1. Review Statistics and Probability.pdf
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The document provides an overview of statistics, emphasizing key concepts such as populations, samples, descriptive and inferential statistics, central tendency, and variability. It discusses various statistical terms, methods for measuring data, and the roles of different types of variables in research. Additionally, it covers the importance of sample size and error margins, culminating in the significance of understanding variability when interpreting statistical results.
1. Review Statistics and Probability.pdf
- 1. Introduction and Descriptive Statistics Review Statistics and Probability Modifiedby: Dr. AchmadNizar Hidayanto Nur FitriahAyuning Budi KhumaisaNuraini
- 2. Learning Outcomes • Review key statistical and research terms 1 • Review the concept of central tendency 2 • Review the concept of variability 3
- 3. Introduction to Statistics PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Eighth Edition by Frederick J Gravetter and Larry B. Wallnau
- 4. 1.1 Statistics, Science and Observations • “Statistics” means “statistical procedures” • Uses of Statistics – Organize and summarize information – Determine exactly what conclusions are justified based on the results that were obtained • Goals of statistical procedures – Accurate and meaningful interpretation – Provide standardized evaluation procedures
- 5. 1.2 Populations and Samples • Population – The set of all the individuals of interest in a particular study – Vary in size; often quite large • Sample – A set of individuals selected from a population – Usually intended to represent the population in a research study
- 6. Figure 1.1 Relationship between population and sample
- 7. Variables and Data • Variable – Characteristic or condition that changes or has different values for different individuals • Data (plural) – Measurements or observations of a variable • Data set – A collection of measurements or observations • A datum (singular) – A single measurement or observation – Commonly called a score or raw score
- 8. Parameters and Statistics • Parameter – A value, usually a numerical value, that describes a population – Derived from measurements of the individuals in the population • Statistic – A value, usually a numerical value, that describes a sample – Derived from measurements of the individuals in the sample
- 9. Descriptive & Inferential Statistics • Descriptive statistics – Summarize data – Organize data – Simplify data • Familiar examples – Tables – Graphs – Averages • Inferential statistics – Study samples to make generalizations about the population – Interpret experimental data • Common terminology – “Margin of error” – “Statistically significant”
- 10. Sampling Error • Sample is never identical to population • Sampling Error – The discrepancy, or amount of error, that exists between a sample statistic and the corresponding population parameter • Example: Margin of Error in Polls – “This poll was taken from a sample of registered voters and has a margin of error of plus-or-minus 4 percentage points” (Box 1.1)
- 11. Figure 1.2 A demonstration of sampling error
- 12. Figure 1.3 Role of statistics in experimental research
- 13. 1.3 Data Structures, Research Methods, and Statistics • Individual Variables – A variable is observed – “Statistics” describe the observed variable – Category and/or numerical variables – Descriptive statistics • Relationships between variables – Two variables observed and measured – One of two possible data structures used to determine what type of relationship exists
- 14. Relationships Between Variables • Data Structure I: The Correlational Method – One group of participants – Measurement of two variables for each participant – Goal is to describe type and magnitude of the relationship – Patterns in the data reveal relationships – Non-experimental method of study
- 15. Figure 1.4 Data structures for studies evaluating the relationship between variables
- 16. Correlational Method Limitations • Can demonstrate the existence of a relationship • Does not provide an explanation for the relationship • Most importantly, does not demonstrate a cause-and-effect relationship between the two variables
- 17. Relationships Between Variables • Data Structure II: Comparing two (or more) groups of Scores – One variable defines the groups – Scores are measured on second variable – Both experimental and non-experimental studies use this structure
- 18. Figure 1.5 Data structure for studies comparing groups
- 19. Experimental Method • Goal of Experimental Method – To demonstrate a cause-and-effect relationship • Manipulation – The level of one variable is determined by the experimenter • Control rules out influence of other variables – Participant variables – Environmental variables
- 20. Figure 1.6 The structure of an experiment
- 21. Independent/Dependent Variables • Independent Variable is the variable manipulated by the researcher – Independent because no other variable in the study influences its value • Dependent Variable is the one observed to assess the effect of treatment – Dependent because its value is thought to depend on the value of the independent variable
- 22. Experimental Method: Control • Methods of control – Random assignment of subjects – Matching of subjects – Holding level of some potentially influential variables constant • Control condition – Individuals do not receive the experimental treatment. – They either receive no treatment or they receive a neutral, placebo treatment – Purpose: to provide a baseline for comparison with the experimental condition • Experimental condition – Individuals do receive the experimental treatment
- 23. Non-experimental Methods • Non-equivalent Groups – Researcher compares groups – Researcher cannot control who goes into which group • Pre-test / Post-test – Individuals measured at two points in time – Researcher cannot control influence of the passage of time • Independent variable is quasi-independent
- 24. Figure 1.7 Two examples of non-experimental studies Insert NEW Figure 1.7
- 25. 1.4 Variables and Measurement • Scores are obtained by observing and measuring variables that scientists use to help define and explain external behaviors • The process of measurement consists of applying carefully defined measurement procedures for each variable
- 26. Constructs & Operational Definitions • Constructs – Internal attributes or characteristics that cannot be directly observed – Useful for describing and explaining behavior • Operational – Identifies the set of operations required to measure an external (observable) behavior – Uses the resulting measurements as both a definition and a measurement of a hypothetical construct
- 27. Discrete and Continuous Variables • Discrete variable – Has separate, indivisible categories – No values can exist between two neighboring categories • Continuous variable – Have an infinite number of possible values between any two observed values – Every interval is divisible into an infinite number of equal parts
- 28. Figure 1.8 Example: Continuous Measurement
- 29. Real Limits of Continuous Variables • Real Limits are the boundaries of each interval representing scores measured on a continuous number line – The real limit separating two adjacent scores is exactly halfway between the two scores – Each score has two real limits • The upper real limit marks the top of the interval • The lower real limit marks the bottom of the interval
- 30. Scales of Measurement • Measurement assigns individuals or events to categories – The categories can simply be names such as male/female or employed/unemployed – They can be numerical values such as 68 inches or 175 pounds • The complete set of categories makes up a scale of measurement • Relationships between the categories determine different types of scales
- 31. Scales of Measurement Scale Characteristics Examples Nominal •Label and categorize •No quantitative distinctions •Gender •Diagnosis •Experimental or Control Ordinal •Categorizes observations •Categories organized by size or magnitude •Rank in class •Clothing sizes (S,M,L,XL) •Olympic medals Interval •Ordered categories •Interval between categories of equal size •Arbitrary or absent zero point •Temperature •IQ •Golf scores (above/below par) Ratio •Ordered categories •Equal interval between categories •Absolute zero point •Number of correct answers •Time to complete task •Gain in height since last year
- 32. Central Tendency PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Seventh Edition by Frederick J Gravetter and Larry B. Wallnau
- 33. 1.5 Overview of central tendency • Central tendency – A single score to define the “center” of a distribution • Purpose: find the single score that is most typical or best represents the entire group
- 34. Figure 1.9 What is the “center” of each distribution?
- 35. 1.6 The Mean • The mean is the sum of all the scores divided by the number of scores in the data. Population Mean Sample Mean N X n X M
- 36. The Mean: Three definitions • Sum of the scores divided by the number of scores in the data • Amount each individual receives when total is divided equally among all: M = ∑X / n • The balance point for the distribution
- 37. Figure 1.10
- 38. Computing the Mean from a Frequency Distribution Table Quiz Score (X) f fX 10 1 10 9 2 18 8 4 32 7 0 0 6 1 6 Total n = Σf = 8 ΣfX = 66 M = ??
- 39. The Weighted Mean • Combine two sets of scores • Three steps: – Determine the combined sum of all the scores – Determine the combined number of scores – Divide the sum of scores by the total number of scores 2 1 2 1 mean (weighted) overall n n X X M
- 40. Characteristics of the Mean • Changing the value of any score changes the mean. • Introducing a new score or removing a score usually changes the mean. • Adding or subtracting a constant from each score changes the mean by the same constant. • Multiplying or dividing each score by a constant multiplies or divides the mean by that constant.
- 41. Figure 1.11
- 42. 1.7 The Median • The median is the midpoint of the scores in a distribution when they are listed in order from smallest to largest. • The median divides the scores into two groups of equal size.
- 43. Figure 1.12
- 44. Figure 1.13
- 45. The Precise Median for a Continuous Variable • A continuous variable can be infinitely divided • The precise median is located in the interval defined by the real limits of the value. • It may be necessary to determine the fraction of the interval needed to divide the distribution exactly in half. • interval in the number 50% reach to needed number fraction
- 46. Figure 1.14
- 47. Median, Mean, and Middle • Mean is the balance point of a distribution – Defined by distances – Often is not the midpoint of the scores • Median is the midpoint of a distribution – Defined by number of scores – Often is not the balance point of the scores • Both measure central tendency, using two different concepts of middle or “central.”
- 48. Figure 1.15
- 49. 1.8 The Mode • The mode is the score or category that has the greatest frequency of any in the frequency distribution – Can be used with any scale of measurement – Corresponds to an actual score in the data – The only one used with nominal data • It is possible to have more than one mode
- 50. Figure 1.16
- 51. 1.9 Selecting a Measure of Central Tendency Measure of Central Tendency Appropriate to choose when … Should not be used when… Mean No situation precludes it •Extreme scores •Skewed distribution •Undetermined values •Open-ended distribution •Ordinal scale •Nominal scale Median •Extreme scores •Skewed distribution •Undetermined values •Open-ended distribution •Ordinal scale •Nominal scale Mode •Nominal scales •Discrete variables •Describing shape •Interval or ratio data, except to accompany mean or median
- 52. Figure 1.17
- 53. Figure 1.18 Means or Medians in a Line Graph
- 54. Figure 1.19 Means or Medians in a Bar Graph
- 55. • Symmetrical distributions – Mean and median have same value – If exactly one mode, it has same value as the mean and the median – Distribution may have more than one mode, or no mode at all 1.10 Central Tendency and the Shape of the Distribution
- 56. Figure 1.20
- 57. Central Tendency in Skewed Distributions • Mean is found far toward the long tail (positive or negative) • Median is found toward the long tail, but not as far as the mean • Mode is found near the piled-up scores. • If positively skewed, order from left to right is mode, median, mean; • If negatively skewed, order from left to right is mean, median, mode
- 58. Figure 1.21
- 59. Variability PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Seventh Edition by Frederick J Gravetter and Larry B. Wallnau
- 60. 1.11 Overview • Variability can be defined several ways – A quantitative measure of the differences between scores – Describes the degree to which the scores are spread out or clustered together • Purposes of Measure of Variability – Describe the distribution – Measure how well an individual score represents the distribution
- 61. Figure 1.22 Population Distributions: Height, Weight
- 62. Three Measures of Variability • The Range • The Standard Deviation • The Variance
- 63. 1.12 The Range • The distance covered by the scores in a distribution – From smallest value to highest value • For continuous data, real limits are used • For discrete variables range is number of categories range = URL for Xmax — LRL for Xmin
- 64. 1.13 Standard Deviation and Variance for a Population • Most common and most important measure of variability – A measure of the standard, or average, distance from the mean – Describes whether the scores are clustered closely around the mean or are widely scattered • Calculation differs for population and samples
- 65. Developing the Standard Deviation • Step One: Determine the Deviation Score (distance from the mean) for each score: • Step Two: Calculate Mean (Average) of Deviations – Deviations sum to 0 because M is balance point of the distribution – The Mean (Average) Deviation will always equal 0; another method must be found Deviation score = X — μ
- 66. Developing the Standard Deviation (2) • Step Three: Get rid of negatives in Deviations: – Square each deviation score – Using the squared values, compute the Mean Squared Deviation, known as the Variance – • Variability is now measured in squared units and is called the Variance. Population variance equals the mean squared deviation -- Variance is the average squared distance from the mean
- 67. Developing the Standard Deviation (2) • Step Four: – Variance measures the average squared distance from the mean; not quite on goal • Correct for having squared all the deviations by taking the square root of the variance Variance Deviation Standard
- 68. Figure 1.23 Calculation of the Variance
- 69. Formulas for Population Variance and Standard Deviation • • SS (sum of squares) is the sum of the squared deviations of scores from the mean • Two equations for computing SS scores of number deviations squared of sum Variance
- 70. Two formulas for SS Definitional Formula • Find each deviation score (X–μ) • Square each deviation score, (X–μ)2 • Sum up the squared deviations Computational Formula 2 X SS • Square each score and sum the squared scores • Find the sum of scores, square it, divide by N • Subtract the second part from the first N X X SS 2 2
- 71. Population Variance: Formula and Notation Formula N SS N SS deviation standard variance Notation • Lowercase Greek letter sigma is used to denote the standard deviation of a population: σ • Because the standard deviation is the square root of the variance, we write the variance of a population as σ2
- 72. Figure 1.24 Graphic Representation of Mean and Standard Deviation
- 73. 1.14 Standard Deviation and Variance for a Sample • Goal of inferential statistics: – Draw general conclusions about population – Based on limited information from a sample • Samples differ from the population – Samples have less variability – Computing the Variance and Standard Deviation in the same way as for a population would give a biased estimate of the population values
- 74. Figure 1.25 Population of Adult Heights
- 75. Variance and Standard Deviation for a Sample • Sum of Squares (SS) is computed as before • Formula has n-1 rather than N in the denominator • Notation uses s instead of σ 1 1 2 n SS n SS s sample of deviation standard s sample of variance
- 76. Degrees of Freedom • Population variance – Mean is known – Deviations are computed from a known mean • Sample variance as estimate of population – Population mean is unknown – Using sample mean restricts variability • Degrees of freedom – Number of scores in sample that are independent and free to vary – Degrees of freedom (df) = n – 1
- 77. 1.15 More about Variance and Standard Deviation • Unbiased estimate of a population parameter – Average value of statistic is equal to parameter – Average value uses all possible samples of a particular size n • Biased estimate of a population parameter – Systematically overestimates or underestimates (as with variance) the population parameter
- 78. Table 4.1 Biased & Unbiased Estimates Sample Statistics Sample 1st Score 2nd Score Mean Biased (used n) Unbiased (used n-1) 1 0 0 0.00 0.00 0.00 2 0 3 1.50 2.25 4.50 3 0 9 4.50 20.25 40.50 4 3 0 1.50 2.25 4.50 5 3 3 3.00 0.00 0.00 6 3 9 6.00 9.00 18.00 7 9 0 4.50 20.25 40.50 8 9 3 6.00 9.00 18.00 9 9 9 9.00 0.00 0.00 Totals 36.00 63.00/9 126.00/8 Actual σ2 = 14 This is an adaptation of Table 4.1
- 79. Figure 1.26 Sample of n = 20, M = 36, and s = 4
- 80. Transformations of Scale • Adding a constant to each score – The Mean is changed – The standard deviation is unchanged • Multiplying each score by a constant – The Mean is changed – Standard Deviation is also changed – The Standard Deviation is multiplied by that constant
- 81. Variance and Inferential Statistics • Goal of inferential statistics: To detect meaningful and significant patterns in research results • Variability in the data influences how easy it is to see patterns – High variability obscures patterns that would be visible in low variability samples – Variability is sometimes called error variance
- 82. Figure 1.27 Experiments with high and low variability