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Dr. Ji Son

Shape: Calculating Skewness & Kurtosis

Slide Duration:Table of Contents

I. Introduction

Descriptive Statistics vs. Inferential Statistics

25m 31s

- Intro0:00
- Roadmap0:10
- Roadmap0:11
- Statistics0:35
- Statistics0:36
- Let's Think About High School Science1:12
- Measurement and Find Patterns (Mathematical Formula)1:13
- Statistics = Math of Distributions4:58
- Distributions4:59
- Problematic… but also GREAT5:58
- Statistics7:33
- How is It Different from Other Specializations in Mathematics?7:34
- Statistics is Fundamental in Natural and Social Sciences7:53
- Two Skills of Statistics8:20
- Description (Exploration)8:21
- Inference9:13
- Descriptive Statistics vs. Inferential Statistics: Apply to Distributions9:58
- Descriptive Statistics9:59
- Inferential Statistics11:05
- Populations vs. Samples12:19
- Populations vs. Samples: Is it the Truth?12:20
- Populations vs. Samples: Pros & Cons13:36
- Populations vs. Samples: Descriptive Values16:12
- Putting Together Descriptive/Inferential Stats & Populations/Samples17:10
- Putting Together Descriptive/Inferential Stats & Populations/Samples17:11
- Example 1: Descriptive Statistics vs. Inferential Statistics19:09
- Example 2: Descriptive Statistics vs. Inferential Statistics20:47
- Example 3: Sample, Parameter, Population, and Statistic21:40
- Example 4: Sample, Parameter, Population, and Statistic23:28

II. About Samples: Cases, Variables, Measurements

About Samples: Cases, Variables, Measurements

32m 14s

- Intro0:00
- Data0:09
- Data, Cases, Variables, and Values0:10
- Rows, Columns, and Cells2:03
- Example: Aircrafts3:52
- How Do We Get Data?5:38
- Research: Question and Hypothesis5:39
- Research Design7:11
- Measurement7:29
- Research Analysis8:33
- Research Conclusion9:30
- Types of Variables10:03
- Discrete Variables10:04
- Continuous Variables12:07
- Types of Measurements14:17
- Types of Measurements14:18
- Types of Measurements (Scales)17:22
- Nominal17:23
- Ordinal19:11
- Interval21:33
- Ratio24:24
- Example 1: Cases, Variables, Measurements25:20
- 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

III. Visualizing Distributions

Introduction to Excel

8m 9s

- Intro0:00
- Before Visualizing Distribution0:10
- Excel0:11
- Excel: Organization0:45
- Workbook0:46
- Column x Rows1:50
- Tools: Menu Bar, Standard Toolbar, and Formula Bar3:00
- Excel + Data6:07
- Exce and Data6:08

Frequency Distributions in Excel

39m 10s

- Intro0:00
- Roadmap0:08
- Data in Excel and Frequency Distributions0:09
- Raw Data to Frequency Tables0:42
- Raw Data to Frequency Tables0:43
- Frequency Tables: Using Formulas and Pivot Tables1:28
- Example 1: Number of Births7:17
- Example 2: Age Distribution20:41
- Example 3: Height Distribution27:45
- Example 4: Height Distribution of Males32:19

Frequency Distributions and Features

25m 29s

- Intro0:00
- Roadmap0:10
- Data in Excel, Frequency Distributions, and Features of Frequency Distributions0:11
- Example #11:35
- Uniform1:36
- Example #22:58
- Unimodal, Skewed Right, and Asymmetric2:59
- Example #36:29
- Bimodal6:30
- Example #4a8:29
- Symmetric, Unimodal, and Normal8:30
- Point of Inflection and Standard Deviation11:13
- Example #4b12:43
- Normal Distribution12:44
- Summary13:56
- Uniform, Skewed, Bimodal, and Normal13:57
- Sketch Problem 1: Driver's License17:34
- Sketch Problem 2: Life Expectancy20:01
- Sketch Problem 3: Telephone Numbers22:01
- Sketch Problem 4: Length of Time Used to Complete a Final Exam23:43

Dotplots and Histograms in Excel

42m 42s

- Intro0:00
- Roadmap0:06
- Roadmap0:07
- Previously1:02
- Data, Frequency Table, and visualization1:03
- Dotplots1:22
- Dotplots Excel Example1:23
- Dotplots: Pros and Cons7:22
- Pros and Cons of Dotplots7:23
- Dotplots Excel Example Cont.9:07
- Histograms12:47
- Histograms Overview12:48
- Example of Histograms15:29
- Histograms: Pros and Cons31:39
- Pros31:40
- Cons32:31
- Frequency vs. Relative Frequency32:53
- Frequency32:54
- Relative Frequency33:36
- Example 1: Dotplots vs. Histograms34:36
- Example 2: Age of Pennies Dotplot36:21
- Example 3: Histogram of Mammal Speeds38:27
- Example 4: Histogram of Life Expectancy40:30

Stemplots

12m 23s

- Intro0:00
- Roadmap0:05
- Roadmap0:06
- What Sets Stemplots Apart?0:46
- Data Sets, Dotplots, Histograms, and Stemplots0:47
- Example 1: What Do Stemplots Look Like?1:58
- Example 2: Back-to-Back Stemplots5:00
- Example 3: Quiz Grade Stemplot7:46
- Example 4: Quiz Grade & Afterschool Tutoring Stemplot9:56

Bar Graphs

22m 49s

- Intro0:00
- Roadmap0:05
- Roadmap0:08
- Review of Frequency Distributions0:44
- Y-axis and X-axis0:45
- Types of Frequency Visualizations Covered so Far2:16
- Introduction to Bar Graphs4:07
- Example 1: Bar Graph5:32
- Example 1: Bar Graph5:33
- 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 Gender14:02
- Example 3: Cases, Variables, and Frequency Visualization16:34
- Example 4: What Kind of Graphs are Shown Below?19:29

IV. Summarizing Distributions

Central Tendency: Mean, Median, Mode

38m 50s

- Intro0:00
- Roadmap0:07
- Roadmap0:08
- Central Tendency 10:56
- Way to Summarize a Distribution of Scores0:57
- Mode1:32
- Median2:02
- Mean2:36
- Central Tendency 23:47
- Mode3:48
- Median4:20
- Mean5:25
- Summation Symbol6:11
- Summation Symbol6:12
- Population vs. Sample10:46
- Population vs. Sample10:47
- Excel Examples15:08
- Finding Mode, Median, and Mean in Excel15:09
- Median vs. Mean21:45
- Effect of Outliers21:46
- Relationship Between Parameter and Statistic22:44
- Type of Measurements24:00
- Which Distributions to Use With24:55
- Example 1: Mean25:30
- Example 2: Using Summation Symbol29:50
- Example 3: Average Calorie Count32:50
- Example 4: Creating an Example Set35:46

Variability

42m 40s

- Intro0:00
- Roadmap0:05
- Roadmap0:06
- Variability (or Spread)0:45
- Variability (or Spread)0:46
- Things to Think About5:45
- Things to Think About5:46
- Range, Quartiles and Interquartile Range6:37
- Range6:38
- Interquartile Range8:42
- Interquartile Range Example10:58
- Interquartile Range Example10:59
- Variance and Standard Deviation12:27
- Deviations12:28
- Sum of Squares14:35
- Variance16:55
- Standard Deviation17:44
- Sum of Squares (SS)18:34
- Sum of Squares (SS)18:35
- Population vs. Sample SD22:00
- Population vs. Sample SD22:01
- Population vs. Sample23:20
- Mean23:21
- SD23:51
- Example 1: Find the Mean and Standard Deviation of the Variable Friends in the Excel File27:21
- Example 2: Find the Mean and Standard Deviation of the Tagged Photos in the Excel File35:25
- Example 3: Sum of Squares38:58
- Example 4: Standard Deviation41:48

Five Number Summary & Boxplots

57m 15s

- Intro0:00
- Roadmap0:06
- Roadmap0:07
- Summarizing Distributions0:37
- Shape, Center, and Spread0:38
- 5 Number Summary1:14
- Boxplot: Visualizing 5 Number Summary3:37
- Boxplot: Visualizing 5 Number Summary3:38
- Boxplots on Excel9:01
- Using 'Stocks' and Using Stacked Columns9:02
- Boxplots on Excel Example10:14
- When are Boxplots Useful?32:14
- Pros32:15
- Cons32:59
- How to Determine Outlier Status33:24
- Rule of Thumb: Upper Limit33:25
- Rule of Thumb: Lower Limit34:16
- Signal Outliers in an Excel Data File Using Conditional Formatting34:52
- Modified Boxplot48:38
- Modified Boxplot48:39
- Example 1: Percentage Values & Lower and Upper Whisker49:10
- Example 2: Boxplot50:10
- Example 3: Estimating IQR From Boxplot53:46
- Example 4: Boxplot and Missing Whisker54:35

Shape: Calculating Skewness & Kurtosis

41m 51s

- Intro0:00
- Roadmap0:16
- Roadmap0:17
- Skewness Concept1:09
- Skewness Concept1:10
- Calculating Skewness3:26
- Calculating Skewness3:27
- Interpreting Skewness7:36
- Interpreting Skewness7:37
- Excel Example8:49
- Kurtosis Concept20:29
- Kurtosis Concept20:30
- Calculating Kurtosis24:17
- Calculating Kurtosis24:18
- Interpreting Kurtosis29:01
- Leptokurtic29:35
- Mesokurtic30:10
- Platykurtic31:06
- Excel Example32:04
- Example 1: Shape of Distribution38:28
- Example 2: Shape of Distribution39:29
- Example 3: Shape of Distribution40:14
- Example 4: Kurtosis41:10

Normal Distribution

34m 33s

- Intro0:00
- Roadmap0:13
- Roadmap0:14
- What is a Normal Distribution0:44
- The Normal Distribution As a Theoretical Model0:45
- Possible Range of Probabilities3:05
- Possible Range of Probabilities3:06
- What is a Normal Distribution5:07
- Can Be Described By5:08
- Properties5:49
- 'Same' Shape: Illusion of Different Shape!7:35
- 'Same' Shape: Illusion of Different Shape!7:36
- Types of Problems13:45
- Example: Distribution of SAT Scores13:46
- Shape Analogy19:48
- Shape Analogy19:49
- Example 1: The Standard Normal Distribution and Z-Scores22:34
- Example 2: The Standard Normal Distribution and Z-Scores25:54
- Example 3: Sketching and Normal Distribution28:55
- Example 4: Sketching and Normal Distribution32:32

Standard Normal Distributions & Z-Scores

41m 44s

- Intro0:00
- Roadmap0:06
- Roadmap0:07
- A Family of Distributions0:28
- Infinite Set of Distributions0:29
- Transforming Normal Distributions to 'Standard' Normal Distribution1:04
- Normal Distribution vs. Standard Normal Distribution2:58
- Normal Distribution vs. Standard Normal Distribution2:59
- Z-Score, Raw Score, Mean, & SD4:08
- Z-Score, Raw Score, Mean, & SD4:09
- Weird Z-Scores9:40
- Weird Z-Scores9:41
- Excel16:45
- For Normal Distributions16:46
- For Standard Normal Distributions19:11
- Excel Example20:24
- Types of Problems25:18
- Percentage Problem: P(x)25:19
- Raw Score and Z-Score Problems26:28
- Standard Deviation Problems27:01
- Shape Analogy27:44
- Shape Analogy27:45
- Example 1: Deaths Due to Heart Disease vs. Deaths Due to Cancer28:24
- Example 2: Heights of Male College Students33:15
- Example 3: Mean and Standard Deviation37:14
- Example 4: Finding Percentage of Values in a Standard Normal Distribution37:49

Normal Distribution: PDF vs. CDF

55m 44s

- Intro0:00
- Roadmap0:15
- Roadmap0:16
- Frequency vs. Cumulative Frequency0:56
- Frequency vs. Cumulative Frequency0:57
- Frequency vs. Cumulative Frequency4:32
- Frequency vs. Cumulative Frequency Cont.4:33
- Calculus in Brief6:21
- Derivative-Integral Continuum6:22
- PDF10:08
- PDF for Standard Normal Distribution10:09
- PDF for Normal Distribution14:32
- Integral of PDF = CDF21:27
- Integral of PDF = CDF21:28
- Example 1: Cumulative Frequency Graph23:31
- Example 2: Mean, Standard Deviation, and Probability24:43
- Example 3: Mean and Standard Deviation35:50
- Example 4: Age of Cars49:32

V. Linear Regression

Scatterplots

47m 19s

- Intro0:00
- Roadmap0:04
- Roadmap0:05
- Previous Visualizations0:30
- Frequency Distributions0:31
- Compare & Contrast2:26
- Frequency Distributions Vs. Scatterplots2:27
- Summary Values4:53
- Shape4:54
- Center & Trend6:41
- Spread & Strength8:22
- Univariate & Bivariate10:25
- Example Scatterplot10:48
- Shape, Trend, and Strength10:49
- Positive and Negative Association14:05
- Positive and Negative Association14:06
- Linearity, Strength, and Consistency18:30
- Linearity18:31
- Strength19:14
- Consistency20:40
- Summarizing a Scatterplot22:58
- Summarizing a Scatterplot22:59
- Example 1: Gapminder.org, Income x Life Expectancy26:32
- Example 2: Gapminder.org, Income x Infant Mortality36:12
- Example 3: Trend and Strength of Variables40:14
- Example 4: Trend, Strength and Shape for Scatterplots43:27

Regression

32m 2s

- Intro0:00
- Roadmap0:05
- Roadmap0:06
- Linear Equations0:34
- Linear Equations: y = mx + b0:35
- Rough Line5:16
- Rough Line5:17
- Regression - A 'Center' Line7:41
- Reasons for Summarizing with a Regression Line7:42
- Predictor and Response Variable10:04
- Goal of Regression12:29
- Goal of Regression12:30
- Prediction14:50
- Example: Servings of Mile Per Year Shown By Age14:51
- Intrapolation17:06
- Extrapolation17:58
- Error in Prediction20:34
- Prediction Error20:35
- Residual21:40
- Example 1: Residual23:34
- Example 2: Large and Negative Residual26:30
- Example 3: Positive Residual28:13
- Example 4: Interpret Regression Line & Extrapolate29:40

Least Squares Regression

56m 36s

- Intro0:00
- Roadmap0:13
- Roadmap0:14
- Best Fit0:47
- Best Fit0:48
- 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 Line4:51
- Quantitative Properties of Regression Line4:52
- So How do we Find Such a Line?6:49
- SSEs of Different Line Equations & Lowest SSE6:50
- Carl Gauss' Method8:01
- How Do We Find Slope (b1)11:00
- How Do We Find Slope (b1)11:01
- Hoe Do We Find Intercept15:11
- Hoe Do We Find Intercept15:12
- 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 It26:31
- Example 3: Summarize the Scatterplot and Find the Regression Line.34:31
- Example 4: Examine the Mean of Residuals43:52

Correlation

43m 58s

- Intro0:00
- Roadmap0:05
- Roadmap0:06
- Summarizing a Scatterplot Quantitatively0:47
- Shape0:48
- Trend1:11
- Strength: Correlation ®1:45
- Correlation Coefficient ( r )2:30
- Correlation Coefficient ( r )2:31
- Trees vs. Forest11:59
- Trees vs. Forest12:00
- Calculating r15:07
- Average Product of z-scores for x and y15:08
- Relationship between Correlation and Slope21:10
- Relationship between Correlation and Slope21:11
- Example 1: Find the Correlation between Grams of Fat and Cost24:11
- Example 2: Relationship between r and b130:24
- Example 3: Find the Regression Line33:35
- Example 4: Find the Correlation Coefficient for this Set of Data37:37

Correlation: r vs. r-squared

52m 52s

- Intro0:00
- Roadmap0:07
- Roadmap0:08
- R-squared0:44
- What is the Meaning of It? Why Squared?0:45
- Parsing Sum of Squared (Parsing Variability)2:25
- SST = SSR + SSE2:26
- What is SST and SSE?7:46
- What is SST and SSE?7:47
- r-squared18:33
- Coefficient of Determination18:34
- 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 Data23:56
- 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 137:29
- Example 4: Find the r-squared for This Set of Data39:55

Transformations of Data

27m 8s

- Intro0:00
- Roadmap0:05
- Roadmap0:06
- Why Transform?0:26
- Why Transform?0:27
- Shape-preserving vs. Shape-changing Transformations5:14
- Shape-preserving = Linear Transformations5:15
- Shape-changing Transformations = Non-linear Transformations6:20
- Common Shape-Preserving Transformations7:08
- Common Shape-Preserving Transformations7:09
- Common Shape-Changing Transformations8:59
- Powers9:00
- Logarithms9:39
- Change Just One Variable? Both?10:38
- Log-log Transformations10:39
- Log Transformations14:38
- Example 1: Create, Graph, and Transform the Data Set15:19
- Example 2: Create, Graph, and Transform the Data Set20:08
- Example 3: What Kind of Model would You Choose for this Data?22:44
- Example 4: Transformation of Data25:46

VI. Collecting Data in an Experiment

Sampling & Bias

54m 44s

- Intro0:00
- Roadmap0:05
- Roadmap0:06
- Descriptive vs. Inferential Statistics1:04
- Descriptive Statistics: Data Exploration1:05
- Example2:03
- To tackle Generalization…4:31
- Generalization4:32
- Sampling6:06
- 'Good' Sample6:40
- Defining Samples and Populations8:55
- Population8:56
- Sample11:16
- Why Use Sampling?13:09
- Why Use Sampling?13:10
- Goal of Sampling: Avoiding Bias15:04
- What is Bias?15:05
- Where does Bias Come from: Sampling Bias17:53
- Where does Bias Come from: Response Bias18:27
- Sampling Bias: Bias from Bas Sampling Methods19:34
- Size Bias19:35
- Voluntary Response Bias21:13
- Convenience Sample22:22
- Judgment Sample23:58
- Inadequate Sample Frame25:40
- Response Bias: Bias from 'Bad' Data Collection Methods28:00
- Nonresponse Bias29:31
- Questionnaire Bias31:10
- Incorrect Response or Measurement Bias37:32
- 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

- Intro0:00
- Roadmap0:05
- Roadmap0:06
- Biased vs. Unbiased Sampling Methods0:32
- Biased Sampling0:33
- Unbiased Sampling1:13
- Probability Sampling Methods2:31
- Simple Random2:54
- Stratified Random Sampling4:06
- Cluster Sampling5:24
- Two-staged Sampling6:22
- Systematic Sampling7:25
- Example 1: Which Type(s) of Sampling was this?8:33
- Example 2: Describe How to Take a Two-Stage Sample from this Book10:16
- Example 3: Sampling Methods11:58
- Example 4: Cluster Sample Plan12:48

Research Design

53m 54s

- Intro0:00
- Roadmap0:06
- Roadmap0:07
- Descriptive vs. Inferential Statistics0:51
- Descriptive Statistics: Data Exploration0:52
- Inferential Statistics1:02
- Variables and Relationships1:44
- Variables1:45
- Relationships2:49
- Not Every Type of Study is an Experiment…4:16
- Category I - Descriptive Study4:54
- Category II - Correlational Study5:50
- Category III - Experimental, Quasi-experimental, Non-experimental6:33
- Category III7:42
- Experimental, Quasi-experimental, and Non-experimental7:43
- Why CAN'T the Other Strategies Determine Causation?10:18
- Third-variable Problem10:19
- Directionality Problem15:49
- What Makes Experiments Special?17:54
- Manipulation17:55
- Control (and Comparison)21:58
- Methods of Control26:38
- Holding Constant26:39
- Matching29:11
- Random Assignment31:48
- Experiment Terminology34:09
- 'true' Experiment vs. Study34:10
- Independent Variable (IV)35:16
- Dependent Variable (DV)35:45
- Factors36:07
- Treatment Conditions36:23
- Levels37:43
- Confounds or Extraneous Variables38:04
- Blind38:38
- Blind Experiments38:39
- Double-blind Experiments39:29
- How Categories Relate to Statistics41:35
- Category I - Descriptive Study41:36
- Category II - Correlational Study42:05
- Category III - Experimental, Quasi-experimental, Non-experimental42:43
- Example 1: Research Design43:50
- Example 2: Research Design47:37
- Example 3: Research Design50:12
- Example 4: Research Design52:00

Between and Within Treatment Variability

41m 31s

- Intro0:00
- Roadmap0:06
- Roadmap0:07
- Experimental Designs0:51
- Experimental Designs: Manipulation & Control0:52
- Two Types of Variability2:09
- Between Treatment Variability2:10
- Within Treatment Variability3:31
- Updated Goal of Experimental Design5:47
- Updated Goal of Experimental Design5:48
- Example: Drugs and Driving6:56
- Example: Drugs and Driving6:57
- Different Types of Random Assignment11:27
- All Experiments11:28
- Completely Random Design12:02
- Randomized Block Design13:19
- Randomized Block Design15:48
- Matched Pairs Design15:49
- Repeated Measures Design19:47
- Between-subject Variable vs. Within-subject Variable22:43
- Completely Randomized Design22:44
- Repeated Measures Design25:03
- Example 1: Design a Completely Random, Matched Pair, and Repeated Measures Experiment26:16
- Example 2: Block Design31:41
- Example 3: Completely Randomized Designs35:11
- Example 4: Completely Random, Matched Pairs, or Repeated Measures Experiments?39:01

VII. Review of Probability Axioms

Sample Spaces

37m 52s

- Intro0:00
- Roadmap0:07
- Roadmap0:08
- Why is Probability Involved in Statistics0:48
- Probability0:49
- Can People Tell the Difference between Cheap and Gourmet Coffee?2:08
- Taste Test with Coffee Drinkers3:37
- If No One can Actually Taste the Difference3:38
- If Everyone can Actually Taste the Difference5:36
- Creating a Probability Model7:09
- Creating a Probability Model7:10
- D'Alembert vs. Necker9:41
- D'Alembert vs. Necker9:42
- Problem with D'Alembert's Model13:29
- Problem with D'Alembert's Model13:30
- Covering Entire Sample Space15:08
- Fundamental Principle of Counting15:09
- Where Do Probabilities Come From?22:54
- Observed Data, Symmetry, and Subjective Estimates22:55
- Checking whether Model Matches Real World24:27
- Law of Large Numbers24:28
- Example 1: Law of Large Numbers27:46
- Example 2: Possible Outcomes30:43
- Example 3: Brands of Coffee and Taste33:25
- Example 4: How Many Different Treatments are there?35:33

Addition Rule for Disjoint Events

20m 29s

- Intro0:00
- Roadmap0:08
- Roadmap0:09
- Disjoint Events0:41
- Disjoint Events0:42
- Meaning of 'or'2:39
- In Regular Life2:40
- In Math/Statistics/Computer Science3:10
- Addition Rule for Disjoin Events3:55
- 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 Rule5:41
- General Addition Rule5:42
- Generalized Addition Rule8:31
- 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 Party15:17
- Example 4: Home Owner's Insurance18:30

Conditional Probability

57m 19s

- Intro0:00
- Roadmap0:05
- Roadmap0:06
- 'or' vs. 'and' vs. Conditional Probability1:07
- 'or' vs. 'and' vs. Conditional Probability1:08
- 'and' vs. Conditional Probability5:57
- P (M or L)5:58
- P (M and L)8:41
- P (M|L)11:04
- P (L|M)12:24
- Tree Diagram15:02
- Tree Diagram15:03
- Defining Conditional Probability22:42
- Defining Conditional Probability22:43
- Common Contexts for Conditional Probability30:56
- Medical Testing: Positive Predictive Value30:57
- Medical Testing: Sensitivity33:03
- Statistical Tests34:27
- Example 1: Drug and Disease36:41
- Example 2: Marbles and Conditional Probability40:04
- Example 3: Cards and Conditional Probability45:59
- Example 4: Votes and Conditional Probability50:21

Independent Events

24m 27s

- Intro0:00
- Roadmap0:05
- Roadmap0:06
- Independent Events & Conditional Probability0:26
- Non-independent Events0:27
- Independent Events2:00
- Non-independent and Independent Events3:08
- Non-independent and Independent Events3:09
- Defining Independent Events5:52
- Defining Independent Events5:53
- Multiplication Rule7:29
- Previously…7:30
- But with Independent Evens8:53
- Example 1: Which of These Pairs of Events are Independent?11:12
- Example 2: Health Insurance and Probability15:12
- Example 3: Independent Events17:42
- Example 4: Independent Events20:03

VIII. Probability Distributions

Introduction to Probability Distributions

56m 45s

- Intro0:00
- Roadmap0:08
- Roadmap0:09
- Sampling vs. Probability0:57
- Sampling0:58
- Missing1:30
- What is Missing?3:06
- Insight: Probability Distributions5:26
- Insight: Probability Distributions5:27
- What is a Probability Distribution?7:29
- From Sample Spaces to Probability Distributions8:44
- Sample Space8:45
- Probability Distribution of the Sum of Two Die11:16
- The Random Variable17:43
- The Random Variable17:44
- Expected Value21:52
- Expected Value21:53
- Example 1: Probability Distributions28:45
- Example 2: Probability Distributions35:30
- Example 3: Probability Distributions43:37
- Example 4: Probability Distributions47:20

Expected Value & Variance of Probability Distributions

53m 41s

- Intro0:00
- Roadmap0:06
- Roadmap0:07
- Discrete vs. Continuous Random Variables1:04
- Discrete vs. Continuous Random Variables1:05
- Mean and Variance Review4:44
- Mean: Sample, Population, and Probability Distribution4:45
- Variance: Sample, Population, and Probability Distribution9:12
- Example Situation14:10
- Example Situation14:11
- Some Special Cases…16:13
- Some Special Cases…16:14
- Linear Transformations19:22
- Linear Transformations19:23
- What Happens to Mean and Variance of the Probability Distribution?20:12
- n Independent Values of X25:38
- n Independent Values of X25:39
- Compare These Two Situations30:56
- Compare These Two Situations30:57
- Two Random Variables, X and Y32:02
- Two Random Variables, X and Y32:03
- Example 1: Expected Value & Variance of Probability Distributions35:35
- Example 2: Expected Values & Standard Deviation44:17
- Example 3: Expected Winnings and Standard Deviation48:18

Binomial Distribution

55m 15s

- Intro0:00
- Roadmap0:05
- Roadmap0:06
- Discrete Probability Distributions1:42
- Discrete Probability Distributions1:43
- Binomial Distribution2:36
- Binomial Distribution2:37
- Multiplicative Rule Review6:54
- Multiplicative Rule Review6:55
- How Many Outcomes with k 'Successes'10:23
- Adults and Bachelor's Degree: Manual List of Outcomes10:24
- P (X=k)19:37
- Putting Together # of Outcomes with the Multiplicative Rule19:38
- Expected Value and Standard Deviation in a Binomial Distribution25:22
- Expected Value and Standard Deviation in a Binomial Distribution25:23
- Example 1: Coin Toss33:42
- Example 2: College Graduates38:03
- Example 3: Types of Blood and Probability45:39
- Example 4: Expected Number and Standard Deviation51:11

IX. Sampling Distributions of Statistics

Introduction to Sampling Distributions

48m 17s

- Intro0:00
- Roadmap0:08
- Roadmap0:09
- Probability Distributions vs. Sampling Distributions0:55
- Probability Distributions vs. Sampling Distributions0:56
- Same Logic3:55
- Logic of Probability Distribution3:56
- Example: Rolling Two Die6:56
- Simulating Samples9:53
- To Come Up with Probability Distributions9:54
- In Sampling Distributions11:12
- Connecting Sampling and Research Methods with Sampling Distributions12:11
- Connecting Sampling and Research Methods with Sampling Distributions12:12
- Simulating a Sampling Distribution14:14
- Experimental Design: Regular Sleep vs. Less Sleep14:15
- Logic of Sampling Distributions23:08
- Logic of Sampling Distributions23:09
- General Method of Simulating Sampling Distributions25:38
- General Method of Simulating Sampling Distributions25:39
- Questions that Remain28:45
- Questions that Remain28:46
- Example 1: Mean and Standard Error of Sampling Distribution30:57
- Example 2: What is the Best Way to Describe Sampling Distributions?37:12
- Example 3: Matching Sampling Distributions38:21
- Example 4: Mean and Standard Error of Sampling Distribution41:51

Sampling Distribution of the Mean

1h 8m 48s

- Intro0:00
- Roadmap0:05
- Roadmap0:06
- Special Case of General Method for Simulating a Sampling Distribution1:53
- Special Case of General Method for Simulating a Sampling Distribution1:54
- Computer Simulation3:43
- Using Simulations to See Principles behind Shape of SDoM15:50
- Using Simulations to See Principles behind Shape of SDoM15:51
- Conditions17:38
- Using Simulations to See Principles behind Center (Mean) of SDoM20:15
- Using Simulations to See Principles behind Center (Mean) of SDoM20:16
- Conditions: Does n Matter?21:31
- Conditions: Does Number of Simulation Matter?24:37
- Using Simulations to See Principles behind Standard Deviation of SDoM27:13
- Using Simulations to See Principles behind Standard Deviation of SDoM27:14
- Conditions: Does n Matter?34:45
- Conditions: Does Number of Simulation Matter?36:24
- Central Limit Theorem37:13
- SHAPE38:08
- CENTER39:34
- SPREAD39:52
- Comparing Population, Sample, and SDoM43:10
- Comparing Population, Sample, and SDoM43:11
- 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 Average55:25
- Example 2: Mean Sampling Distribution and Standard Error59:07
- Example 3: Sampling Distribution of the Mean1:01:04

Sampling Distribution of Sample Proportions

54m 37s

- Intro0:00
- Roadmap0:06
- Roadmap0:07
- 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
- Notation3:34
- Population Proportion and Sample Proportion Notations3:35
- What's the Difference?9:19
- SDoM vs. SDoSP: Type of Data9:20
- SDoM vs. SDoSP: Shape11:24
- SDoM vs. SDoSP: Center12:30
- SDoM vs. SDoSP: Spread15:34
- Binomial Distribution vs. Sampling Distribution of Sample Proportions19:14
- Binomial Distribution vs. SDoSP: Type of Data19:17
- Binomial Distribution vs. SDoSP: Shape21:07
- Binomial Distribution vs. SDoSP: Center21:43
- Binomial Distribution vs. SDoSP: Spread24:08
- Example 1: Sampling Distribution of Sample Proportions26:07
- Example 2: Sampling Distribution of Sample Proportions37:58
- Example 3: Sampling Distribution of Sample Proportions44:42
- Example 4: Sampling Distribution of Sample Proportions45:57

X. Inferential Statistics

Introduction to Confidence Intervals

42m 53s

- Intro0:00
- Roadmap0:06
- Roadmap0:07
- Inferential Statistics0:50
- Inferential Statistics0:51
- 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 - Goal7:56
- When We Don't Know m but know s7:57
- When We Don't Know18:27
- When We Don't Know m nor s18:28
- Example 1: Confidence Intervals26:18
- Example 2: Confidence Intervals29:46
- Example 3: Confidence Intervals32:18
- Example 4: Confidence Intervals38:31

t Distributions

1h 2m 6s

- Intro0:00
- Roadmap0:04
- Roadmap0:05
- 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: Commonality3:03
- z-score and t-score: Formulas3:34
- z-score and t-score: Difference5:22
- Why not z? (Why t?)7:24
- Why not z? (Why t?)7:25
- But Don't Worry!15:13
- Gossett and t-distributions15:14
- Rules of t Distributions17:05
- t-distributions are More Normal as n Gets Bigger17:06
- t-distributions are a Family of Distributions18:55
- Degrees of Freedom (df)20:02
- Degrees of Freedom (df)20:03
- t Family of Distributions24:07
- t Family of Distributions : df = 2 , 4, and 6024:08
- df = 6029:16
- df = 229:59
- How to Find It?31:01
- 'Student's t-distribution' or 't-distribution'31:02
- Excel Example33:06
- Example 1: Which Distribution Do You Use? Z or t?45:26
- Example 2: Friends on Facebook47:41
- Example 3: t Distributions52:15
- Example 4: t Distributions , confidence interval, and mean55:59

Introduction to Hypothesis Testing

1h 6m 33s

- Intro0:00
- Roadmap0:06
- Roadmap0:07
- Issues to Overcome in Inferential Statistics1:35
- Issues to Overcome in Inferential Statistics1:36
- What Happens When We Don't Know What the Population Looks Like?2:57
- How Do We Know whether a sample is Sufficiently Unlikely3:43
- Hypothesizing a Population6:44
- Hypothesizing a Population6:45
- Null Hypothesis8:07
- Alternative Hypothesis8:56
- Hypotheses11:58
- Hypotheses11:59
- Errors in Hypothesis Testing14:22
- Errors in Hypothesis Testing14:23
- Steps of Hypothesis Testing21:15
- Steps of Hypothesis Testing21:16
- Single Sample HT ( When Sigma Available)26:08
- Example: Average Facebook Friends26:09
- Step127:08
- Step 227:58
- Step 328:17
- Step 432:18
- Single Sample HT (When Sigma Not Available)36:33
- Example: Average Facebook Friends36:34
- Step1: Hypothesis Testing36:58
- Step 2: Significance Level37:25
- Step 3: Decision Stage37:40
- Step 4: Sample41:36
- Sigma and p-value45:04
- Sigma and p-value45:05
- On tailed vs. Two Tailed Hypotheses45:51
- Example 1: Hypothesis Testing48:37
- Example 2: Heights of Women in the US57:43
- Example 3: Select the Best Way to Complete This Sentence1:03:23

Confidence Intervals for the Difference of Two Independent Means

55m 14s

- Intro0:00
- Roadmap0:14
- Roadmap0:15
- One Mean vs. Two Means1:17
- One Mean vs. Two Means1:18
- Notation2:41
- A Sample! A Set!2:42
- Mean of X, Mean of Y, and Difference of Two Means3:56
- SE of X4:34
- SE of Y6:28
- 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 Hypothesis15:01
- Standard Error17:39
- When can We Construct a CI for the Difference between Two Means?21:28
- Three Conditions21:29
- Finding CI23:56
- One Mean CI23:57
- Two Means CI25:45
- Finding t29:16
- Finding t29:17
- Interpreting CI30:25
- Interpreting CI30:26
- Better Estimate of s (s pool)34:15
- Better Estimate of s (s pool)34:16
- Example 1: Confidence Intervals42:32
- Example 2: SE of the Difference52:36

Hypothesis Testing for the Difference of Two Independent Means

50m

- Intro0:00
- Roadmap0:06
- Roadmap0:07
- The Goal of Hypothesis Testing0:56
- One Sample and Two Samples0:57
- 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
- Shape6:47
- Mean for the Null Hypothesis7:26
- 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 CI10:08
- Three Conditions10:09
- Steps of Hypothesis Testing11:04
- Steps of Hypothesis Testing11:05
- Formulas that Go with Steps of Hypothesis Testing13:21
- Step 113:25
- Step 214:18
- Step 315:00
- Step 416:57
- Example 1: Hypothesis Testing for the Difference of Two Independent Means18:47
- Example 2: Hypothesis Testing for the Difference of Two Independent Means33:55
- Example 3: Hypothesis Testing for the Difference of Two Independent Means44:22

Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means

1h 14m 11s

- Intro0:00
- Roadmap0:09
- Roadmap0:10
- The Goal of Hypothesis Testing1:27
- One Sample and Two Samples1:28
- Independent Samples vs. Paired Samples3:16
- Independent Samples vs. Paired Samples3:17
- Which is Which?5:20
- Independent SAMPLES vs. Independent VARIABLES7:43
- independent SAMPLES vs. Independent VARIABLES7:44
- T-tests Always…10:48
- T-tests Always…10:49
- Notation for Paired Samples12:59
- Notation for Paired Samples13:00
- Steps of Hypothesis Testing for Paired Samples16:13
- Steps of Hypothesis Testing for Paired Samples16:14
- Rules of the SDoD (Adding on Paired Samples)18:03
- Shape18:04
- Mean for the Null Hypothesis18:31
- Standard Error for Independent Samples (When Variance is Homogenous)19:25
- Standard Error for Paired Samples20:39
- Formulas that go with Steps of Hypothesis Testing22:59
- Formulas that go with Steps of Hypothesis Testing23:00
- Confidence Intervals for Paired Samples30:32
- Confidence Intervals for Paired Samples30:33
- Example 1: Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means32:28
- Example 2: Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means44:02
- Example 3: Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means52:23

Type I and Type II Errors

31m 27s

- Intro0:00
- Roadmap0:18
- Roadmap0:19
- 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 Facebook7:01
- Two Sample t-test: Friends on Facebook13:46
- Usually, Lots of Overlap between Null and Alternative Distributions16:59
- Overlap between Null and Alternative Distributions17:00
- How Distributions and 'Box' Fit Together22:45
- How Distributions and 'Box' Fit Together22:46
- Example 1: Types of Errors25:54
- Example 2: Types of Errors27:30
- Example 3: What is the Danger of the Type I Error?29:38

Effect Size & Power

44m 41s

- Intro0:00
- Roadmap0:05
- Roadmap0:06
- Distance between Distributions: Sample t0:49
- Distance between Distributions: Sample t0:50
- Problem with Distance in Terms of Standard Error2:56
- Problem with Distance in Terms of Standard Error2:57
- 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 Size6:09
- Rules of Effect Size6:10
- Why Do We Need Effect Size?8:21
- Tells You the Practical Significance8:22
- HT can be Deceiving…10:25
- Important Note10:42
- What is Power?11:20
- What is Power?11:21
- Why Do We Need Power?14:19
- Conditional Probability and Power14:20
- 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 Power32:48
- Example 1: Effect Size & Power35:40
- Example 2: Effect Size & Power37:38
- Example 3: Effect Size & Power40:55

XI. Analysis of Variance

F-distributions

24m 46s

- Intro0:00
- Roadmap0:04
- Roadmap0:05
- Z- & T-statistic and Their Distribution0:34
- Z- & T-statistic and Their Distribution0:35
- F-statistic4:55
- The F Ration ( the Variance Ratio)4:56
- F-distribution12:29
- F-distribution12:30
- s and p-value15:00
- s and p-value15:01
- Example 1: Why Does F-distribution Stop At 0 But Go On Until Infinity?18:33
- Example 2: F-distributions19:29
- Example 3: F-distributions and Heights21:29

ANOVA with Independent Samples

1h 9m 25s

- Intro0:00
- Roadmap0:05
- Roadmap0:06
- The Limitations of t-tests1:12
- The Limitations of t-tests1:13
- Two Major Limitations of Many t-tests3:26
- Two Major Limitations of Many t-tests3:27
- Ronald Fisher's Solution… F-test! New Null Hypothesis4:43
- Ronald Fisher's Solution… F-test! New Null Hypothesis (Omnibus Test - One Test to Rule Them All!)4:44
- Analysis of Variance (ANoVA) Notation7:47
- Analysis of Variance (ANoVA) Notation7:48
- Partitioning (Analyzing) Variance9:58
- Total Variance9:59
- Within-group Variation14:00
- Between-group Variation16:22
- Time out: Review Variance & SS17:05
- Time out: Review Variance & SS17:06
- F-statistic19:22
- The F Ratio (the Variance Ratio)19:23
- S²bet = SSbet / dfbet22:13
- 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 / dfw26:05
- 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 ANOVA29:25
- Chart of Independent Samples ANOVA29:26
- Example 1: Who Uploads More Photos: Unknown Ethnicity, Latino, Asian, Black, or White Facebook Users?35:52
- Hypotheses35:53
- Significance Level39:40
- Decision Stage40:05
- Calculate Samples' Statistic and p-Value44:10
- Reject or Fail to Reject H055:54
- Example 2: ANOVA with Independent Samples58:21

Repeated Measures ANOVA

1h 15m 13s

- Intro0:00
- Roadmap0:05
- Roadmap0:06
- The Limitations of t-tests0:36
- 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 Hypothesis5:50
- Analyze Variance7:27
- Independent Samples vs. Repeated Measures9:12
- Same Start9:13
- Independent Samples ANOVA10:43
- Repeated Measures ANOVA12:00
- Independent Samples ANOVA16:00
- Same Start: All the Variance Around Grand Mean16:01
- Independent Samples16:23
- Repeated Measures ANOVA18:18
- Same Start: All the Variance Around Grand Mean18:19
- Repeated Measures18:33
- Repeated Measures F-statistic21:22
- The F Ratio (The Variance Ratio)21:23
- S²bet = SSbet / dfbet23:07
- 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 resid25:46
- What is This?25:47
- So What is SS resid?26:44
- So What is the df resid?27:36
- SS subj and df subj28:11
- 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 total31:42
- 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 ANOVA33:19
- Chart of Repeated Measures ANOVA: F and Between-samples Variability33:20
- Chart of Repeated Measures ANOVA: Total Variability, Within-subject (case) Variability, Residual Variability35:50
- Example 1: Which is More Prevalent on Facebook: Tagged, Uploaded, Mobile, or Profile Photos?40:25
- Hypotheses40:26
- Significance Level41:46
- Decision Stage42:09
- Calculate Samples' Statistic and p-Value46:18
- Reject or Fail to Reject H057:55
- Example 2: Repeated Measures ANOVA58:57
- Example 3: What's the Problem with a Bunch of Tiny t-tests?1:13:59

XII. Chi-square Test

Chi-Square Goodness-of-Fit Test

58m 23s

- Intro0:00
- Roadmap0:05
- Roadmap0:06
- 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-Fit7:23
- HT in General7:24
- Goodness-of-Fit HT8:26
- Hypotheses about Proportions12:17
- Null Hypothesis12:18
- Alternative Hypothesis13:23
- Example14:38
- Chi-Square Statistic17:52
- Chi-Square Statistic17:53
- Chi-Square Distributions24:31
- Chi-Square Distributions24:32
- Conditions for Chi-Square28:58
- Condition 128:59
- Condition 230:20
- Condition 330:32
- Condition 431:47
- Example 1: Chi-Square Goodness-of-Fit Test32:23
- Example 2: Chi-Square Goodness-of-Fit Test44:34
- 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

- Intro0:00
- Roadmap0:09
- Roadmap0:10
- Goodness-of-Fit vs. Homogeneity1:13
- Goodness-of-Fit HT1:14
- Homogeneity2:00
- Analogy2:38
- Hypotheses About Proportions5:00
- Null Hypothesis5:01
- Alternative Hypothesis6:11
- Example6:33
- Chi-Square Statistic10:12
- Same as Goodness-of-Fit Test10:13
- Set Up Data12:28
- Setting Up Data Example12:29
- Expected Frequency16:53
- Expected Frequency16:54
- Chi-Square Distributions & df19:26
- Chi-Square Distributions & df19:27
- Conditions for Test of Homogeneity20:54
- Condition 120:55
- Condition 221:39
- Condition 322:05
- Condition 422:23
- Example 1: Chi-Square Test of Homogeneity22:52
- Example 2: Chi-Square Test of Homogeneity32:10

XIII. Overview of Statistics

Overview of Statistics

18m 11s

- Intro0:00
- Roadmap0:07
- Roadmap0:08
- The Statistical Tests (HT) We've Covered0:28
- The Statistical Tests (HT) We've Covered0:29
- Organizing the Tests We've Covered…1:08
- One Sample: Continuous DV and Categorical DV1:09
- Two Samples: Continuous DV and Categorical DV5:41
- More Than Two Samples: Continuous DV and Categorical DV8:21
- The Following Data: OK Cupid10:10
- The Following Data: OK Cupid10:11
- Example 1: Weird-MySpace-Angle Profile Photo10:38
- Example 2: Geniuses12:30
- Example 3: Promiscuous iPhone Users13:37
- Example 4: Women, Aging, and Messaging16:07

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0 answers

Post by Srikanth C on April 22, 2015

I couldn't get the point where we can use any number to divide the heart of skewness. If (n-1) is 2, then we are literally dividing the skewness value from the formula by half and if (n-1) is 3 we are making it one-third, so are we not reducing the overall value of skewness by increasing n? So, will it be correct to say we can choose any number for n-1 to divide while calculating skewness?

1 answer

Last reply by: Manoj Joseph

Wed May 1, 2013 11:23 PM

Post by Manoj Joseph on May 1, 2013

I think you are making a wrong mistake to SPSS file instead of Excel shell

0 answers

Post by Kambiz Khosrowshahi on March 28, 2013

In the equation for skewness, I dont understand why it's ok to multiply it by 1/n-1. Isnt the correct equation that "heart"? If so, then by multiplying it by 1/n-1, wont it produce the wrong skewness?

1 answer

Last reply by: Professor Son

Wed Aug 15, 2012 2:15 PM

Post by KIM CARTER on April 27, 2012

Love all your work!!!!!!!!The correct spelling of (you wrote)asymtotic it is actually asymptotic. (Smile)