  Dr. Ji Son

Standard Normal Distributions & Z-Scores

Slide Duration:

Section 1: Introduction
Descriptive Statistics vs. Inferential Statistics

25m 31s

Intro
0:00
0:10
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

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
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
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
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
0:06
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
0:05
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
7:46
Example 4: Quiz Grade & Afterschool Tutoring Stemplot
9:56
Bar Graphs

22m 49s

Intro
0:00
0:05
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
0:07
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
0:05
0:06
0:45
0:46
5:45
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
0:06
0:07
Summarizing Distributions
0:37
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
0:16
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
0:13
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
0:06
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
0:15
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
0:04
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
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
0:05
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
0:13
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
0:05
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
0:07
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
0:05
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
0:05
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
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
0:05
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
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
0:06
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
0:06
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
0:07
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

20m 29s

Intro
0:00
0:08
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
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
5:41
5:42
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
0:05
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
0:05
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
0:08
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
0:06
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
0:05
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
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
0:08
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
0:05
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
39:52
Comparing Population, Sample, and SDoM
43:10
Comparing Population, Sample, and SDoM
43:11
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
0:06
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
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
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
0:06
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
0:04
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
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
0:06
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
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
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
0:14
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
0:06
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
0:09
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
0:18
0:19
Errors and Relationship to HT and the Sample Statistic?
1:11
Errors and Relationship to HT and the Sample Statistic?
1:12
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
0:05
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
0:04
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
0:05
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
0:05
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
0:05
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
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
0:09
0:10
Goodness-of-Fit vs. Homogeneity
1:13
Goodness-of-Fit HT
1:14
Homogeneity
2:00
Analogy
2:38
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
0:07
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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• Related Books 0 answersPost by Ryan Reddell on April 30, 2014All these questions with no answers.  Isn't asking questions answered part of the monthly fee? 0 answersPost by Marc Patrone on February 18, 2014In my exam I won't be allowed to use excel or any other program so I would appreciate doing thisthe long way. Thanks. 0 answersPost by Michelle Greene on October 7, 2013Example 2 HELP!  When calculating manually the z-score I do get .7037, however I am not getting .24 from the table.  What am I doing wrong?  We cannot use excel so calculator or manual way would be be great. thanks. 0 answersPost by Michelle Greene on October 7, 2013yes,Please tell us how to solve on the TI 83.  On the exams we only have access to our TI83 calculator and not excel. 0 answersPost by Abdulaziz Baathman on April 2, 20134. A normal random variable x has mean Âµ = 1.20 and standard deviation Ïƒ = 0.15. Find the probabilities these x-values:a) 1.00 < x < 1.10b) 1.35 < x < 1.50c) x > 1.385. A normal random variable x has mean 35 and standard deviation 10. Find a value of x that has area 0.01 to its right. pls help me 0 answersPost by Matthew Compton on February 28, 2013Is there any helpful links to use for TI calcuators.Since the lectures ignore the needs of your customers that do not use excel. 0 answersPost by Kristen Gravlee on September 30, 2012Where is normdist on the calc? 0 answersPost by Mariya Kossidi on September 26, 2012teach us on normal calculator, because on the exams students are allowed to have only calculators 0 answersPost by Kathryn Connor on March 13, 2012Please show how to do the calculations on a TI-84 calc 1 answerLast reply by: Kristen GravleeSun Sep 30, 2012 9:36 PMPost by Kamal Almarzooq on January 22, 2012you depend on excel too much :(what about the students who don't have itteach us how to find it in the normal calculations

Standard Normal Distributions & Z-Scores

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
• A Family of Distributions 0:28
• Infinite Set of Distributions
• Transforming Normal Distributions to 'Standard' Normal Distribution
• Normal Distribution vs. Standard Normal Distribution 2:58
• Normal Distribution vs. Standard Normal Distribution
• Z-Score, Raw Score, Mean, & SD 4:08
• Z-Score, Raw Score, Mean, & SD
• Weird Z-Scores 9:40
• Weird Z-Scores
• Excel 16:45
• For Normal Distributions
• For Standard Normal Distributions
• Excel Example
• Types of Problems 25:18
• Percentage Problem: P(x)
• Raw Score and Z-Score Problems
• Standard Deviation Problems
• Shape Analogy 27:44
• Shape Analogy
• 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

Transcription: Standard Normal Distributions & Z-Scores

Welcome to www.educator.com.0000

Now let us talk about standard normal distributions and z scores.0003

First we are going to contrast the normal distribution against standard normal distribution.0008

It is pretty because just by knowing the normal distribution you already really know the standard normal distribution.0015

Then we are going to talk about some normal distribution problems and contrast that with standard normal distribution problems.0021

Before we talked about how the normal distribution really is a family of problems, it is not just one shape,0031

but it could be stretched or shrunk on that x axis.0037

And because of that it is actually an infinite set of distribution with all the different means and different standard deviations.0042

You could have means of 10 and deviation of 10.0051

That is one way of distribution.0053

Another one could have a mean of 1 and a stdev of 2.0055

There is like an infinite number of these.0059

They all fit the empirical rule.0062

Because it is problematic to work with like everything will one of these different normal distribution0066

and they have thought of this transformative system where you transform all normal distribution into what is called the standard normal distribution.0072

And there are what we are doing is where making that mean 0 and the stdev become 1.0081

This way we do not have to worry about the actual values.0089

We do not have to worry about the mean of actually 50.0090

We just have to know the mean is in the middle.0095

And the standard deviation by being 1 makes it really easy for us to label this stuff.0102

In a standard normal distribution, what we really done is we have transformed the values into z scores or what we call standard deviation.0111

We have normalized everything we do not care that the standard deviation is actually 10.0133

We just care how many is the stdev the way they are.0142

If you are 1 standard deviation away, 2 standard deviation away, and because of that we have changed this x axis0146

instead of actually putting the values we are now putting the z scores.0152

That is the only difference between the normal distribution and the standard normal distribution.0156

In the standard normal distribution we basically ignore the values and we only use the z scores.0160

We have everything in terms of standard deviations.0168

Are you 1 standard deviation away, 1/2 standard deviation away?0171

That is what we care about.0175

In a normal distribution, which I will draw in blue, in a regular normal distribution, we usually have is probability, which is the area.0180

This is represented by the area.0193

The raw score, with the mean and standard deviations and actual values.0197

Things like 150, 450.0207

We have the z score, 0, -1, 1.0212

That is what we think about a normal distribution, a regular normal distribution.0220

What we do in a standard normal deviation is we ignore that part.0224

It is the same thing.0230

You still have probability and we still have the z scores, right?0231

Now all the scores are z scores because we do not have any other scores other than that.0235

That is the only thing that is different about a standard normal distribution.0240

Now let us talk about the z score, raw score, and the mean, and the stdev, and the relationship that we have to each other.0250

Before we really did it in an intuitive way.0257

For instance, if our mean was 10, the mean is 10 and our standard deviation is 5.0262

How do we know where to draw these notches?0276

Well if it is a normal distribution we know that this is approximately this area should be About 68% more than half.0280

What you could do is look for the point of inflection and often in my drawings the point of inflections is hard to see0289

but in a nicer drawing you could see the point of inflection.0296

What we could see is 1 standard deviation away means that it is a distance of 5.0300

What would this value be right here?0309

Since it is 5 on the negative side, all we have to do is subtract 5 and it is pretty intuitive.0312

If we would want to go another five steps to the left and that would be 0.0318

If we wanted to go yet another 5 steps to the left that would be -5.0323

If we wanted to go yet another 5 steps to the left that would be -10 and so on and so forth.0328

Same thing on the positive side, if we wanted to go 5 steps to the right and that would be 15, 20, 25, 30 and so forth.0335

On to infinity and negative infinity. Now before we just did it in a very intuitive way and these notches just corresponds with z scores.0348

This is 1 standard deviation out, 2 standard deviations out three standard deviations out,0363

but there is systematic relationship that we could exploit here in turn with the formula between the z scores, raw score, and mean, and standard deviation.0373

If we wanted to go 2 standard deviations out and find what this value was, in order to find the raw score what we do is take the mean0384

and then add the z score and multiply by the stdev.0402

This actually works with the negative side too because when we have a z score of -1 then you would subtract the stdev instead of adding to the mean.0408

This formula ends up being good for us.0422

Instead of raw score, we usually call the raw score the axis.0425

Instead of writing out mean we would write mule.0429

Then we could keep z and write sigma as our stdev.0436

Now we see our formula here that allows us to go from the mean, z score and standard deviation into a raw score.0442

We could use this formula to find anything, any of those four things.0452

If we wanted to find the z score for instance, and we did some algebraic to this0457

and we would just see that is actually the distance between the raw score and the mean divided by the standard deviation.0465

Let us think for a second about what that means.0477

That means the z score is take this distance and cut it open to these chunks.0481

How many of those chunks do you have right?0487

the z score really0489

is telling you how many standard deviations are you away from the mean?0490

It is totally relative to the mean and standard deviation.0495

Also you can use this very same formula to solve for the mean and standard deviation.0501

We could do that over here right.0509

Let us say we wanted to solve for the mean, what would that look like?0513

All we have to do is move this over so that would be x - z score multiplied by the standard deviation.0519

Let us say we wanted to find standard deviation, what would we do then?0528

That is pretty easy as well.0535

All we have to do is you could just take this formula and swap these guys and that would be stdev = x to find out what is z.0538

Here we see that using this very simple formula and you could just remember one of these0552

because having one of them, you can derive the other ones from it.0559

Just by having one of these formulas if you had any 3 of these you could solve for the fourth one.0563

You do not even have to do the algebraic transformations.0571

You could just plug them in and find out what is missing.0574

So far we have only talked about the z scores for its nice and even like 1, -1, 2, -2.0583

We have not talked about weird z scores.0592

For instance, what about a z score of something like .5?0594

What would be the area over here?0608

Some of you maybe tempted to take .34 and divide it in half, but as you will see from this picture that would not work.0615

You are not really dividing this area in half, this area still ends up being slightly bigger because it is taller than this area.0625

In this area have been chunk out of it.0634

You cannot just divide it by half.0640

That is not going to be a good strategy.0642

That would not give you the right area.0647

How do we deal with this?0650

Well, there are two ways of dealing with this these weird z scores and how to go from these weird z scores to probabilities that we do not know yet.0651

That are not nice than my 34, 13 ½ .0660

There are two ways of doing it.0665

One way is by looking at up on a table.0666

Often there are tables in the back of your text book or even in the back of an EP statistics0670

like Princeton Review book or something that show you the transformation from weird z scores like .5 and .67, 1.9 probability.0675

That is one way of doing it using z tables.0690

The second way of doing it is by using Excel or your calculator.0694

If you do not have a fancy TI something calculator, it should also come with similar functions to Excel.0703

I’m going to show you those two methods of how to go from weird z scores to probabilities of their z scores and vice versa.0710

How to go from the probabilities, like weird probabilities like 50%.0720

We do not know where like 51% but we do not know what this value would be for 51%.0725

how to go from those weird probabilities into that weird z scores.0733

First, let us talk about the method by using the tables in the back of your butt.01225.1 Usually it is the first table you will see back there, table A or something like that and let us break it down.0739

A lot of tables looks somewhat like this one here might look slightly different than probably roughly similar.0752

And what it shows you up here is exactly what probability is plotted down here.0760

What it shows you is the probability that shown certain on the negative side.0765

Everything below the z score and this is what we call the cumulative probability because you are accumulating it as we go right.0771

It is adding up all the probabilities on this side.0783

This is showing you the cumulative probability at the z score.0786

The table entry for z is the probabilities lying below z.0793

Here the z scores and these are the probabilities.0801

Now for the weird z scores, what would really be helpful is if we had z scores of - .34, -.341, -3.42, -3.43.0804

We had all of these decimal places.0827

It would be really nice if we had all these different z scores.0829

That would probably be a skinny list of a whole bunch of z scores and a skinny list of a whole bunch of different probabilities.0833

That would be a very inefficient use of space.0843

What a lot of tables do is they put the z score up to like here, it is up on the tens place in this side and then we put the hundreds on this dimension.0847

In order to find the probability for the z score -3.45, you have to find -3.4 and then go to .05.0861

It is like you add this and you stick it on there if you added it would not work because it is negative.0876

Here that would be -3.45 and at -3.45 the cumulative probability is .0003.0887

Notice that it is a very small probability, it is not 0 but it is very, very small.0902

And let us do another example.0907

Let us say we wanted to know the probability at the z score -2.48.0909

We go to -2.4 and then also go to 8 and that would be our probability .0067 less than -.1%.0919

What if we wanted to find out the upper side?0933

We are like your only giving me the cumulative probability on the lower side.0938

What if we wanted to find out this probability at 2.48?0943

What would we do then?0955

Because the normal distribution and standard normal distribution to be is perfectly symmetrical0956

what we would see is that if we know it for the negative side, we can just flip it over on the positive side.0965

The negative side over here at -2.48 is a probability of .00066 then this 2 is also .0066 because it is perfectly symmetrical.0970

You do not need to have both the positive and negative sides.0986

Oftentimes tables might just give you the positive side or the negative side.0991

Sometimes, they will give you both, but you do not actually need both, you could just figure it out from there.0997

Let us talk about how to do that with Excel.1007

For Excel they are nice functions that Excel prewritten out for us so that we do not have to actually look it up on a table.1009

For normal distributions, when you know the mean and the standard deviation, there are ways to go from the raw score to the area underneath the curve.1018

So basically it is very similar to that picture we found that z score.1035

What it will give you is this cumulative probability.1040

That entire area below that value.1045

The normal distribution, you need to enter in the mean and the standard deviation.1050

In order to go from the score into the probability you would use the normdis function and1057

you would put in the score, but you would also put in the mean, the standard deviation.1066

and there is another thing, you also have to put in shrew in order to get the cumulative probability because this is sort of asking cumulative probability.1072

and that will give you the probability of that score.1086

The cumulative probability of that score.1090

Norm inverse is just the inverse of that.1092

Here we give it the probability and it is been stuck out the score.1097

Here you give it the probability, the mean, and the standard deviation, and it spits back out to you the score.1101

Notice that these two are inverses that is why one of them is called norm in because in one, you get the probability and the other you get the score.1112

These are our 2 flipping around values.1127

For standard normal distribution, you do not need to enter in the mean and standard deviation because all standard normal distribution means are 0.1132

The mean of 0 the stdev is 1.1145

It makes it a lot simpler for us.1148

The only difference between a normal distribution and standard normal distribution is this one little letter.1152

All you have to do is remember to enter in norm s dis and here you could just put in the score, forget everything else and it will spit out the probability.1158

For norm set in, it is exactly the opposite where you put in the probability and a spit back out of the score.1172

Another handy little functions might be the functions standardized.1188

Here you put in the raw score, the mean, and the standard deviation, and it will give you the z score.1192

The standardized function simply uses the formula that we have talked about earlier where1204

in order to give you z score, it takes the raw score or x – mule ÷ stdev.1213

Given that, let us look at a few examples and do them in Excel.1226

The distribution of SAT scores for Math for incoming students to the University was approximately normal.1230

That is again important, it has to say approximately normal.1237

watch out for that.1241

I do not know if you have tricky instructors or tests but sometimes they might give you a problem that looks like a normal distribution problem,1242

but it does not say it is normally distributed.1247

With the mule of 550 and a standard deviation of 100, what percentage of scores with 400 or below.1251

Note that 400 if you transform it into a z score would not it be nice even z score like -1 or -2 is actually like -1.51260

That is going to be a little bit difficult for us to use just the empirical rule.1276

That is where we have to use something like either the table in the back of your book or Excel.1279

Here I just have an empty Excel sheet and one thing I’m going to do is I’m going to use my columns1286

to denote probability raw score z scores just because that can help me keep track of what I’m doing.1300

I’m also going to write down my mean and standard deviation just to help make things a little easier.1308

550, 100, and my probability where x is less than 400, that is where I’m going for.1316

My raw score is 400 and my z score.1332

I can just do this in my head it is just the distance 150 ÷ 100.1344

It is on the negative side so it is -1.5.1354

But just to practice using Excel, let us use our standardized function.1357

Excel will guide you into what you exactly need.1367

You need your x value for raw score.1370

The mean and the standard deviation.1374

I’m going to close my parentheses and I will get -1.5.1379

The other way you could do this in Excel is do the formula because you know that you need the raw score – mean / stdev.1384

You will get the same thing but I just wanted to illustrate for you that is what the standardized function is doing.1400

Let us see how that now you need the probability, I could find the probability in 2 ways.1406

One, I could use this standard normal distribution or I could use the regular normal distribution.1418

Let us do both.1423

First, I know that is the norm dist, I do not need the inverse one because I have the raw score, I need the probability.1425

Here I know I need my x value and I know I need the mean, stdev.1437

I’m just going to ask you it is cumulative?1445

Let me just write should.1448

That gives us .0668 and that makes sense if we think about where -1.5 was, it is in between -2 and -1.1451

If it was at -1, it would be like 16%.1468

If it is all the way at -2, it will be like 2%.1472

6% sounds like it is in between 2% and 16%.1476

We could also do this by using norms set dist.1483

For that, all you need is the z score.1493

I will put in my z score and I should get the exact same thing because the z score corresponds with the raw score.1496

They are right above each other.1505

Probabilities at that point should be exactly the same.1508

Now let us talk about the different types of problems.1519

Now that you know z score and standardized normal distribution, what kinds of problems might come in your way?1522

The first sign of information you do not always need these numbers because the first set of information1529

is the mean, stdev, probability, raw score or x, the z score or z.1535

That is the first set.1546

Anyone of these things can be missing and you could find it.1548

What is missing here?1552

Here we have that same prompt, it says what percentages of scores where 400 and below?1554

That is percentages of scores.1561

Or it might ask what percent of scores fall below the z score of -1.5.1563

Here we see that they are both about percentage of score.1575

What might be missing is dist.1579

If you have either one of these, and this 2, you could find it no problem.1582

Here is another set of problem, same prompt but what is missing here?1588

Here it says what math scores separates the lowest 10% than the rest or what z scores separates the lowest 25% from the rest?1595

Here you have the probabilities and you have the mean of stdev but one of these 2 things might be missing.1605

I think of this as the score missing problems.1613

We have probability missing problems and we have score missing problems.1616

The last set looks like this.1621

Note that this prompt does not give you the stdev but it does give you the z score and the raw score so that you could find the stdev here.1625

Or it gives you the percentages and the raw score and then you find the stdev.1637

Here something like a stdev is missing.1645

You could also find the mean to be missing as well and they would look similar to this problem.1648

As long as you have some combination of the other values in place, you could actually figure out what is missing.1654

Now let us add in the shape analogy that we did in the previous lesson.1666

I’m going to skip over a lot of this because it all stays the same.1671

They are only parts where they drive your attention to is right here.1674

Before we could only look at our probabilities and raw scores but now we have extended our repeater.1677

We could look at probabilities to raw scores to z scores and even find the missing mean and stdev.1685

That is all done to a combination of the z scores formula as well as either the tables or Excel.1693

Let us do some problems.1705

Here is example 1.1707

In the US, the distribution of deaths due to heart disease 289 deaths per 100,000 stdev of 54 and cancer a mean of 200 stdev of 31 are roughly normal.1709

We know that we could use our normal distribution stuff.1728

In California, 254 deaths are form heart disease and 166 deaths are from cancer per 100 residents.1730

Which rate is more extreme compared to the rest of the states, the average for the US?1740

We have California’s death rate from heart disease and from cancer.1748

We want to know which of these are really extreme?1756

One way that we could find out is by finding out the z scores.1759

I am going to get my Excel out.1766

The US mean for heart disease. I will use this row for heart disease and this row for cancer.1780

The US mean is 289 and the stdev is 54.1791

For cancer, it is 231.1809

We want to know how extreme the California deaths are.1813

It is hard to compare with just the numbers because even though 240 is less than 289, and 166 deaths is less than 200,1817

we are wondering how far away from the mean are you?1833

One way that we could do that is by using z scores because z scores will give you the distance in terms of the stdev.1838

Because these populations have very different standard deviation, that is worth knowing.1845

Here is California, heart disease and cancer.1852

In California the mean is 240 and 166.1865

What we might want to know is the z score.1875

What is the z score?1879

I might just put standardized and put in my x, put in the theoretical mean that I want to compare it too and my stdev.1882

Obviously, I could also do my (x – mean) ÷ stdev.1897

Here we see that the z score for heart disease is about 1 stdev away, -.9.1908

How far down is cancer?1918

How much less is cancer?1921

How more healthy are Californian in terms of cancer?1925

For here I will put in my regular formula, just so that we practice that too1929

My cancer x – theoretical population ÷ stdev.1935

I want it in terms of standard deviation step.1944

We find out that cancer and this is probably because stdev is smaller and actually sort of out is farther than heart disease.1947

The cancer rate is more extreme in a positive way.1962

It is more extremely low than heart disease, although they are very close.1968

Here the trick is find rates in terms of stdev or z scores.1977

Example 2, heights of male college students in the US are approximately normal, estimate the percentage of these males that are at least 6ft tall.1995

It will help if we just sketch this out briefly.2012

Here is my mean as 70.1, my stdev is 2.7, what percentage of these males are at least 6 ft tall?2016

What helps is if you transform this 6ft into inches, that is 72 inches.2027

Where is 72 inches?2043

That might be something like this.2045

What percentage of males are at least 6 ft tall?2048

I want all of these people because all of these people are at least 6 ft tall.2052

They might be 7 ft tall but they are at least 6 ft tall.2058

That is what I really want.2061

What might be helpful is instead of my raw score, if I could find my z score and then I can look it up on the table in the book.2063

Or I could use Excel.2072

I’m going to use Excel to help me.2075

Let us find the z score for this.03449.5 In order to find z score, let me write the formula here.2086

In order to find my z score for 72 inches that would be 72 which is my (x – mean 70.1) ÷ stdev 2.7.2095

Because I want to know how many jumps away, my jumps are 2.7.2115

Now I will pull up my Excel and I could just put in (72 – 70.1) ÷ 2.7.2127

I find that my z score is .70.2144

In order to find the area, the cumulative area, remember cumulative area means this side.2151

Excel is going to give me this side but that is now I want.2159

I want this side.2163

I might put in 1 – this area in order to get that area.2165

1 – and will just put in my norms dist.2172

Thankfully Excel gives you a little hint if you are a little off.2190

What I get is .24, about 24% of these males are at least 6 ft tall.2195

If I want to write it all out I would write my probability where x is greater than 72 inches is .24.2223

This is example 3, in a standard normal distribution where P(z score) < .41 = .659, what is the mean and stdev?2235

Actually this is a tricky question, before you go often trying to find the z score and all the stuff, note that it says standard normal distribution.2249

Every standard normal distribution, mean is 0, stdev 1.2257

Example 4, find the percentage of values in a standard normal distribution that fall between -.1.446 and 1.46.2266

This is nice if we would sketch this out so that we know what to expect.2283

Here is 0, and this is a standard normal distribution that is why I know that it is 0.2288

Here is 1, here is where 1.46 is.2295

A little bit less than1/2, probably a little bit too much less than half.2308

I want to find this area right in between.2314

Usually we look at the area between -1 and 1, and we know that 68%.2328

We know that it is a little bit more than 68%.2340

It is not quite like 95%, it is not quite that high but it is somewhere in between 68 and 95%.2342

Let us try to figure this out.2352

One way we could do it is by using our Excel or by looking it up in our book.2354

In our book and in Excel, our problem is that they will give you the cumulative distribution.2363

They will give you this area.2369

What we really want is this area.2372

What do we do?2376

One thing we might want to do is just use our ability to reason so this is 50% of that curve.2378

If we take 50% and take away this area, we could use norms dist and put in -1.46 then that should give us this area right here.2388

This whole area is this.2405

This area is in blue.2414

That should give us the red area.2418

Here I’m going to put in 50% - norms dist – 1.46.2421

I want to take that area and multiply it by 2 because I want the other side too and it is perfectly symmetrical.2442

We do not have to do any work just by multiply by 2.2452

Then I would get .8557 so a little bit more cloes to 86%.2455

That makes sense.2465

Since it is more than 68%, it is less than 95% right?2466

80 and 86%.2470

My answer is where -1.46 my z falls in between this values, .86.2475

That is example 4 and that is it for the standard normal distribution.2496

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