Chapter 1: The Nature of Social Research PDF

Fox/Levin/Forde, Elementary Statistics in Social Research, 12e Chapter 1: Why the Social Researcher Uses Statistics © 2014 by Pearson Higher Education, Inc 1 Upper Saddle River, New Jersey 0...

Fox/Levin/Forde, Elementary Statistics in Social Research, 12e Chapter 1: Why the Social Researcher Uses Statistics © 2014 by Pearson Higher Education, Inc 1 Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes Understand who the consumers and 1.1 producers of social research are Internal Use 1.1 The Nature of Social Research Consumers of Producers of Social Research Social Research The General Public Academics Agency Administrators Private Sector Investigators Policymakers Government Agencies Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes Understand the core concepts and 1.2 terms of social research Internal Use 1.2 Some Key Terms and Concepts Variable – a characteristic that differs or varies from one individual to another or from one point in time to another Constant – a characteristic that does not vary from one individual to another or from one point in time to another Unit of Observation – the element that is being studied Hypothesis – a statement of a relationship between two or more variables Independent variable – the presumed cause Dependent variable – the presumed effect or outcome 5 Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes Distinguish between the various forms 1.3 of social research Internal Use 1.3 Forms of Research The Experiment A type of research where the researcher manipulates one or more independent variables Experimental Group – the group that is manipulated Control Group – the group that is not manipulated All other initial differences between the experimental and control groups are eliminated by random assignment to these groups Internal Use 1.3 Forms of Research The Survey Retrospective Research The effects of independent variables on dependent variables are recorded after they have occurred Variables are not manipulated and subjects are not assigned to groups at random It is much more difficult to establish cause and effect Benefits Investigates a greater number of independent variables More representative – results can be generalized Internal Use 1.3 Forms of Research Content Analysis Objectively describes the content of previously produced messages May study the content of books, magazines, newspapers, films, radio, broadcasts, photographs, cartoons, letters, music, etc. Internal Use 1.3 Forms of Research Participant Observation Research where the researcher actually participates in the daily life of the people under study Either openly or covertly Internal Use 1.3 Forms of Research Secondary Analysis Research done using data collected by another researcher Benefit Cost effective Limitations Limited to what is available No control over what was asked, how it was asked, or why it was asked Internal Use 1.3 Forms of Research Meta-Analysis Research that combines the results obtained in a number of previous studies Effect size – a measure of the extent to which a relationship exists in the population Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes Understand why social researchers 1.4 test hypotheses Internal Use 1.4 Why Test Hypotheses? Commonsense observations are often based on narrow, The acceptance of biased preconceptions invalid conclusions and personal experiences Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes Understand the stages of social 1.5 research Internal Use 1.5 The Stages of Social Research Develop Develop Collect Analyze Hypotheses Instruments Data Data Interpret and Communicate Results Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes Distinguish between the three levels 1.6 of measurement Internal Use 1.6 Levels of Measurement Interval/ Nominal Ordinal Ratio Naming or Labeling Ordering of Categories Ordering and Exact Distances There are different ways to measure the same variable Ordinal variables can be treated as interval/ratio variables if the distances between response categories are assumed to be equal Internal Use 1.6 Figure 1.1 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes Distinguish between the descriptive 1.7 and decision-making functions of statistics Internal Use 1.7 The Functions of Statistics Description Decisions Frequency and grouped- Inferences and generalizations frequency distributions from a sample to a population Graphs and tables Testing hypotheses regarding the nature of social reality Arithmetic averages 21  Fox/Levin/Forde, Elementary Statistics in Social Research, 12e Chapter 2: Organizing the Data © 2014 by Pearson Higher Education, Inc 22 Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use 2.1 Introduction Formulas and statistical techniques are used by researchers to: Organize raw data Test hypotheses Raw data is often difficult to synthesize Frequency tables make raw data easier to understand 23 Internal Use 2.1 Frequency Distributions of Nominal Data Characteristics of a frequency distribution of nominal data: Responses of Young Boys to Title Removal of Toy Consists of two columns: Response of Child f Left column: Cry 25 characteristics (e.g., Response of Child) Express Anger 15 Right column: frequency Withdraw 5 (f) Ply with another toy 5 N=50 24 Internal Use 2.1 Comparing Distributions Comparisons clarify results, add information, and allow for comparisons Response to Removal of Toy by Gender of Child Gender of Child Response of Child Male Female Cry 25 28 Express Anger 15 3 Withdraw 5 4 Play with another toy 5 15 Total 50 50 25 Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes Calculate proportions, percentages, 2.2 ratios, and rates Internal Use 2.2 Proportions and Percentages Allows for a comparison of groups of different sizes Proportion – number of cases compared to the total size P= f of distribution N Percentage – the frequency of f occurrence of a category per % = (100) N 100 cases 27 Internal Use 2.2 Ratio and Rates Ratio – compares the frequency f1 of one category to another Ratio = f2 Rate – compares between  f actual cases  Rate = (1,000 )   actual and potential cases  f potential cases  28 Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes Create simple and grouped 2.3 frequency distributions Internal Use 2.3 TABLE 2.4 The Distribution of Marital Status Shown Three Ways Marital Status f Marital Status f Marital Status f Married 30 Single 20 Previously married 10 Single 20 Previously married 10 Married 30 Previously married 10 Married 30 Single 20 Total 60 Total 60 Total 60 Table 2.4 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use 2.3 TABLE 2.5 A Frequency Distribution of Attitudes toward a Proposed Tuition Hike on a College Campus: Incorrect and Correct Presentations Attitude toward Attitude toward a Tuition Hike f a Tuition Hike f Slightly favorable 2 Strongly favorable 0 Somewhat unfavorable 21 Somewhat favorable 1 Strongly favorable 0 Slightly favorable 2 Slightly unfavorable 4 Slightly unfavorable 4 Strongly unfavorable 10 Somewhat unfavorable 21 Somewhat favorable 1 Strongly unfavorable 10 Total 38 Total 38 INCORRECT CORRECT Table 2.5 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use Grouped Frequency Distribution of 2.3 Interval Data Used to clarify the presentation of interval-level scores spread over a wide range Class Intervals Smaller categories or groups containing more than one score Class interval size determined by the number of score values it contains 32 Internal Use TABLE 2.7 Grouped Frequency 2.3 Distribution of Final-Examination Grades for 71 Students Class Interval f % 95–99 3 4.23 90–94 2 2.82 85–89 4 5.63 80–84 7 9.86 75–79 12 16.90 70–74 17 23.94 65–69 12 16.90 60–64 5 7.04 55–59 5 7.04 50–54 4 5.63 Total 71 100 a a The percentages as they appear add to only 99.99%. We write the sum as 100% instead, because we know that.01% was lost in rounding. Table 2.7 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use 2.3 Class Limits and the Midpoint Class Limits The point halfway between i =U −L adjacent intervals Upper and lower limits i = size of a class interval – Distance from upper and U = upper limit of a class interval lower limit determines the size of class interval L = lower limit of a class interval The Midpoint The middlemost score value in a class interval – The sum of the lowest and highest value in a class interval divided by two lowest score value + highest score value m= 2 34 Internal Use 2.3 Cumulative Distributions Cumulative Frequencies Total number of cases having a given score or a score that is lower – Shown as cf – Obtained by the sum of frequencies in that category plus all lower categories’ frequencies Cumulative Percentage Percentage of cases having a given score or a score that is lower cf c % = (100 ) N 35 Internal Use TABLE 2.7 Grouped Frequency 2.3 Distribution of Final-Examination Grades for 71 Students Class Interval f % 95–99 3 4.23 90–94 2 2.82 85–89 4 5.63 80–84 7 9.86 75–79 12 16.90 70–74 17 23.94 65–69 12 16.90 60–64 5 7.04 55–59 5 7.04 50–54 4 5.63 Total 71 100 a a The percentages as they appear add to only 99.99%. We write the sum as 100% instead, because we know that.01% was lost in rounding. Table 2.7 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use 2.3 Percentiles The percentage of cases falling at or below a given score Deciles – points that divide a distribution into 10 equally sized portions Quartiles – points that divide a distribution into quarters Median – the point that divides a distribution in two, half above it and half below it 37 Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes 2.4 Create cross-tabulations Internal Use 2.4 Table 2.17 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use 2.4 Percents within Cross-Tabulations f Total Percents: total % = (100 ) Ntotal f Row Percents: row % = (100 ) Nrow f Column Percents: column% = (100 ) Ncolumn The choice comes down to which is more relevant to the purpose of the analysis If the independent variable is on the rows, use row percents If the independent variable is on the columns, use column percents If the independent variable is unclear, use whichever percent is most meaningful 40 Internal Use 2.4 Table 2.18 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes Distinguish between various forms of 2.5 graphic presentations Internal Use 2.5 Figure 2.4 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use 2.5 Figure 2.6 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use 2.5 Figure 2.9 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use 2.5 Figure 2.11 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use 2.5 Figure 2.12 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use 2.5 Figure 2.14 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use 2.5 Figure 2.15 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved  Fox/Levin/Forde, Elementary Statistics in Social Research, 12e Chapter 3: Measures of Central Tendency © 2014 by Pearson Higher Education, Inc 50 Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use CHAPTER OBJECTIVES 3.1 Calculate the mode, the median, and the mean 3.2 Calculate deviations 3.3 Calculate the weighted mean Calculate the mode, the median, and the mean from a simple 3.4 frequency distribution Understand what influences a researcher’s decision to use a 3.5 specific measure of central tendency © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes Calculate the mode, the median, 3.1 and the mean Internal Use 3.1 Introduction Measures of Central Tendency Mode Median Mean Internal Use 3.1 The Mode The most frequently occurring value in a distribution Example: 20, 21, 30, 20, 22, 20, 21, 20 – Mode = 20 Sometimes there is more than one mode – Example: 96, 91, 96, 90, 93, 90, 96, 90 – Mode = 90 and 96 This is a bimodal distribution The mode is the only measure of central tendency appropriate for nominal-level variables 54 Internal Use 3.1 Figure 3.1 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use 3.1 The Median The middlemost case in a distribution Appropriate for ordinal or interval level data N +1 Position of Median = How to find the median: 2 – Cases must be ordered – If there are an odd number of cases, there will be a single middlemost case – If there are an even number of cases, there will be two middlemost cases – The halfway point between these two cases should be used as the median 56 Internal Use 3.1 The Median: Example 1 What is the median of the following distribution: 1, 5, 2, 9, 13, 11, 4 Step 1: Sort distribution from lowest to highest 1, 2, 4, 5, 9, 11, 13 Step 2: Locate the position of the median N +1 7 +1 8 Position of Median = = = =4 2 2 2 Step 3: Locate the median 1, 2, 4, 5, 9, 11, 13 Internal Use 3.1 The Median: Example 2 What is the median of the following distribution: 4, 3, 1, 1, 6, 2, 2, 4 Step 1: Sort distribution from lowest to highest 1, 1, 2, 2, 3, 4, 4, 6 Step 2: Locate the position of the median N +1 8 +1 9 Position of Median = = = = 4.5 2 2 2 Step 3: Locate the median 1, 1, 2, 2, 3, 4, 4, 6 Step 4: Take the halfway point between the two cases Median = 2.5 Internal Use 3.1 The Mean The “center of gravity” of a distribution Appropriate for interval/level data X= X N X = mean = sum X = raw scores in a set of scores N = total number of scores in a set 59 Internal Use 3.1 Figure 3.2 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use 3.1 The Mean: Example What is the mean of the following distribution: 4, 8, 11, 2 X=  X N X= ( 4 + 8 + 11 + 2 ) 4 25 X= 4 X = 6.25 years Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes 3.2 Calculate deviations Internal Use 3.2 Deviations The distance and direction of any raw score from the mean Deviation = X − X The sum of the deviations that fall above the mean is equal in absolute value to the sum of the deviations that fall below the mean. 63 Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes 3.3 Calculate the weighted mean Internal Use 3.3 The Weighted Mean The “mean of the means” The overall mean for a number of groups Xw =  N group X group Ntotal X w = weighted mean X group = mean of a particular group Ngroup = number in a particular group Ntotal = number in all groups combined 65 Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes Calculate the mode, the median, 3.4 and the mean from a simple frequency distribution Internal Use Obtaining the Mode, Median, and Mean 3.4 from a Simple Frequency Distribution X f cf fX 31 1 25 31 30 1 24 30 29 1 23 29 28 0 22 0 N + 1 25 + 1 26 27 2 22 54 Position of the Mdn = = = =13 26 3 20 78 2 2 2 25 1 17 25 24 1 16 24 X 23 2 15 46 Mdn 22 2 13 44 X =  fX 575 = = 23 21 2 11 42 20 3 9 60 N 25 Mo 19 4 6 76 18 2 2 36 67 Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes Understand what influences a 3.5 researcher’s decision to use a specific measure of central tendency Internal Use 3.5 Comparing the Mode, Median, and Mean Three factors in choosing a measure of central tendency Level of Shape of Research Measurement Distribution Objective Internal Use 3.5 Level of Measurement Level of Mode Median Mean Measurement Nominal Yes No No Ordinal Yes Yes No Interval Yes Yes Yes 70 Internal Use 3.5 Shape of the Distribution Symmetrical Distributions The mode, median, and mean have identical values Skewed Distributions The mode is the peak of the curve The mean is closer to the tail The median falls between the two Bimodal Distributions Both modes should be used to describe the data 71 Internal Use 3.5 Figure 3.3 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use 3.5 Figure 3.4 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use 3.5 Research Objective Fast and Simple Research → Mode Skewed Distribution → Median Advanced Statistics Analysis → Mean 74  Fox/Levin/Forde, Elementary Statistics in Social Research, 12e Chapter 4: Measures of Variability © 2014 by Pearson Higher Education, Inc 75 Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use CHAPTER OBJECTIVES 4.1 Calculate the range and inter-quartile range 4.2 Calculate the variance and standard deviation Obtain the variance and standard deviation from a simple 4.3 frequency distribution 4.4 Understand the meaning of the standard deviation 4.5 Calculate the coefficient of variation 4.6 Use box plots to visualize distributions © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes Calculate the range and inter- 4.1 quartile rage Internal Use 4.1 Introduction Measures of Measures of Central Tendency Variability Summarizes what is average or Summarizes how scores are typical of a distribution scattered around the center of the distribution 78 Internal Use 4.1 The Range The difference between the highest and lowest scores in a distribution R = H −L R = range H = highest score in a distribution L = lowest score in a distribution Provides a crude measure of variation – Outliers affect interpretation 79 Internal Use 4.1 The Inter-Quartile Range The difference between the score at the first quartile and the score at the third quartile IQR = Q3 − Q1 IQR = inter-quartile range Q1 = the score value at or below which 25% of the cases fall Q3 = the score value at or below which 75% of the cases fall Manages the effects of extreme outliers – Sensitive to the way in which scores are concentrated around the center of the distribution 80 Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes Calculate the variance and standard 4.2 deviation Internal Use 4.2 The Variance We need a measure of variability that takes into account every score Deviation: the distance of any given raw score from the mean Squaring deviations eliminates the minus signs Summing the squared deviations and dividing by N gives us the average of the squared deviations ( ) 2 X−X s2 = N s 2 = variance ( ) 2 X−X = sum of the squared deviations from the mean N = total number of scores 82 Internal Use 4.2 The Standard Deviation With the variance, the unit of measurement is squared It is difficult to interpret squared units We can remove the squared units by taking the square root of both sides of the equation This will give us the standard deviation ( ) 2 X−X s= N 83 Internal Use 4.2 The Raw-Score Formulas There is an easier way to calculate the variance and standard deviation Using raw scores s 2 = X 2 − X2 N s 2 = variance s = standard deviation N = total number of scores X 2 = mean squared s= X 2 − X2 N 84 Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes Obtain the variance and standard 4.3 deviation from a simple frequency distribution Internal Use 4.3 Example Obtaining the variance and standard deviation from a simple frequency distribution X f fX fX2 31 1 31 961 X=  fX = 575 = 23 30 1 30 900 N 25 29 1 29 841 28 0 0 0 2 27 2 54 1,458 X = (23)2 = 529 26 3 78 2,028 25 1 25 625 24 1 24 576 s 2 =  fX 2 2 −X = 13,589 − 529 = 543.56 − 529 = 14.56 23 2 46 1,058 N 25 22 2 44 968 21 2 42 882 20 3 60 1,200 s=  fX 2 2 − X = 14.56 = 3.82 19 4 76 1,444 N 18 2 36 648 575 13,589 Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes Understand the meaning of the 4.4 standard deviation Internal Use 4.4 The Meaning of the Standard Deviation The standard deviation converts the variance to units we can understand But, how do we interpret this new score? The standard deviation represents the average variability in a distribution – It is the average deviations from the mean The greater the variability, the larger the standard deviation Allows for a comparison between a given raw score in a set against a standardized measure 88 Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes 4.5 Calculate the coefficient of variation Internal Use 4.5 The Coefficient of Variation Used to compare the variability for two or more characteristics that have been measured in different units The coefficient of variation is based on the size of the standard deviation Its value is independent of the unit of the measurement s  CV = 100   X CV = coefficient of variation s = standard deviation X = mean 90 Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes 4.6 Use box plots to visualize distributions Internal Use 4.6 Figure 4.4 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use 4.6 Figure 4.5 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved  Fox/Levin/Forde, Elementary Statistics in Social Research, 12e Chapter 5: Probability and the Normal Curve © 2014 by Pearson Higher Education, Inc 94 Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use CHAPTER OBJECTIVES 5.1 Calculate probabilities and understand the rules of probability 5.2 Understand the concept of a probability distribution 5.3 List the characteristics of the normal curve 5.4 Understand the area under the normal curve 5.5 Calculate and use z scores © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes Calculate probabilities and 5.1 understand the rules of probability Internal Use 5.1 Probability The relative likelihood of occurrence of any given outcome P=0 P =.5 P=1 The outcome is The outcome is The outcome is Impossible as likely to certain happen as not happen Internal Use 5.1 The Rules of Probability Probability Number of times the outcome or event can occur P (F) = Total number of times any outcome or event can occur Converse Rule: The probability that something will not occur ( ) P F = 1 − P (F ) Addition Rule: The probability of obtaining one of several different and distinct outcomes P ( A or B) = P ( A ) + P (B) Multiplication Rule: The probability of obtaining two or more outcomes in combination P ( A and B) = P ( A ) X P (B) Internal Use 5.1 Probability: Example Heads or Tails? Let’s flip a coin two times: 1 Probability of heads on the first flip: =.5 2 1 Probability of heads on the second flip: =.5 2 Probability of getting heads on both flip: (.5 )(.5 ) =.25 Internal Use 5.1 Probability: Example Heads or Tails? What is the probability that one flip in two flips will land on heads? P (HT ) + P ( TH) = P (H) P ( T ) + P ( T ) P (H) = (.50 )(.50 ) + (.50 )(.50 ) =.25 +.25 =.50 Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes Understand the concept of a 5.2 probability distribution Internal Use 5.2 Probability Distributions Directly analogous to a frequency distribution Except it is based on probability theory Standard Mean = μ Deviation = σ 10 2 Internal Use 5.2 Figure 5.1 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes List the characteristics of the normal 5.3 curve Internal Use 5.3 Characteristics of the Normal Curve Smooth Symmetrical Unimodal Mean = Median = Mode Infinite in Both Directions Probability Distribution Mean = μ; Standard Deviation = σ Areas Under the Curve = 100% 10 5 Internal Use 5.3 The Reality of the Normal Curve The normal curve is a theoretical ideal Some variables do not conform to the normal curve Many distributions are skewed, multi-modal, and symmetrical but not bell-shaped Assuming normality when it does not exist can impact the validity of our conclusions 10 6 Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes Understand the area under the 5.4 normal curve Internal Use 5.4 Figure 5.5 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use 5.4 Figure 5.6 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use 5.4 Figure 5.7 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use 5.4 Figure 5.8 © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Internal Use Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes 5.5 Calculate and use z scores Internal Use 5.5 Standard Scores and the Normal Curve It is possible to determine the area under the curve for any sigma distance from the mean This distance is called a z score Indicates direction and distance that any raw score deviates from the mean in sigma units X − z=   = mean of a distribution  = standard deviation of a distribution z = standard score 11 3 Internal Use Finding Probability under the Normal 5.5 Curve When the normal curve is used in conjunction with z scores and Table A in Appendix C, we can determine the probability of obtaining any raw score (X) in a distribution The converse, addition, and multiplication rules still apply We can also reverse this process to calculate score values from particular portions of area or percentages X =  + z 11 4