Statistics
Chapter 1: Introduction to Statistics..........................................................................................4
1.1 Statistics Science and Observations................................................................................. 4
1.2 Data Structures, Research Methods, and Statistics.......................................................... 5
1.3 Variables and Measurement.............................................................................................. 8
1.4 Statistical notes..................................................................................................................9
Summary:...........................................................................................................................9
Chapter 2: Frequency distributions......................................................................................... 11
2.1 Frequency Distribution and Frequency Distribution Tables..............................................11
2.2 Grouped Frequency Distribution Tables...........................................................................11
2.3 Frequency Distribution Graphs........................................................................................ 12
2.4 Percentile, Percentile rank and Interpolation................................................................... 14
Summary:.........................................................................................................................15
Chapter 3: Central Tendency.................................................................................................... 16
3.1 Overview..........................................................................................................................16
3.2 The Mean.........................................................................................................................16
3.3 The Median......................................................................................................................17
3.4 The Mode.........................................................................................................................18
3.6 Central Tendency and the Shape of the Distribution....................................................... 20
Summary:.........................................................................................................................21
Chapter 4: Variability................................................................................................................. 22
4.1 Introduction of Variability.................................................................................................22
4.2 Defining Standard Deviation and Variance..................................................................... 23
4.3 Measuring Variance and Standard Deviation for a Population....................................... 23
4.4 Measuring Standard Deviation and Variance for a Sample............................................ 24
4.5 Sample Variance as an Unbiased Statistic..................................................................... 26
4.6 More about Variance and Standard Deviation................................................................ 26
Summary:.........................................................................................................................27
Chapter 5: z-Scores: Location of Scores and Standardized Distributions.......................... 28
5.1 Introduction to z-Scores...................................................................................................28
5.2 z-Scores and Location in a Distribution........................................................................... 29
5.3 Other Relationships Between z, X, and.......................................................................... 29
5.4 Using z-Scores to Standardize a Distribution.................................................................. 29
5.5 Other Standardized Distributions Based on z-Scores..................................................... 30
5.6 Computing z-Scores for Samples.................................................................................... 30
5.7 Looking Ahead to Inferential Statistics.............................................................................31
Summary:.........................................................................................................................31
Chapter 6: Probability............................................................................................................... 32
6.1 Introduction to Probability................................................................................................ 32
, 6.2 Probability and the Normal Distribution........................................................................... 34
6.3 Probabilities and Proportions for Scores from a Normal Distribution...............................35
6.4 Probability and the Binomial Distribution......................................................................... 35
6.5 Looking Ahead to Inferential Statistics.............................................................................36
Summary:.........................................................................................................................37
Chapter 7:................................................................................................................................... 38
7.1 Samples, Populations, and the Distribution of Sample Means........................................ 38
7.2 The Distribution of Sample Means for any Population and any Sample Size..................39
7.3 Probability and the Distribution of Sample Means........................................................... 41
7.4 More about Standard Error.............................................................................................. 42
7.5 Looking Ahead to Inferential Statistics.............................................................................43
Summary:.........................................................................................................................43
Chapter 8:................................................................................................................................... 44
8.1 The Logic of Hypothesis Testing......................................................................................44
8.2 Uncertainty and Errors in Hypothesis Testing..................................................................47
8.3 More about Hypothesis Tests.......................................................................................... 49
8.4 Directional (One-Tailed) Hypothesis Tests.......................................................................51
8.5 Concerns about Hypothesis Testing: Measuring Effect Size........................................... 52
8.6 Statistical Power.............................................................................................................. 53
Summary:.........................................................................................................................54
Chapter 9:................................................................................................................................... 55
9.1: The t Statistic: An alternative to z................................................................................... 55
9.2: Hypothesis Tests with the t Statistic................................................................................58
9.3: Measuring Effect Size for the t Statistic.......................................................................... 59
9.4: Directional Hypotheses and One-Tailed Tests................................................................ 61
Summary:.........................................................................................................................62
Chapter 10:................................................................................................................................. 63
10.1 Introduction to the Independent-Measures Design........................................................63
10.2 The Null Hypothesis and the Independent-Measures t Statistic.................................... 63
10.3 Hypothesis Tests with the Independent-Measures t Statistic.........................................66
10.4 Effect Size and Confidence Intervals for the Independent Measures t.......................... 68
10.5 The Role of Sample Variance and Sample Size in the Independent-Measures t Test.. 70
Summary:.........................................................................................................................70
Chapter 11:................................................................................................................................. 72
11.1 Introduction to Repeated-Measures Designs.................................................................72
11.2 The t Statistic for a Repeated-Measures Research Design........................................... 72
11.3 Hypothesis Tests for the Repeated-Measures Design................................................... 74
11.4 Effect Size and Confidence Intervals for the Repeated-Measures t.............................. 74
11.5 Comparing Repeated- and Independent-Measures Designs.........................................76
Summary:.........................................................................................................................78
Chapter 15:................................................................................................................................. 79
,15.1 Introduction.................................................................................................................... 79
15.2 The Pearson Correlation................................................................................................79
15.3 Using and Interpreting the Pearson Correlation............................................................ 81
15.4 Hypothesis Tests with the Pearson Correlation............................................................. 83
15.5 Alternatives to the Pearson Correlation......................................................................... 85
Summary:.........................................................................................................................88
, Chapter 1: Introduction to Statistics
1.1 Statistics Science and Observations
Statistics = shortened version of statistical procedures
Statistics: set of mathematical procedure for organizing, summarizing and interpreting information
Statistics serve two general proposes:
1. Used to organize and summarize the information so that the researcher can see what happened
in the research study and can communicate the results to others.
2. Help the researcher to answer the questions that initiated the research by determining exactly
what general conclusions are justified based on the specific results that were obtained.
Statistical procedures help ensure that the information or observations are presented and interpreted in
an accurate and informative way → help researchers bring order out of chaos.
Statistics provide researchers with a set of standardized techniques that are recognized and understood
throughout the scientific community → statistical methods used by one researcher will be familiar to other
researchers, who can accurately interpret the statistical analyses with a full understanding of how the
analysis was done and what the results signify.
Population: set of all individuals of interest in a particular study
Sample: Set of individual selected from a population usually intended to represent the population in a
research study
- Relationship between a population and a sample
Population and sample can also refer to scores: populations or samples of scores. Because research
typically involves measuring each individual to obtain a score, every sample (or population) of individuals
produces a corresponding sample (or population) of scores.
Variable: Characteristic or condition that changes or has different value for different individuals. Variable
can be:
- Characteristics that differ from one individual to another
- Environmental conditions that change
To demonstrate changes in variables, it is necessary to make measurements of the variables being
examined.
- Data (plural) are measurements or observations.
- Data set is a collection of measurements or observations.
- Datum (singular) is a single measurement or observation and is commonly called a score or raw
score.
Chapter 1: Introduction to Statistics..........................................................................................4
1.1 Statistics Science and Observations................................................................................. 4
1.2 Data Structures, Research Methods, and Statistics.......................................................... 5
1.3 Variables and Measurement.............................................................................................. 8
1.4 Statistical notes..................................................................................................................9
Summary:...........................................................................................................................9
Chapter 2: Frequency distributions......................................................................................... 11
2.1 Frequency Distribution and Frequency Distribution Tables..............................................11
2.2 Grouped Frequency Distribution Tables...........................................................................11
2.3 Frequency Distribution Graphs........................................................................................ 12
2.4 Percentile, Percentile rank and Interpolation................................................................... 14
Summary:.........................................................................................................................15
Chapter 3: Central Tendency.................................................................................................... 16
3.1 Overview..........................................................................................................................16
3.2 The Mean.........................................................................................................................16
3.3 The Median......................................................................................................................17
3.4 The Mode.........................................................................................................................18
3.6 Central Tendency and the Shape of the Distribution....................................................... 20
Summary:.........................................................................................................................21
Chapter 4: Variability................................................................................................................. 22
4.1 Introduction of Variability.................................................................................................22
4.2 Defining Standard Deviation and Variance..................................................................... 23
4.3 Measuring Variance and Standard Deviation for a Population....................................... 23
4.4 Measuring Standard Deviation and Variance for a Sample............................................ 24
4.5 Sample Variance as an Unbiased Statistic..................................................................... 26
4.6 More about Variance and Standard Deviation................................................................ 26
Summary:.........................................................................................................................27
Chapter 5: z-Scores: Location of Scores and Standardized Distributions.......................... 28
5.1 Introduction to z-Scores...................................................................................................28
5.2 z-Scores and Location in a Distribution........................................................................... 29
5.3 Other Relationships Between z, X, and.......................................................................... 29
5.4 Using z-Scores to Standardize a Distribution.................................................................. 29
5.5 Other Standardized Distributions Based on z-Scores..................................................... 30
5.6 Computing z-Scores for Samples.................................................................................... 30
5.7 Looking Ahead to Inferential Statistics.............................................................................31
Summary:.........................................................................................................................31
Chapter 6: Probability............................................................................................................... 32
6.1 Introduction to Probability................................................................................................ 32
, 6.2 Probability and the Normal Distribution........................................................................... 34
6.3 Probabilities and Proportions for Scores from a Normal Distribution...............................35
6.4 Probability and the Binomial Distribution......................................................................... 35
6.5 Looking Ahead to Inferential Statistics.............................................................................36
Summary:.........................................................................................................................37
Chapter 7:................................................................................................................................... 38
7.1 Samples, Populations, and the Distribution of Sample Means........................................ 38
7.2 The Distribution of Sample Means for any Population and any Sample Size..................39
7.3 Probability and the Distribution of Sample Means........................................................... 41
7.4 More about Standard Error.............................................................................................. 42
7.5 Looking Ahead to Inferential Statistics.............................................................................43
Summary:.........................................................................................................................43
Chapter 8:................................................................................................................................... 44
8.1 The Logic of Hypothesis Testing......................................................................................44
8.2 Uncertainty and Errors in Hypothesis Testing..................................................................47
8.3 More about Hypothesis Tests.......................................................................................... 49
8.4 Directional (One-Tailed) Hypothesis Tests.......................................................................51
8.5 Concerns about Hypothesis Testing: Measuring Effect Size........................................... 52
8.6 Statistical Power.............................................................................................................. 53
Summary:.........................................................................................................................54
Chapter 9:................................................................................................................................... 55
9.1: The t Statistic: An alternative to z................................................................................... 55
9.2: Hypothesis Tests with the t Statistic................................................................................58
9.3: Measuring Effect Size for the t Statistic.......................................................................... 59
9.4: Directional Hypotheses and One-Tailed Tests................................................................ 61
Summary:.........................................................................................................................62
Chapter 10:................................................................................................................................. 63
10.1 Introduction to the Independent-Measures Design........................................................63
10.2 The Null Hypothesis and the Independent-Measures t Statistic.................................... 63
10.3 Hypothesis Tests with the Independent-Measures t Statistic.........................................66
10.4 Effect Size and Confidence Intervals for the Independent Measures t.......................... 68
10.5 The Role of Sample Variance and Sample Size in the Independent-Measures t Test.. 70
Summary:.........................................................................................................................70
Chapter 11:................................................................................................................................. 72
11.1 Introduction to Repeated-Measures Designs.................................................................72
11.2 The t Statistic for a Repeated-Measures Research Design........................................... 72
11.3 Hypothesis Tests for the Repeated-Measures Design................................................... 74
11.4 Effect Size and Confidence Intervals for the Repeated-Measures t.............................. 74
11.5 Comparing Repeated- and Independent-Measures Designs.........................................76
Summary:.........................................................................................................................78
Chapter 15:................................................................................................................................. 79
,15.1 Introduction.................................................................................................................... 79
15.2 The Pearson Correlation................................................................................................79
15.3 Using and Interpreting the Pearson Correlation............................................................ 81
15.4 Hypothesis Tests with the Pearson Correlation............................................................. 83
15.5 Alternatives to the Pearson Correlation......................................................................... 85
Summary:.........................................................................................................................88
, Chapter 1: Introduction to Statistics
1.1 Statistics Science and Observations
Statistics = shortened version of statistical procedures
Statistics: set of mathematical procedure for organizing, summarizing and interpreting information
Statistics serve two general proposes:
1. Used to organize and summarize the information so that the researcher can see what happened
in the research study and can communicate the results to others.
2. Help the researcher to answer the questions that initiated the research by determining exactly
what general conclusions are justified based on the specific results that were obtained.
Statistical procedures help ensure that the information or observations are presented and interpreted in
an accurate and informative way → help researchers bring order out of chaos.
Statistics provide researchers with a set of standardized techniques that are recognized and understood
throughout the scientific community → statistical methods used by one researcher will be familiar to other
researchers, who can accurately interpret the statistical analyses with a full understanding of how the
analysis was done and what the results signify.
Population: set of all individuals of interest in a particular study
Sample: Set of individual selected from a population usually intended to represent the population in a
research study
- Relationship between a population and a sample
Population and sample can also refer to scores: populations or samples of scores. Because research
typically involves measuring each individual to obtain a score, every sample (or population) of individuals
produces a corresponding sample (or population) of scores.
Variable: Characteristic or condition that changes or has different value for different individuals. Variable
can be:
- Characteristics that differ from one individual to another
- Environmental conditions that change
To demonstrate changes in variables, it is necessary to make measurements of the variables being
examined.
- Data (plural) are measurements or observations.
- Data set is a collection of measurements or observations.
- Datum (singular) is a single measurement or observation and is commonly called a score or raw
score.
