:00
Roadmap
0:05
Roadmap
0:06
What Sets Stemplots Apart?
0:46
Data Sets, Dotplots, Histograms, and Stemplots
0:47
Example 1: What Do Stemplots Look Like?
1:58
Example 2: Back-to-Back Stemplots
5:00
Example 3: Quiz Grade Stemplot
7:46
Example 4: Quiz Grade & Afterschool Tutoring Stemplot
9:56
Bar Graphs

22m 49s

Intro
0:00
Roadmap
0:05
Roadmap
0:08
Review of Frequency Distributions
0:44
Y-axis and X-axis
0:45
Types of Frequency Visualizations Covered so Far
2:16
Introduction to Bar Graphs
4:07
Example 1: Bar Graph
5:32
Example 1: Bar Graph
5:33
Do Shapes, Center, and Spread of Distributions Apply to Bar Graphs?
11:07
Do Shapes, Center, and Spread of Distributions Apply to Bar Graphs?
11:08
Example 2: Create a Frequency Visualization for Gender
14:02
Example 3: Cases, Variables, and Frequency Visualization
16:34
Example 4: What Kind of Graphs are Shown Below?
19:29
Section 4: Summarizing Distributions
Central Tendency: Mean, Median, Mode

38m 50s

Intro
0:00
Roadmap
0:07
Roadmap
0:08
Central Tendency 1
0:56
Way to Summarize a Distribution of Scores
0:57
Mode
1:32
Median
2:02
Mean
2:36
Central Tendency 2
3:47
Mode
3:48
Median
4:20
Mean
5:25
Summation Symbol
6:11
Summation Symbol
6:12
Population vs. Sample
10:46
Population vs. Sample
10:47
Excel Examples
15:08
Finding Mode, Median, and Mean in Excel
15:09
Median vs. Mean
21:45
Effect of Outliers
21:46
Relationship Between Parameter and Statistic
22:44
Type of Measurements
24:00
Which Distributions to Use With
24:55
Example 1: Mean
25:30
Example 2: Using Summation Symbol
29:50
Example 3: Average Calorie Count
32:50
Example 4: Creating an Example Set
35:46
Variability

42m 40s

Intro
0:00
Roadmap
0:05
Roadmap
0:06
Variability (or Spread)
0:45
Variability (or Spread)
0:46
Things to Think About
5:45
Things to Think About
5:46
Range, Quartiles and Interquartile Range
6:37
Range
6:38
Interquartile Range
8:42
Interquartile Range Example
10:58
Interquartile Range Example
10:59
Variance and Standard Deviation
12:27
Deviations
12:28
Sum of Squares
14:35
Variance
16:55
Standard Deviation
17:44
Sum of Squares (SS)
18:34
Sum of Squares (SS)
18:35
Population vs. Sample SD
22:00
Population vs. Sample SD
22:01
Population vs. Sample
23:20
Mean
23:21
SD
23:51
Example 1: Find the Mean and Standard Deviation of the Variable Friends in the Excel File
27:21
Example 2: Find the Mean and Standard Deviation of the Tagged Photos in the Excel File
35:25
Example 3: Sum of Squares
38:58
Example 4: Standard Deviation
41:48
Five Number Summary & Boxplots

57m 15s

Intro
0:00
Roadmap
0:06
Roadmap
0:07
Summarizing Distributions
0:37
Shape, Center, and Spread
0:38
5 Number Summary
1:14
Boxplot: Visualizing 5 Number Summary
3:37
Boxplot: Visualizing 5 Number Summary
3:38
Boxplots on Excel
9:01
Using 'Stocks' and Using Stacked Columns
9:02
Boxplots on Excel Example
10:14
When are Boxplots Useful?
32:14
Pros
32:15
Cons
32:59
How to Determine Outlier Status
33:24
Rule of Thumb: Upper Limit
33:25
Rule of Thumb: Lower Limit
34:16
Signal Outliers in an Excel Data File Using Conditional Formatting
34:52
Modified Boxplot
48:38
Modified Boxplot
48:39
Example 1: Percentage Values & Lower and Upper Whisker
49:10
Example 2: Boxplot
50:10
Example 3: Estimating IQR From Boxplot
53:46
Example 4: Boxplot and Missing Whisker
54:35
Shape: Calculating Skewness & Kurtosis

41m 51s

Intro
0:00
Roadmap
0:16
Roadmap
0:17
Skewness Concept
1:09
Skewness Concept
1:10
Calculating Skewness
3:26
Calculating Skewness
3:27
Interpreting Skewness
7:36
Interpreting Skewness
7:37
Excel Example
8:49
Kurtosis Concept
20:29
Kurtosis Concept
20:30
Calculating Kurtosis
24:17
Calculating Kurtosis
24:18
Interpreting Kurtosis
29:01
Leptokurtic
29:35
Mesokurtic
30:10
Platykurtic
31:06
Excel Example
32:04
Example 1: Shape of Distribution
38:28
Example 2: Shape of Distribution
39:29
Example 3: Shape of Distribution
40:14
Example 4: Kurtosis
41:10
Normal Distribution

34m 33s

Intro
0:00
Roadmap
0:13
Roadmap
0:14
What is a Normal Distribution
0:44
The Normal Distribution As a Theoretical Model
0:45
Possible Range of Probabilities
3:05
Possible Range of Probabilities
3:06
What is a Normal Distribution
5:07
Can Be Described By
5:08
Properties
5:49
'Same' Shape: Illusion of Different Shape!
7:35
'Same' Shape: Illusion of Different Shape!
7:36
Types of Problems
13:45
Example: Distribution of SAT Scores
13:46
Shape Analogy
19:48
Shape Analogy
19:49
Example 1: The Standard Normal Distribution and Z-Scores
22:34
Example 2: The Standard Normal Distribution and Z-Scores
25:54
Example 3: Sketching and Normal Distribution
28:55
Example 4: Sketching and Normal Distribution
32:32
Standard Normal Distributions & Z-Scores

41m 44s

Intro
0:00
Roadmap
0:06
Roadmap
0:07
A Family of Distributions
0:28
Infinite Set of Distributions
0:29
Transforming Normal Distributions to 'Standard' Normal Distribution
1:04
Normal Distribution vs. Standard Normal Distribution
2:58
Normal Distribution vs. Standard Normal Distribution
2:59
Z-Score, Raw Score, Mean, & SD
4:08
Z-Score, Raw Score, Mean, & SD
4:09
Weird Z-Scores
9:40
Weird Z-Scores
9:41
Excel
16:45
For Normal Distributions
16:46
For Standard Normal Distributions
19:11
Excel Example
20:24
Types of Problems
25:18
Percentage Problem: P(x)
25:19
Raw Score and Z-Score Problems
26:28
Standard Deviation Problems
27:01
Shape Analogy
27:44
Shape Analogy
27:45
Example 1: Deaths Due to Heart Disease vs. Deaths Due to Cancer
28:24
Example 2: Heights of Male College Students
33:15
Example 3: Mean and Standard Deviation
37:14
Example 4: Finding Percentage of Values in a Standard Normal Distribution
37:49
Normal Distribution: PDF vs. CDF

55m 44s

Intro
0:00
Roadmap
0:15
Roadmap
0:16
Frequency vs. Cumulative Frequency
0:56
Frequency vs. Cumulative Frequency
0:57
Frequency vs. Cumulative Frequency
4:32
Frequency vs. Cumulative Frequency Cont.
4:33
Calculus in Brief
6:21
Derivative-Integral Continuum
6:22
PDF
10:08
PDF for Standard Normal Distribution
10:09
PDF for Normal Distribution
14:32
Integral of PDF = CDF
21:27
Integral of PDF = CDF
21:28
Example 1: Cumulative Frequency Graph
23:31
Example 2: Mean, Standard Deviation, and Probability
24:43
Example 3: Mean and Standard Deviation
35:50
Example 4: Age of Cars
49:32
Section 5: Linear Regression
Scatterplots

47m 19s

Intro
0:00
Roadmap
0:04
Roadmap
0:05
Previous Visualizations
0:30
Frequency Distributions
0:31
Compare & Contrast
2:26
Frequency Distributions Vs. Scatterplots
2:27
Summary Values
4:53
Shape
4:54
Center & Trend
6:41
Spread & Strength
8:22
Univariate & Bivariate
10:25
Example Scatterplot
10:48
Shape, Trend, and Strength
10:49
Positive and Negative Association
14:05
Positive and Negative Association
14:06
Linearity, Strength, and Consistency
18:30
Linearity
18:31
Strength
19:14
Consistency
20:40
Summarizing a Scatterplot
22:58
Summarizing a Scatterplot
22:59
Example 1: Gapminder.org, Income x Life Expectancy
26:32
Example 2: Gapminder.org, Income x Infant Mortality
36:12
Example 3: Trend and Strength of Variables
40:14
Example 4: Trend, Strength and Shape for Scatterplots
43:27
Regression

32m 2s

Intro
0:00
Roadmap
0:05
Roadmap
0:06
Linear Equations
0:34
Linear Equations: y = mx + b
0:35
Rough Line
5:16
Rough Line
5:17
Regression - A 'Center' Line
7:41
Reasons for Summarizing with a Regression Line
7:42
Predictor and Response Variable
10:04
Goal of Regression
12:29
Goal of Regression
12:30
Prediction
14:50
Example: Servings of Mile Per Year Shown By Age
14:51
Intrapolation
17:06
Extrapolation
17:58
Error in Prediction
20:34
Prediction Error
20:35
Residual
21:40
Example 1: Residual
23:34
Example 2: Large and Negative Residual
26:30
Example 3: Positive Residual
28:13
Example 4: Interpret Regression Line & Extrapolate
29:40
Least Squares Regression

56m 36s

Intro
0:00
Roadmap
0:13
Roadmap
0:14
Best Fit
0:47
Best Fit
0:48
Sum of Squared Errors (SSE)
1:50
Sum of Squared Errors (SSE)
1:51
Why Squared?
3:38
Why Squared?
3:39
Quantitative Properties of Regression Line
4:51
Quantitative Properties of Regression Line
4:52
So How do we Find Such a Line?
6:49
SSEs of Different Line Equations & Lowest SSE
6:50
Carl Gauss' Method
8:01
How Do We Find Slope (b1)
11:00
How Do We Find Slope (b1)
11:01
Hoe Do We Find Intercept
15:11
Hoe Do We Find Intercept
15:12
Example 1: Which of These Equations Fit the Above Data Best?
17:18
Example 2: Find the Regression Line for These Data Points and Interpret It
26:31
Example 3: Summarize the Scatterplot and Find the Regression Line.
34:31
Example 4: Examine the Mean of Residuals
43:52
Correlation

43m 58s

Intro
0:00
Roadmap
0:05
Roadmap
0:06
Summarizing a Scatterplot Quantitatively
0:47
Shape
0:48
Trend
1:11
Strength: Correlation ®
1:45
Correlation Coefficient ( r )
2:30
Correlation Coefficient ( r )
2:31
Trees vs. Forest
11:59
Trees vs. Forest
12:00
Calculating r
15:07
Average Product of z-scores for x and y
15:08
Relationship between Correlation and Slope
21:10
Relationship between Correlation and Slope
21:11
Example 1: Find the Correlation between Grams of Fat and Cost
24:11
Example 2: Relationship between r and b1
30:24
Example 3: Find the Regression Line
33:35
Example 4: Find the Correlation Coefficient for this Set of Data
37:37
Correlation: r vs. r-squared

52m 52s

Intro
0:00
Roadmap
0:07
Roadmap
0:08
R-squared
0:44
What is the Meaning of It? Why Squared?
0:45
Parsing Sum of Squared (Parsing Variability)
2:25
SST = SSR + SSE
2:26
What is SST and SSE?
7:46
What is SST and SSE?
7:47
r-squared
18:33
Coefficient of Determination
18:34
If the Correlation is Strong…
20:25
If the Correlation is Strong…
20:26
If the Correlation is Weak…
22:36
If the Correlation is Weak…
22:37
Example 1: Find r-squared for this Set of Data
23:56
Example 2: What Does it Mean that the Simple Linear Regression is a 'Model' of Variance?
33:54
Example 3: Why Does r-squared Only Range from 0 to 1
37:29
Example 4: Find the r-squared for This Set of Data
39:55
Transformations of Data

27m 8s

Intro
0:00
Roadmap
0:05
Roadmap
0:06
Why Transform?
0:26
Why Transform?
0:27
Shape-preserving vs. Shape-changing Transformations
5:14
Shape-preserving = Linear Transformations
5:15
Shape-changing Transformations = Non-linear Transformations
6:20
Common Shape-Preserving Transformations
7:08
Common Shape-Preserving Transformations
7:09
Common Shape-Changing Transformations
8:59
Powers
9:00
Logarithms
9:39
Change Just One Variable? Both?
10:38
Log-log Transformations
10:39
Log Transformations
14:38
Example 1: Create, Graph, and Transform the Data Set
15:19
Example 2: Create, Graph, and Transform the Data Set
20:08
Example 3: What Kind of Model would You Choose for this Data?
22:44
Example 4: Transformation of Data
25:46
Section 6: Collecting Data in an Experiment
Sampling & Bias

54m 44s

Intro
0:00
Roadmap
0:05
Roadmap
0:06
Descriptive vs. Inferential Statistics
1:04
Descriptive Statistics: Data Exploration
1:05
Example
2:03
To tackle Generalization…
4:31
Generalization
4:32
Sampling
6:06
'Good' Sample
6:40
Defining Samples and Populations
8:55
Population
8:56
Sample
11:16
Why Use Sampling?
13:09
Why Use Sampling?
13:10
Goal of Sampling: Avoiding Bias
15:04
What is Bias?
15:05
Where does Bias Come from: Sampling Bias
17:53
Where does Bias Come from: Response Bias
18:27
Sampling Bias: Bias from Bas Sampling Methods
19:34
Size Bias
19:35
Voluntary Response Bias
21:13
Convenience Sample
22:22
Judgment Sample
23:58
Inadequate Sample Frame
25:40
Response Bias: Bias from 'Bad' Data Collection Methods
28:00
Nonresponse Bias
29:31
Questionnaire Bias
31:10
Incorrect Response or Measurement Bias
37:32
Example 1: What Kind of Biases?
40:29
Example 2: What Biases Might Arise?
44:46
Example 3: What Kind of Biases?
48:34
Example 4: What Kind of Biases?
51:43
Sampling Methods

14m 25s

Intro
0:00
Roadmap
0:05
Roadmap
0:06
Biased vs. Unbiased Sampling Methods
0:32
Biased Sampling
0:33
Unbiased Sampling
1:13
Probability Sampling Methods
2:31
Simple Random
2:54
Stratified Random Sampling
4:06
Cluster Sampling
5:24
Two-staged Sampling
6:22
Systematic Sampling
7:25
Example 1: Which Type(s) of Sampling was this?
8:33
Example 2: Describe How to Take a Two-Stage Sample from this Book
10:16
Example 3: Sampling Methods
11:58
Example 4: Cluster Sample Plan
12:48
Research Design

53m 54s

Intro
0:00
Roadmap
0:06
Roadmap
0:07
Descriptive vs. Inferential Statistics
0:51
Descriptive Statistics: Data Exploration
0:52
Inferential Statistics
1:02
Variables and Relationships
1:44
Variables
1:45
Relationships
2:49
Not Every Type of Study is an Experiment…
4:16
Category I - Descriptive Study
4:54
Category II - Correlational Study
5:50
Category III - Experimental, Quasi-experimental, Non-experimental
6:33
Category III
7:42
Experimental, Quasi-experimental, and Non-experimental
7:43
Why CAN'T the Other Strategies Determine Causation?
10:18
Third-variable Problem
10:19
Directionality Problem
15:49
What Makes Experiments Special?
17:54
Manipulation
17:55
Control (and Comparison)
21:58
Methods of Control
26:38
Holding Constant
26:39
Matching
29:11
Random Assignment
31:48
Experiment Terminology
34:09
'true' Experiment vs. Study
34:10
Independent Variable (IV)
35:16
Dependent Variable (DV)
35:45
Factors
36:07
Treatment Conditions
36:23
Levels
37:43
Confounds or Extraneous Variables
38:04
Blind
38:38
Blind Experiments
38:39
Double-blind Experiments
39:29
How Categories Relate to Statistics
41:35
Category I - Descriptive Study
41:36
Category II - Correlational Study
42:05
Category III - Experimental, Quasi-experimental, Non-experimental
42:43
Example 1: Research Design
43:50
Example 2: Research Design
47:37
Example 3: Research Design
50:12
Example 4: Research Design
52:00
Between and Within Treatment Variability

41m 31s

Intro
0:00
Roadmap
0:06
Roadmap
0:07
Experimental Designs
0:51
Experimental Designs: Manipulation & Control
0:52
Two Types of Variability
2:09
Between Treatment Variability
2:10
Within Treatment Variability
3:31
Updated Goal of Experimental Design
5:47
Updated Goal of Experimental Design
5:48
Example: Drugs and Driving
6:56
Example: Drugs and Driving
6:57
Different Types of Random Assignment
11:27
All Experiments
11:28
Completely Random Design
12:02
Randomized Block Design
13:19
Randomized Block Design
15:48
Matched Pairs Design
15:49
Repeated Measures Design
19:47
Between-subject Variable vs. Within-subject Variable
22:43
Completely Randomized Design
22:44
Repeated Measures Design
25:03
Example 1: Design a Completely Random, Matched Pair, and Repeated Measures Experiment
26:16
Example 2: Block Design
31:41
Example 3: Completely Randomized Designs
35:11
Example 4: Completely Random, Matched Pairs, or Repeated Measures Experiments?
39:01
Section 7: Review of Probability Axioms
Sample Spaces

37m 52s

Intro
0:00
Roadmap
0:07
Roadmap
0:08
Why is Probability Involved in Statistics
0:48
Probability
0:49
Can People Tell the Difference between Cheap and Gourmet Coffee?
2:08
Taste Test with Coffee Drinkers
3:37
If No One can Actually Taste the Difference
3:38
If Everyone can Actually Taste the Difference
5:36
Creating a Probability Model
7:09
Creating a Probability Model
7:10
D'Alembert vs. Necker
9:41
D'Alembert vs. Necker
9:42
Problem with D'Alembert's Model
13:29
Problem with D'Alembert's Model
13:30
Covering Entire Sample Space
15:08
Fundamental Principle of Counting
15:09
Where Do Probabilities Come From?
22:54
Observed Data, Symmetry, and Subjective Estimates
22:55
Checking whether Model Matches Real World
24:27
Law of Large Numbers
24:28
Example 1: Law of Large Numbers
27:46
Example 2: Possible Outcomes
30:43
Example 3: Brands of Coffee and Taste
33:25
Example 4: How Many Different Treatments are there?
35:33
Addition Rule for Disjoint Events

20m 29s

Intro
0:00
Roadmap
0:08
Roadmap
0:09
Disjoint Events
0:41
Disjoint Events
0:42
Meaning of 'or'
2:39
In Regular Life
2:40
In Math/Statistics/Computer Science
3:10
Addition Rule for Disjoin Events
3:55
If A and B are Disjoint: P (A and B)
3:56
If A and B are Disjoint: P (A or B)
5:15
General Addition Rule
5:41
General Addition Rule
5:42
Generalized Addition Rule
8:31
If A and B are not Disjoint: P (A or B)
8:32
Example 1: Which of These are Mutually Exclusive?
10:50
Example 2: What is the Probability that You will Have a Combination of One Heads and Two Tails?
12:57
Example 3: Engagement Party
15:17
Example 4: Home Owner's Insurance
18:30
Conditional Probability

57m 19s

Intro
0:00
Roadmap
0:05
Roadmap
0:06
'or' vs. 'and' vs. Conditional Probability
1:07
'or' vs. 'and' vs. Conditional Probability
1:08
'and' vs. Conditional Probability
5:57
P (M or L)
5:58
P (M and L)
8:41
P (M|L)
11:04
P (L|M)
12:24
Tree Diagram
15:02
Tree Diagram
15:03
Defining Conditional Probability
22:42
Defining Conditional Probability
22:43
Common Contexts for Conditional Probability
30:56
Medical Testing: Positive Predictive Value
30:57
Medical Testing: Sensitivity
33:03
Statistical Tests
34:27
Example 1: Drug and Disease
36:41
Example 2: Marbles and Conditional Probability
40:04
Example 3: Cards and Conditional Probability
45:59
Example 4: Votes and Conditional Probability
50:21
Independent Events

24m 27s

Intro
0:00
Roadmap
0:05
Roadmap
0:06
Independent Events & Conditional Probability
0:26
Non-independent Events
0:27
Independent Events
2:00
Non-independent and Independent Events
3:08
Non-independent and Independent Events
3:09
Defining Independent Events
5:52
Defining Independent Events
5:53
Multiplication Rule
7:29
Previously…
7:30
But with Independent Evens
8:53
Example 1: Which of These Pairs of Events are Independent?
11:12
Example 2: Health Insurance and Probability
15:12
Example 3: Independent Events
17:42
Example 4: Independent Events
20:03
Section 8: Probability Distributions
Introduction to Probability Distributions

56m 45s

Intro
0:00
Roadmap
0:08
Roadmap
0:09
Sampling vs. Probability
0:57
Sampling
0:58
Missing
1:30
What is Missing?
3:06
Insight: Probability Distributions
5:26
Insight: Probability Distributions
5:27
What is a Probability Distribution?
7:29
From Sample Spaces to Probability Distributions
8:44
Sample Space
8:45
Probability Distribution of the Sum of Two Die
11:16
The Random Variable
17:43
The Random Variable
17:44
Expected Value
21:52
Expected Value
21:53
Example 1: Probability Distributions
28:45
Example 2: Probability Distributions
35:30
Example 3: Probability Distributions
43:37
Example 4: Probability Distributions
47:20
Expected Value & Variance of Probability Distributions

53m 41s

Intro
0:00
Roadmap
0:06
Roadmap
0:07
Discrete vs. Continuous Random Variables
1:04
Discrete vs. Continuous Random Variables
1:05
Mean and Variance Review
4:44
Mean: Sample, Population, and Probability Distribution
4:45
Variance: Sample, Population, and Probability Distribution
9:12
Example Situation
14:10
Example Situation
14:11
Some Special Cases…
16:13
Some Special Cases…
16:14
Linear Transformations
19:22
Linear Transformations
19:23
What Happens to Mean and Variance of the Probability Distribution?
20:12
n Independent Values of X
25:38
n Independent Values of X
25:39
Compare These Two Situations
30:56
Compare These Two Situations
30:57
Two Random Variables, X and Y
32:02
Two Random Variables, X and Y
32:03
Example 1: Expected Value & Variance of Probability Distributions
35:35
Example 2: Expected Values & Standard Deviation
44:17
Example 3: Expected Winnings and Standard Deviation
48:18
Binomial Distribution

55m 15s

Intro
0:00
Roadmap
0:05
Roadmap
0:06
Discrete Probability Distributions
1:42
Discrete Probability Distributions
1:43
Binomial Distribution
2:36
Binomial Distribution
2:37
Multiplicative Rule Review
6:54
Multiplicative Rule Review
6:55
How Many Outcomes with k 'Successes'
10:23
Adults and Bachelor's Degree: Manual List of Outcomes
10:24
P (X=k)
19:37
Putting Together # of Outcomes with the Multiplicative Rule
19:38
Expected Value and Standard Deviation in a Binomial Distribution
25:22
Expected Value and Standard Deviation in a Binomial Distribution
25:23
Example 1: Coin Toss
33:42
Example 2: College Graduates
38:03
Example 3: Types of Blood and Probability
45:39
Example 4: Expected Number and Standard Deviation
51:11
Section 9: Sampling Distributions of Statistics
Introduction to Sampling Distributions

48m 17s

Intro
0:00
Roadmap
0:08
Roadmap
0:09
Probability Distributions vs. Sampling Distributions
0:55
Probability Distributions vs. Sampling Distributions
0:56
Same Logic
3:55
Logic of Probability Distribution
3:56
Example: Rolling Two Die
6:56
Simulating Samples
9:53
To Come Up with Probability Distributions
9:54
In Sampling Distributions
11:12
Connecting Sampling and Research Methods with Sampling Distributions
12:11
Connecting Sampling and Research Methods with Sampling Distributions
12:12
Simulating a Sampling Distribution
14:14
Experimental Design: Regular Sleep vs. Less Sleep
14:15
Logic of Sampling Distributions
23:08
Logic of Sampling Distributions
23:09
General Method of Simulating Sampling Distributions
25:38
General Method of Simulating Sampling Distributions
25:39
Questions that Remain
28:45
Questions that Remain
28:46
Example 1: Mean and Standard Error of Sampling Distribution
30:57
Example 2: What is the Best Way to Describe Sampling Distributions?
37:12
Example 3: Matching Sampling Distributions
38:21
Example 4: Mean and Standard Error of Sampling Distribution
41:51
Sampling Distribution of the Mean

1h 8m 48s

Intro
0:00
Roadmap
0:05
Roadmap
0:06
Special Case of General Method for Simulating a Sampling Distribution
1:53
Special Case of General Method for Simulating a Sampling Distribution
1:54
Computer Simulation
3:43
Using Simulations to See Principles behind Shape of SDoM
15:50
Using Simulations to See Principles behind Shape of SDoM
15:51
Conditions
17:38
Using Simulations to See Principles behind Center (Mean) of SDoM
20:15
Using Simulations to See Principles behind Center (Mean) of SDoM
20:16
Conditions: Does n Matter?
21:31
Conditions: Does Number of Simulation Matter?
24:37
Using Simulations to See Principles behind Standard Deviation of SDoM
27:13
Using Simulations to See Principles behind Standard Deviation of SDoM
27:14
Conditions: Does n Matter?
34:45
Conditions: Does Number of Simulation Matter?
36:24
Central Limit Theorem
37:13
SHAPE
38:08
CENTER
39:34
SPREAD
39:52
Comparing Population, Sample, and SDoM
43:10
Comparing Population, Sample, and SDoM
43:11
Answering the 'Questions that Remain'
48:24
What Happens When We Don't Know What the Population Looks Like?
48:25
Can We Have Sampling Distributions for Summary Statistics Other than the Mean?
49:42
How Do We Know whether a Sample is Sufficiently Unlikely?
53:36
Do We Always Have to Simulate a Large Number of Samples in Order to get a Sampling Distribution?
54:40
Example 1: Mean Batting Average
55:25
Example 2: Mean Sampling Distribution and Standard Error
59:07
Example 3: Sampling Distribution of the Mean
1:01:04
Sampling Distribution of Sample Proportions

54m 37s

Intro
0:00
Roadmap
0:06
Roadmap
0:07
Intro to Sampling Distribution of Sample Proportions (SDoSP)
0:51
Categorical Data (Examples)
0:52
Wish to Estimate Proportion of Population from Sample…
2:00
Notation
3:34
Population Proportion and Sample Proportion Notations
3:35
What's the Difference?
9:19
SDoM vs. SDoSP: Type of Data
9:20
SDoM vs. SDoSP: Shape
11:24
SDoM vs. SDoSP: Center
12:30
SDoM vs. SDoSP: Spread
15:34
Binomial Distribution vs. Sampling Distribution of Sample Proportions
19:14
Binomial Distribution vs. SDoSP: Type of Data
19:17
Binomial Distribution vs. SDoSP: Shape
21:07
Binomial Distribution vs. SDoSP: Center
21:43
Binomial Distribution vs. SDoSP: Spread
24:08
Example 1: Sampling Distribution of Sample Proportions
26:07
Example 2: Sampling Distribution of Sample Proportions
37:58
Example 3: Sampling Distribution of Sample Proportions
44:42
Example 4: Sampling Distribution of Sample Proportions
45:57
Section 10: Inferential Statistics
Introduction to Confidence Intervals

42m 53s

Intro
0:00
Roadmap
0:06
Roadmap
0:07
Inferential Statistics
0:50
Inferential Statistics
0:51
Two Problems with This Picture…
3:20
Two Problems with This Picture…
3:21
Solution: Confidence Intervals (CI)
4:59
Solution: Hypotheiss Testing (HT)
5:49
Which Parameters are Known?
6:45
Which Parameters are Known?
6:46
Confidence Interval - Goal
7:56
When We Don't Know m but know s
7:57
When We Don't Know
18:27
When We Don't Know m nor s
18:28
Example 1: Confidence Intervals
26:18
Example 2: Confidence Intervals
29:46
Example 3: Confidence Intervals
32:18
Example 4: Confidence Intervals
38:31
t Distributions

1h 2m 6s

Intro
0:00
Roadmap
0:04
Roadmap
0:05
When to Use z vs. t?
1:07
When to Use z vs. t?
1:08
What is z and t?
3:02
z-score and t-score: Commonality
3:03
z-score and t-score: Formulas
3:34
z-score and t-score: Difference
5:22
Why not z? (Why t?)
7:24
Why not z? (Why t?)
7:25
But Don't Worry!
15:13
Gossett and t-distributions
15:14
Rules of t Distributions
17:05
t-distributions are More Normal as n Gets Bigger
17:06
t-distributions are a Family of Distributions
18:55
Degrees of Freedom (df)
20:02
Degrees of Freedom (df)
20:03
t Family of Distributions
24:07
t Family of Distributions : df = 2 , 4, and 60
24:08
df = 60
29:16
df = 2
29:59
How to Find It?
31:01
'Student's t-distribution' or 't-distribution'
31:02
Excel Example
33:06
Example 1: Which Distribution Do You Use? Z or t?
45:26
Example 2: Friends on Facebook
47:41
Example 3: t Distributions
52:15
Example 4: t Distributions , confidence interval, and mean
55:59
Introduction to Hypothesis Testing

1h 6m 33s

Intro
0:00
Roadmap
0:06
Roadmap
0:07
Issues to Overcome in Inferential Statistics
1:35
Issues to Overcome in Inferential Statistics
1:36
What Happens When We Don't Know What the Population Looks Like?
2:57
How Do We Know whether a sample is Sufficiently Unlikely
3:43
Hypothesizing a Population
6:44
Hypothesizing a Population
6:45
Null Hypothesis
8:07
Alternative Hypothesis
8:56
Hypotheses
11:58
Hypotheses
11:59
Errors in Hypothesis Testing
14:22
Errors in Hypothesis Testing
14:23
Steps of Hypothesis Testing
21:15
Steps of Hypothesis Testing
21:16
Single Sample HT ( When Sigma Available)
26:08
Example: Average Facebook Friends
26:09
Step1
27:08
Step 2
27:58
Step 3
28:17
Step 4
32:18
Single Sample HT (When Sigma Not Available)
36:33
Example: Average Facebook Friends
36:34
Step1: Hypothesis Testing
36:58
Step 2: Significance Level
37:25
Step 3: Decision Stage
37:40
Step 4: Sample
41:36
Sigma and p-value
45:04
Sigma and p-value
45:05
On tailed vs. Two Tailed Hypotheses
45:51
Example 1: Hypothesis Testing
48:37
Example 2: Heights of Women in the US
57:43
Example 3: Select the Best Way to Complete This Sentence
1:03:23
Confidence Intervals for the Difference of Two Independent Means

55m 14s

Intro
0:00
Roadmap
0:14
Roadmap
0:15
One Mean vs. Two Means
1:17
One Mean vs. Two Means
1:18
Notation
2:41
A Sample! A Set!
2:42
Mean of X, Mean of Y, and Difference of Two Means
3:56
SE of X
4:34
SE of Y
6:28
Sampling Distribution of the Difference between Two Means (SDoD)
7:48
Sampling Distribution of the Difference between Two Means (SDoD)
7:49
Rules of the SDoD (similar to CLT!)
15:00
Mean for the SDoD Null Hypothesis
15:01
Standard Error
17:39
When can We Construct a CI for the Difference between Two Means?
21:28
Three Conditions
21:29
Finding CI
23:56
One Mean CI
23:57
Two Means CI
25:45
Finding t
29:16
Finding t
29:17
Interpreting CI
30:25
Interpreting CI
30:26
Better Estimate of s (s pool)
34:15
Better Estimate of s (s pool)
34:16
Example 1: Confidence Intervals
42:32
Example 2: SE of the Difference
52:36
Hypothesis Testing for the Difference of Two Independent Means

50m

Intro
0:00
Roadmap
0:06
Roadmap
0:07
The Goal of Hypothesis Testing
0:56
One Sample and Two Samples
0:57
Sampling Distribution of the Difference between Two Means (SDoD)
3:42
Sampling Distribution of the Difference between Two Means (SDoD)
3:43
Rules of the SDoD (Similar to CLT!)
6:46
Shape
6:47
Mean for the Null Hypothesis
7:26
Standard Error for Independent Samples (When Variance is Homogenous)
8:18
Standard Error for Independent Samples (When Variance is not Homogenous)
9:25
Same Conditions for HT as for CI
10:08
Three Conditions
10:09
Steps of Hypothesis Testing
11:04
Steps of Hypothesis Testing
11:05
Formulas that Go with Steps of Hypothesis Testing
13:21
Step 1
13:25
Step 2
14:18
Step 3
15:00
Step 4
16:57
Example 1: Hypothesis Testing for the Difference of Two Independent Means
18:47
Example 2: Hypothesis Testing for the Difference of Two Independent Means
33:55
Example 3: Hypothesis Testing for the Difference of Two Independent Means
44:22
Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means

1h 14m 11s

Intro
0:00
Roadmap
0:09
Roadmap
0:10
The Goal of Hypothesis Testing
1:27
One Sample and Two Samples
1:28
Independent Samples vs. Paired Samples
3:16
Independent Samples vs. Paired Samples
3:17
Which is Which?
5:20
Independent SAMPLES vs. Independent VARIABLES
7:43
independent SAMPLES vs. Independent VARIABLES
7:44
T-tests Always…
10:48
T-tests Always…
10:49
Notation for Paired Samples
12:59
Notation for Paired Samples
13:00
Steps of Hypothesis Testing for Paired Samples
16:13
Steps of Hypothesis Testing for Paired Samples
16:14
Rules of the SDoD (Adding on Paired Samples)
18:03
Shape
18:04
Mean for the Null Hypothesis
18:31
Standard Error for Independent Samples (When Variance is Homogenous)
19:25
Standard Error for Paired Samples
20:39
Formulas that go with Steps of Hypothesis Testing
22:59
Formulas that go with Steps of Hypothesis Testing
23:00
Confidence Intervals for Paired Samples
30:32
Confidence Intervals for Paired Samples
30:33
Example 1: Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means
32:28
Example 2: Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means
44:02
Example 3: Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means
52:23
Type I and Type II Errors

31m 27s

Intro
0:00
Roadmap
0:18
Roadmap
0:19
Errors and Relationship to HT and the Sample Statistic?
1:11
Errors and Relationship to HT and the Sample Statistic?
1:12
Instead of a Box…Distributions!
7:00
One Sample t-test: Friends on Facebook
7:01
Two Sample t-test: Friends on Facebook
13:46
Usually, Lots of Overlap between Null and Alternative Distributions
16:59
Overlap between Null and Alternative Distributions
17:00
How Distributions and 'Box' Fit Together
22:45
How Distributions and 'Box' Fit Together
22:46
Example 1: Types of Errors
25:54
Example 2: Types of Errors
27:30
Example 3: What is the Danger of the Type I Error?
29:38
Effect Size & Power

44m 41s

Intro
0:00
Roadmap
0:05
Roadmap
0:06
Distance between Distributions: Sample t
0:49
Distance between Distributions: Sample t
0:50
Problem with Distance in Terms of Standard Error
2:56
Problem with Distance in Terms of Standard Error
2:57
Test Statistic (t) vs. Effect Size (d or g)
4:38
Test Statistic (t) vs. Effect Size (d or g)
4:39
Rules of Effect Size
6:09
Rules of Effect Size
6:10
Why Do We Need Effect Size?
8:21
Tells You the Practical Significance
8:22
HT can be Deceiving…
10:25
Important Note
10:42
What is Power?
11:20
What is Power?
11:21
Why Do We Need Power?
14:19
Conditional Probability and Power
14:20
Power is:
16:27
Can We Calculate Power?
19:00
Can We Calculate Power?
19:01
How Does Alpha Affect Power?
20:36
How Does Alpha Affect Power?
20:37
How Does Effect Size Affect Power?
25:38
How Does Effect Size Affect Power?
25:39
How Does Variability and Sample Size Affect Power?
27:56
How Does Variability and Sample Size Affect Power?
27:57
How Do We Increase Power?
32:47
Increasing Power
32:48
Example 1: Effect Size & Power
35:40
Example 2: Effect Size & Power
37:38
Example 3: Effect Size & Power
40:55
Section 11: Analysis of Variance
F-distributions

24m 46s

Intro
0:00
Roadmap
0:04
Roadmap
0:05
Z- & T-statistic and Their Distribution
0:34
Z- & T-statistic and Their Distribution
0:35
F-statistic
4:55
The F Ration ( the Variance Ratio)
4:56
F-distribution
12:29
F-distribution
12:30
s and p-value
15:00
s and p-value
15:01
Example 1: Why Does F-distribution Stop At 0 But Go On Until Infinity?
18:33
Example 2: F-distributions
19:29
Example 3: F-distributions and Heights
21:29
ANOVA with Independent Samples

1h 9m 25s

Intro
0:00
Roadmap
0:05
Roadmap
0:06
The Limitations of t-tests
1:12
The Limitations of t-tests
1:13
Two Major Limitations of Many t-tests
3:26
Two Major Limitations of Many t-tests
3:27
Ronald Fisher's Solution… F-test! New Null Hypothesis
4:43
Ronald Fisher's Solution… F-test! New Null Hypothesis (Omnibus Test - One Test to Rule Them All!)
4:44
Analysis of Variance (ANoVA) Notation
7:47
Analysis of Variance (ANoVA) Notation
7:48
Partitioning (Analyzing) Variance
9:58
Total Variance
9:59
Within-group Variation
14:00
Between-group Variation
16:22
Time out: Review Variance & SS
17:05
Time out: Review Variance & SS
17:06
F-statistic
19:22
The F Ratio (the Variance Ratio)
19:23
S²bet = SSbet / dfbet
22:13
What is This?
22:14
How Many Means?
23:20
So What is the dfbet?
23:38
So What is SSbet?
24:15
S²w = SSw / dfw
26:05
What is This?
26:06
How Many Means?
27:20
So What is the dfw?
27:36
So What is SSw?
28:18
Chart of Independent Samples ANOVA
29:25
Chart of Independent Samples ANOVA
29:26
Example 1: Who Uploads More Photos: Unknown Ethnicity, Latino, Asian, Black, or White Facebook Users?
35:52
Hypotheses
35:53
Significance Level
39:40
Decision Stage
40:05
Calculate Samples' Statistic and p-Value
44:10
Reject or Fail to Reject H0
55:54
Example 2: ANOVA with Independent Samples
58:21
Repeated Measures ANOVA

1h 15m 13s

Intro
0:00
Roadmap
0:05
Roadmap
0:06
The Limitations of t-tests
0:36
Who Uploads more Pictures and Which Photo-Type is Most Frequently Used on Facebook?
0:37
ANOVA (F-test) to the Rescue!
5:49
Omnibus Hypothesis
5:50
Analyze Variance
7:27
Independent Samples vs. Repeated Measures
9:12
Same Start
9:13
Independent Samples ANOVA
10:43
Repeated Measures ANOVA
12:00
Independent Samples ANOVA
16:00
Same Start: All the Variance Around Grand Mean
16:01
Independent Samples
16:23
Repeated Measures ANOVA
18:18
Same Start: All the Variance Around Grand Mean
18:19
Repeated Measures
18:33
Repeated Measures F-statistic
21:22
The F Ratio (The Variance Ratio)
21:23
S²bet = SSbet / dfbet
23:07
What is This?
23:08
How Many Means?
23:39
So What is the dfbet?
23:54
So What is SSbet?
24:32
S² resid = SS resid / df resid
25:46
What is This?
25:47
So What is SS resid?
26:44
So What is the df resid?
27:36
SS subj and df subj
28:11
What is This?
28:12
How Many Subject Means?
29:43
So What is df subj?
30:01
So What is SS subj?
30:09
SS total and df total
31:42
What is This?
31:43
What is the Total Number of Data Points?
32:02
So What is df total?
32:34
so What is SS total?
32:47
Chart of Repeated Measures ANOVA
33:19
Chart of Repeated Measures ANOVA: F and Between-samples Variability
33:20
Chart of Repeated Measures ANOVA: Total Variability, Within-subject (case) Variability, Residual Variability
35:50
Example 1: Which is More Prevalent on Facebook: Tagged, Uploaded, Mobile, or Profile Photos?
40:25
Hypotheses
40:26
Significance Level
41:46
Decision Stage
42:09
Calculate Samples' Statistic and p-Value
46:18
Reject or Fail to Reject H0
57:55
Example 2: Repeated Measures ANOVA
58:57
Example 3: What's the Problem with a Bunch of Tiny t-tests?
1:13:59
Section 12: Chi-square Test
Chi-Square Goodness-of-Fit Test

58m 23s

Intro
0:00
Roadmap
0:05
Roadmap
0:06
Where Does the Chi-Square Test Belong?
0:50
Where Does the Chi-Square Test Belong?
0:51
A New Twist on HT: Goodness-of-Fit
7:23
HT in General
7:24
Goodness-of-Fit HT
8:26
Hypotheses about Proportions
12:17
Null Hypothesis
12:18
Alternative Hypothesis
13:23
Example
14:38
Chi-Square Statistic
17:52
Chi-Square Statistic
17:53
Chi-Square Distributions
24:31
Chi-Square Distributions
24:32
Conditions for Chi-Square
28:58
Condition 1
28:59
Condition 2
30:20
Condition 3
30:32
Condition 4
31:47
Example 1: Chi-Square Goodness-of-Fit Test
32:23
Example 2: Chi-Square Goodness-of-Fit Test
44:34
Example 3: Which of These Statements Describe Properties of the Chi-Square Goodness-of-Fit Test?
56:06
Chi-Square Test of Homogeneity

51m 36s

Intro
0:00
Roadmap
0:09
Roadmap
0:10
Goodness-of-Fit vs. Homogeneity
1:13
Goodness-of-Fit HT
1:14
Homogeneity
2:00
Analogy
2:38
Hypotheses About Proportions
5:00
Null Hypothesis
5:01
Alternative Hypothesis
6:11
Example
6:33
Chi-Square Statistic
10:12
Same as Goodness-of-Fit Test
10:13
Set Up Data
12:28
Setting Up Data Example
12:29
Expected Frequency
16:53
Expected Frequency
16:54
Chi-Square Distributions & df
19:26
Chi-Square Distributions & df
19:27
Conditions for Test of Homogeneity
20:54
Condition 1
20:55
Condition 2
21:39
Condition 3
22:05
Condition 4
22:23
Example 1: Chi-Square Test of Homogeneity
22:52
Example 2: Chi-Square Test of Homogeneity
32:10
Section 13: Overview of Statistics
Overview of Statistics

18m 11s

Intro
0:00
Roadmap
0:07
Roadmap
0:08
The Statistical Tests (HT) We've Covered
0:28
The Statistical Tests (HT) We've Covered
0:29
Organizing the Tests We've Covered…
1:08
One Sample: Continuous DV and Categorical DV
1:09
Two Samples: Continuous DV and Categorical DV
5:41
More Than Two Samples: Continuous DV and Categorical DV
8:21
The Following Data: OK Cupid
10:10
The Following Data: OK Cupid
10:11
Example 1: Weird-MySpace-Angle Profile Photo
10:38
Example 2: Geniuses
12:30
Example 3: Promiscuous iPhone Users
13:37
Example 4: Women, Aging, and Messaging
16:07
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Lecture Comments (2)

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Post by Elias Tessema on April 5, 2014

I am having hard time understanding about concordant rate...can you please explain what concordant pair means

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Post by George Kumar on May 11, 2012

Model planes are a good analogy. However, model houses are not a good analogy. Model houses are real. They are sometimes sought after houses.

Correlation: r vs. r-squared

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Roadmap 0:07
    • Roadmap
  • R-squared 0:44
    • What is the Meaning of It? Why Squared?
  • Parsing Sum of Squared (Parsing Variability) 2:25
    • SST = SSR + SSE
  • What is SST and SSE? 7:46
    • What is SST and SSE?
  • r-squared 18:33
    • Coefficient of Determination
  • If the Correlation is Strong… 20:25
    • If the Correlation is Strong…
  • If the Correlation is Weak… 22:36
    • If the Correlation is Weak…
  • Example 1: Find r-squared for this Set of Data 23:56
  • Example 2: What Does it Mean that the Simple Linear Regression is a 'Model' of Variance? 33:54
  • Example 3: Why Does r-squared Only Range from 0 to 1 37:29
  • Example 4: Find the r-squared for This Set of Data 39:55

Transcription: Correlation: r vs. r-squared

Hi and welcome to www.educator.com.0000

We are going to talk about the difference between r and r2.0002

First I’m going to just introduce the quantitative r2 and need to understand it.0010

Why cannot we just square r and be like that is r2.0015

We want to know what the meaning of r2.0019

In order to get to the meaning of r2 we have to understand that sum of squared differences is actually going to split apart it to different ways.0022

We are going to learn how to parse the different parts of the sum of squared differences.0031

Then we are going to talk about what r2 means for a very strong correlation.0035

What r2 maybe for a very weak correlation.0040

One of the reason why practically you will need to understand what r2 is that often when you do regression on the computer,0047

either in SPSS or S data or any of this statistics packages, they will often give you r20056

as one of the output and you might be looking at and me like why are we doing the r2?0064

We want to know what is the meaning of it?0070

Why just r2? Why not just have r?0073

Often if you just find the correlation you will just get r but if you find the regression you will get r2.0076

It is like what is the deal?0084

R2 is really is just r2, but there is a meaning behind it.0087

I want to just stuck and say it is like the difference between feet and feet2.0094

They mean different things.0100

It is not just that you can square the number and be like it is just the numbers squared.0102

It is not just about the number it is also about the actual unit.0109

You have to understand what the unit is because feet is a measurement that examines link but square feet now gives you area.0113

Those are different things.0130

They are obviously related to each other, but they are very different ideas.0132

Because of that you need to also not only know, like how to calculate r2, but also know the meaning of r2.0136

Again in order to understand the meaning of r2 we will need to parse the sum of squares.0148

Remember the sum of squares that we have been talking about is something like x or y and0154

the difference between x and x bar or the difference between y and y bar.0161

Squaring all those and then adding them up, sum of squares.0167

When we say sum of squares you might hear the term that this is about variability.0172

Sum of squares talks about variability and it is because you are always getting that deviation between your data and the mean.0180

Sum of squares is often idea that is highly associated with variability.0189

Another way of thinking about parsing sum of squares is parsing variability because variability comes from a variety of sources.0199

Here we are going to talk about a couple of those sources and how to figure out this variability comes from that but this variability comes from that.0207

When you put it together you have total variability.0216

Now total variability is going to be indicated by SST or sum of squares total.0221

This idea is all the variability in the system.0228

All of the variability.0232

We are going to take that and parse it, split apart into two pieces that are equal pieces but there just 2 different places at that variability comes from.0234

One of the sources of the variability is always from this relationship between X and Y and that can be explained by the regression line.0246

This is sum of squares from the regression and so that can be the idea that sum of squares.0255

This one is going to be the left over sum of squares.0270

There is going to be some variability left over that is not explained by the regression line and that sum of squares error.0276

When we say error, we do not necessarily mean that we made a mistake.0294

It is not that we made a mistake.0300

Error often just means variability that is unexplained.0304

We do know where it came from.0310

We do not know if it is because there was some measurement error.0313

We do not know if there is just noise in the system.0318

We do not know if there is another variable that is causing this variation.0321

Sum of squares error just means variability that we cannot explain.0327

That does not necessarily mean that we made a mistake.0334

Often times that has to be statistics uses that word error but it does not mean that we made a mistake0338

but it means that it just variability that we do not know where it came from.0345

There is no explanation for it.0349

To break this down you could see that this is sum of squares total and that is usually what we get from looking at the difference between y and just the mean.0353

That is like the classic sum of squares because the mean should give us some information about where y is.0365

It is what every single point is going to be at the mean.0372

That is like error but that is the total error.0376

Some of that errors, some of that variation away from the mean can be accounted for by regression like here it is farther and farther up from the mean.0380

The numbers are bigger than the mean and then here the numbers are smaller than the mean.0391

Here this is the residual and this is we have already looked up before.0398

You can also think of it as residual error where it is the rest of the variation that is not accounted for by that nice regression line that we found.0405

We could think of this as the explained variability.0418

This is explained and what explains the variability?0428

The regression line.0434

The regression line says it is been a very systematically like this.0436

The residual is what we call unexplained variability.0441

When another one comes from its real error just variability in the system that is caused by another variable.0447

When you put the explained variability and unexplained variability altogether you will get total variability.0456

Let us break down specifically and mathematically what is sum of squares total or the sum of squares residual or sum of squares are?0469

I will give you a picture of what these things are.0483

First let us talk about sum of squares total.0486

One thing we probably want to do is give a rough idea of what the mean is.0490

Let us say the mean of something like this.0495

I'm just going to call that y bar because that might mean of y roughly.0499

Closer to these points but these guys are sure further down to pin it down.0504

I’m going to call that y bar and I want to know the sum of squares total.0510

Was the total variability that you see here.0518

Because we are squaring all these differences we are not just interested in that residual idea.0522

We interested in the area of little squares.0531

It is not only the distance down but imagine that distance squared and this area.0538

That is the sum of squared variation of one point.0548

Imagine doing that with all of these.0554

You create these squares.0558

Some are big squares, some are little squares and you add up all those different areas.0561

That is sum of squares total.0578

That is the total variation in our data away from the mean.0580

Would not it be nice if all our data looks something like the mean?0585

That would be like I can predict this data but this has more variation.0588

I must give way over to sum of squares error because that is when we actually know.0598

In order to find sum of squared error I need the regression line.0604

I’m just going to draw a regression line like this.0609

It might not be perfect but something like that.0614

Remember how we found residual?0617

To find a residual it is just the difference between my y and y bar that my predicted y hat.0620

These are my y hat and I want to know the difference between them but we are squaring that difference.0635

Instead of just drawing a line we draw a square and imagine getting that area.0643

That is the sum of squared residual or error for one point.0652

We are going to do that with all of the points.0657

Find that area, that area, that area and add up out of all those areas then we get the sum of squared error.0663

The variation away from the regression line.0678

This is our unexplained variation.0685

This is our total variation.0689

Now what is this part?0692

This is the variability that is already accounted for by the regression line.0694

This is the difference between the predicted y and y bar.0700

Here is the idea.0707

If we just have y bar we not have a lot of predicted power.0710

We are just saying our y bar is just average.0715

It is just the average and we only have one guess.0720

The average.0723

If we have the regression line we have a more mere guess.0725

If I know what x is I could tell you more closely what y might be.0730

I will try to redraw my regression line and pretend that is a nice regression.0735

Here is my y hat.0747

Also, here is my y bar.0750

Here what I want to know is how much of the variability is simply accounted for by having this line?0761

Having this line gives us the more predictive power how much of that predictive power is it.0770

We want to know for this point this is now my difference and then I'm just to square that difference.0776

Here is another point but here is the difference.0789

The difference is very like nothing.0795

Here is the difference.0798

It is right here, this difference.0801

Let me give another example like right here for this point this would be the difference.0812

I'm looking at all of these you can think of it as sort of the squared spaces in between my regression line and my main line.0820

I'm looking at that and that gives me how much of my variance in the data is accounted for by the regression line.0830

That is roughly the idea.0840

Let us think about actual formulas and to help us out with that I have a more like nicely drawn variation that my crappy dots0842

but now you could see the square differences between my actual data points and my mean.0854

Here are my square differences.0864

Here is that same data.0866

It is the same data from before, except now we are looking at differences from the regression line not the mean line.0868

Here we are looking at differences between the mean line and the regression line.0880

Let us write these things down in formulas in terms of formulas.0885

In order to find the sum of squares total let us think about what this is as an idea.0890

Okay, we want the sum of squares, so I know it is going to be sum of squares.0896

All of these guys are to be like this I could already write that down.0902

As this r what we call from the sum of squared and here is going to be the sum of something squared.0906

We already know that is going to be the same variability.0926

Here we have for every y give me the difference between that y and the mean and then square it and get that area.0930

Get all these areas and add them up.0942

That just y – y bar.0945

If we want to fill this out, we would know this means for everything single point that we have get y - y bar and then square it and add them up.0953

That is the idea.0963

That is sum of squares total.0965

Sum of squares residual actually let us go over to sum of squares error.0967

I sometimes call it also sum of squares residual because this is the idea of the residual.0975

Remember the residual was y – y hat.0982

And so, we are squaring the difference between y and y hat.0989

That is really easy.1002

Y – y hat.1004

If you want to fill it out, you could obviously put in the (i) as well just so you know you have to do that for every single point.1007

For the sum of squares for the regression I know that is why they call it sum of squares and sum of squares residual because it is confusing for the r.1016

This one is sum of squares regression.1027

I want to think of this guy as the good guy.1034

It is like you want to be able to predict X and Y and this guy helps you because he sucks up some of the variance.1037

This guy is the leftover that I do not know what to do anything about.1043

When we talk about the regression we are talking about the difference between y hat and y bar.1047

That is y hat and y bar.1056

You could obviously do that for each point.1065

There you have it, the formulas for these but if you understand the ideas you could always intercept what is this a picture of?1075

This is a picture of the difference between the data points and y bar.1085

Here is a picture of the difference between the data points and y hat.1091

It may be confusing though which one is y hat?1095

All you should do is go back to the picture and think to yourself by telling a total variance or variance after we have the regression line.1100

Okay, so now that you know as the SST, SSR and SSC now we can talk about r2 because you need those components.1115

R2 is often called the coefficient of determination, not coefficient of correlation squared it is often called the coefficient of determination.1125

One of the reasons that r2 is important is that it has an interpretation.1135

It is actually is talking about the proportion of total variance.1140

Remember variance is standard deviation2.1144

Because we are talking about sum of squared the proportion of that total of variance of y explained by the simple regression model.1149

Here is the idea.1159

It is like here is all that variance and we do not know where that variance comes from.1161

I do not know why they are all varying.1166

We have the regression line.1168

The regression line explains where some of the variation away from the mean comes from.1169

It comes from this relationship of x.1174

Is that regression line is doing a good job then a lot of the total variance is explained by the regression line, that predicted regression y.1178

If the line is not doing a very good job then it does not explain a lot of the variation there is extra variation above and beyond that.1193

All would be very low because only a small portion of that various is accounted for.1207

Given that, let us talk about what a strong r might be and what a weak r might be.1215

If the correlation is very strong let us think about this.1226

Whatever your sum of squares total is they are all variance.1231

Whatever that is this is going to account for a lot of it.1238

Let us say this is like 100% of the variance this accounts for 85% and so this would be small to be 15%.1244

This is of how this works.1258

These two added up, give you the total.1260

If that is true, if the correlation is very strong this should be small and this should be large.1264

If this is small then the proportion of error over the total would be a small number.1275

Here is the formula for r2.1284

R2 is 1 – that proportion of error / the total.1286

This is the unaccounted for error, that leftover error / the total variation.1291

This is the unexplained variation / the total variation.1298

This number should be very, very small and when that number is very small 1 - a very small number is a number very close to 1.1303

R2 is very strong because the maximum r2 could be this 1.1312

This means that if r2 is large this means close to 1 and this means that much of the variation is accounted for by the regression line.1318

The regression line did a great job of explaining variation.1343

As we near the regression line I could tell you I can predict for you y given x.1346

It is doing a good job.1353

On the other hand, if a correlation is weak.1358

If it is weak then this is the correlation how whiny it is.1362

Even if we have a line it does not explain all the variation.1369

There is a lot of leftover variation.1374

That should be low compared to that one.1378

If this is 100% and this is not doing a very good job explaining variation.1383

It only explains 15% of the variation then we have 85% of the variation leftover.1387

If we put the sum of squared error over the total this number should be large.1395

There is a lot of a large proportion of that total variance is still unaccounted for, unexplained.1400

1 - a larger number, one that is closer to 1 this will be a very small number for r2.1407

R2 if it is small this means that not a lot of the variation was accounted for by the regression line.1416

The regression line did not do very good job of explaining the variation in our data.1429

Let us do some example.1438

Previously we work with this data before for the above example data we have already found the regression line and the correlation they give it to us.1440

We could look at this and it has a negative slope and there is more rise than run.1451

Because the cost goes up really fast.1465

For every one that you go if you go up a little bit here.1469

It makes sense that the correlation is negative and strong, it is -.869 that is a pretty strong, very line-y but it had the negative slope.1474

It only gets as far.1489

It is giving us the correlation coefficient, not the coefficient of determination, r2.1492

Find r2 for the set of data and examine whether r2 once we find it in a different way by looking at r2 = 1 - the sum of squared error / sum of squared total.1497

Once we find that examine whether this is also r × r.1514

If you download the examples provided for you below and go to example 1, here is our data and I just provided the graph for you so you could see.1523

I’m just going to move it over to the side because we are not going to need it.1534

Remember that we have this, we are going to need to calculate something in order to find the sum of squared error and the sum of squares total.1546

One thing that I like to do is remind myself if I looked at sum of squared error, if I double clicked on that what would I see inside?1557

Well, we know that the sum of squared error is whatever regression line we have and we need this distance away squared.1568

That is going to be the sum of y - y hat because this is y hat2.1580

I know I’m going to need y hat.1592

What else are we going to need?1596

Sum of squares total is whatever my mean is.1597

Whatever my mean is I’m going to need to know the difference between my data and my mean squared.1603

My data and my mean squared, that is sum of squares total.1612

That I could easily find I should try to find y hat as well.1619

Y hat will be easy to find because we have the regression line.1626

We could just plug-in a whole bunch of x and get each y for all those x.1630

Why do not we start there?1638

Let us find the predicted and then I'm just going to call cost per unit as my y because that was on my y axis.1643

I will talk about predicted cost per unit, predicted CPU.1650

In order to find that I need to put in my regression formula, so that is 795.207 and then subtract 21.514 and Excel will automatically do order of operations for you.1655

Multiplication comes before subtraction.1676

I’m just going to just click in x.1679

Whatever x is this is going to find me the predicted y value.1683

Once I have that I’m just going to drag down this to find all of my predicted CPU.1695

It might be actually be helpful to us to find the sum and averages of all of these.1708

I’m just going to color these in red so that I know is not part of my data.1722

I probably do not need the sum for that.1728

I need the average for these.1730

I’m also going to need the average for these.1738

We have our predicted CPU (cost per unit).1746

That is my y hat.1752

I also find my y bar, my average cost per unit.1754

Let us find the error terms square and also these variations squared.1760

Here I’m just going to write it down for myself as y - the predicted y2 and also my y - y bar2.1773

We could also write CPU - predicted CPU2 or CPU - average CPU2.1796

I am just writing it y just to save space.1805

Let me get my y - the predicted y and all of these squared.1808

Let me also do that for y and y bar.1822

Let me get the parentheses.1825

Y - y bar and all of that squared.1827

Now y bar is never going to change it so I'm just going to lock that down.1841

Once I have that I could just copy and paste these 2 cells all the way down.1850

Once I have that now I could find the sum of the residual squared as well as the sum of these deviations squared.1863

Sum of all these guys and sum of these guys.1884

I have almost everything I need in order to find r2.1897

I have my sum here, my sum here.1902

Let us find r2.1905

R2 is going to be 1 - the sum of squared error ÷ by sum of squares total, that ratio.1910

Let us first just look at the data that we have clicked.1925

This value is smaller than this value.1928

This is 1/6.1932

Because of that 1/6 that is pretty good so we should have about 5/6 should be closer to 1 then to 0.1935

We will get .7 / 6 and so we get a pretty good r2.1947

Notice that r2 is positive even though our slope is negative because r2 does not actually talk about slope.1955

It is just the proportion of variance accounted for by the regression line.1964

It is the same 76% of the total variance is accounted for by that regression line, that majority.1970

And so that is good.1977

Now let us try to put in r × r so we already know what r is.1979

Let us see if r2 will give us .76.1986

So -.8692 we will get something very close and this is probably rounded and so because of that it does not give us precise numbers.1991

We do not have that precision, but is pretty close is still 76%.2009

If you have the actual r that you computed and you squared it, you would get perfectly r2.2015

We found our square for the set of data and examined whether it is r × r and it indeed is.2025

Example 2, the conceptual explanation of r62 is that it is the proportion of total variance of y explained by the simple regression model.2035

A simple regression model we just mean you only have the form y = b knot + b1.2045

It can only be aligned, it can be accrued.2060

That is what we mean by a simple linear regression.2065

What does it mean that the simple linear regression is a model of variance explained by a simple regression model.2069

Let us think about this idea.2086

Here we have our data set.2089

I’m just going to draw some points here.2092

These points do not exactly fall in a line.2099

That line that we made up the regression line, the regression line is really a model.2103

It is not actual data it is a theoretical model that we created from the data.2112

By model just like model airplane or model house, it is not the real houses.2118

It is like a shining example.2133

But not only is it an example, it is idealized.2139

It is the perfect version of the world.2144

If the word are perfect and there was no error that would be a model.2146

When we say a modeling variance we are there is always variance.2152

Where does it come from?2157

When we create a model, we have a little theory of where that variance comes from and in our model here this is our theory that explains the variance.2160

Our theory is that it is a relationship between x and y and it is very small explanation.2185

But it is this relationship between x and y that is where the variation comes from.2192

That is what we mean by the regression is lying as a model of the variance.2197

Now the idea behind r2 is how good is this theory.2204

How good is this model?2211

Does it explain a lot of the total variation or is it a theory that does not really help us out a lot?2213

If we have a big r2, if it is fairly large and this means that our theory is pretty good.2224

Our theory explains a lot of the total variance accounted for the total variance.2231

If our r2 is very small it means our theory was not that great.2237

We had a theory, here is a model but it is not that good.2240

It only explains a little bit of the variance.2244

Example 3, why is r2 only range from 0 to 1?2251

It might be helpful here to start off what r2 is?2256

1 - the sum of squared error / the total sum of squares / the total variance.2262

Now let us think can SSE ever be greater than SST?2272

No it cannot, because SST by definition it equals the sum of squares from a regression and the sum of squared error.2280

This by definition have to be smaller than this and none of these can be negative because they are squared.2292

Whatever it has to be positive numbers it is actually the case that if you add 2 positive numbers together to get another positive sum2299

and that sum has to be greater than or equal to this.2307

Either this is greater than each of these or it is equal to one of them because it could be like this is 0 and this is 100%.2317

There is just actually no way that this could be bigger than 1.2325

Not bigger than 1, bigger than SST?2336

No, cannot be.2348

This proportion have to range between 0 and 1.2351

It got to be 1 or smaller or they could be equal.2360

This could be 0 and this could be 1.2367

There is no way that this could be bigger than this.2372

Because this value only ranges from 0 to 1, 1 - something that ranges from 0 – 1, this whole thing could only range from 0 to 1.2376

Because of that r2 can only range from 0 to 1.2389

Example 4, and this is going to be a do see.2397

Find r2 for this set of data and examine whether this is also r × r.2400

Let us think about what we are going to do.2408

In order to find r × r and so r is the correlation coefficient and that is the sum of the product of z scores z sub x × z sub y and the average product of z scores.2411

We are going to find that.2436

We also have to find r2.2439

In order to find r2 that is 1 - sum of squared error / sum of squared total.2443

In order to find this, we need y hat.2449

In order to find y hat we need the regression line.2454

To find the regression line one thing we could do is once we find a correlation coefficient we could use that in order to find b1.2465

Or obviously we can also just find b1 in other ways too.2482

But this is one is a shortcut and once we find b1 we can find the intercept 1 – b1 × x.2488

We will have a whole bunch of data.2501

We have all this data.2504

Let us get started.2508

If you go to your examples and example 4, here is our data and I’m just going to move this over to the side because we are not going to be needing it for a while.2509

We already can see that it is probably can be a positive correlation if anything.2522

Let us just start by finding the correlation coefficient because it is pretty easy for us to find and once we have that we can find other things.2528

In order to get started on that it often helps to have the sum, the average, and the standard deviation.2539

I’m just going to make these all bolder in red so we know that there are different.2552

I’m going to find the sum for these.2558

We do not need the sum here though but I figured it as well.2564

It is not too hard.2569

There is the average and let us get the standard deviation because we are going to need that for the z score anyway.2570

Great.2580

We go all up now let us find is the scores for TV watching and also the z scores for junk food.2583

It makes sense that there is this more positive correlation.2599

The more TV watch per week perhaps more junk food calories are consumed.2606

Is the correlation strong?2616

I do not know.2619

In order to find the z score we need to have the TV watching data and subtract from that the mean and I want that distance,2620

not in terms of the raw distance, but in terms of standard deviation.2638

How many standard deviations away?2642

All divided by standard deviation.2644

Here I'm just going to lockdown the row.2649

I always use the same mean and standard deviation.2655

Once I have that I could just drag it all the way down and add it while we drag it across.2667

We forgot to find these for junk food calories.2680

Let us just double click on one of these and test it out.2687

Let us see.2692

It gives me the junk food calories - the average / the standard deviation.2693

Perfect.2701

Let us just eyeball this data for a second.2704

We see that roughly half of the z scores are negative and roughly half are positive.2707

Here too roughly half are negative and roughly half are positive.2713

We know that we did a good job at finding z scores.2717

In order to find the average product we are going to need to find the product the z(TV) × z(junk food).2719

This times this and once we have all of that we could sum these and we could find the average.2733

This divided by count how many data points that and then subtract 1.2750

We found the average and that is r.2767

Just regular of r.2770

That r it is .58, so it is not super duper weak but it is not really strongly either.2773

I’m just labeling it so that I know where it is only come out.2782

Once we have r we could find b1, b sub 1.2785

In order to find b sub 1 that will be r × the ratio between the standard deviation for y and standard deviation of x.2804

We have that right over here.2817

standard deviation for y ÷ stdev x, that proportion.2820

And so we get the b1 is 10.75 and once we have b1 we could find b sub 0.2830

Remember, we have the point of averages, but we also have all these points.2844

You can substitute anyone of these points.2851

Any one of the points between x and predicted y.2852

You cannot substitute these points.2858

In order to get the point of averages we will get y – b1 × x.2869

Here we get the intercepts b sub knot or b sub 0 is 186.2881

Now that we have b1 and b0 we can now find predicted y.2891

Let us go up here.2899

To help us out I am just going to color these some color so that we know that this is one is all about finding the correlation coefficient.2904

We found the correlation coefficient.2921

Now what we want to do is find r2.2923

And so in order to find r2 let us think about what we need.2929

We need predicted y, predicted junk food and we could easily find that and once we have that we know we are going to need y - predicted y2.2933

That is our sum of squared error. But we also going to need y - y bar2.2958

That is going to be our total error.2967

Let us start with predicted y.2970

Predicted y is always going to be b sub y + slope × x which is TV watching.2973

We will lock down b sub knot and the slope b sub 1 because do not want that to move.2992

Once we have that we could find (y - the predicted y)2 .3006

And then finally we want to find (y - the average y)2.3033

We want this average to be locked in place in order to move.3052

Once we have all of those 3 pieces we could just do the easy job of copying and pasting all the way down.3062

Once we do that, we could sum these up because we are going to need to have3074

the sum of squared residual I’m going to need the sum of squared deviation from the mean.3081

In order to find the sum I could just copy and paste that.3093

Once we have the sum I can now find r2.3100

I can just put in 1 – SSE / SST.3115

Let us see.3127

I will get .3377.3129

The regression line accounts for about 34% of the variation.3133

Let us see.3142

Is this r × r?3144

Is that going to be the same thing?3148

We have r we can just scroll and we get exactly 34%.3151

If we get a question like this, Excel can help.3160

Thanks for watching www.educator.com.3169

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