1. 1 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Chapter 11 Understanding Statistics in Research

2. 2 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Clinical Uses of Statistics  Reading or critiquing published research  Examining outcomes of nursing practice by analyzing data collected in clinical site  Developing administrative reports with support data  Analyzing research done by nursing staff and other health professionals at a clinical site  Demonstrating a problem or need and conducting a study

3. 3 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Stages in Data Analysis 1.Prepare data for analysis. 2.Describe the sample. 3.Test the reliability of measurement methods. 4.Conduct exploratory analysis. 5.Conduct confirmatory analysis guided by hypotheses, questions, or objectives. 6.Conduct post hoc analyses.

4. 4 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Preparing the Data for Analysis 1.Enter data into the computer using means designed to reduce errors. 2.Clean the data to ensure accuracy. 3.Correct all identified errors. 4.Identify missing data points. 5.Add missing data when possible.

5. 5 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Describing the Sample  Purpose: to obtain as complete a picture of the sample as possible  Determine frequencies of variables related to sample Age Education Gender Health status Ethnicity

6. 6 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Describing the Sample (cont’d)  Examine averages and variation of demographic variables.  If there are study groups, compare using variables such as age, education, health status, gender, and ethnicity.  Determine the comparability of groups.  If groups are not comparable, planned comparative analyses cannot be performed.

7. 7 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Testing Reliability of Measurement  Examine reliability of study scales before testing hypotheses, questions, or objectives using Cronbach’s alpha coefficient.  Values should be at least 0.70.

8. 8 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Cronbach’s Alpha Coefficient  Tests internal consistency of measurement scale  To what extent is the measure a true reflection of subject’s responses?  Reliability of 0.7 is the lowest acceptable alpha.  This means that 70% of the time you can trust the score to accurately reflect what is being measured.

9. 9 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Conducting Exploratory Analysis  Determine the nature of data in variables used to test hypotheses, questions, and objectives.  Identify outliers (subjects or data points with extreme values or values unlike the rest of the sample).  Examine relationships among variables.

10. 10 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Conducting Confirmatory Analyses  Perform analyses designed to test hypothesis, research questions, or objectives.  Generalize findings from sample to appropriate populations (inference).

11. 11 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Performing Post Hoc Analyses  Necessary when ANOVA is used in studies with three or more groups  Necessary with chi-square analyses  Purpose: to determine which groups are significantly different

12. 12 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Probability Theory  Deductive  Used to explain:  Extent of a relationship  Probability of an event occurring  Probability that an event can be accurately predicted  Expressed as lowercase p with values expressed as percents

13. 13 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Probability  If probability is 0.23, then p = 0.23.  There is a 23% probability that a particular event will occur.  Probability is usually expected to be p < 0.05.

14. 14 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Decision Theory  Inductive reasoning  Assumes that all the groups in a study used to test a hypothesis are components of the same population relative to the variables under study.  It is up to the researcher to provide evidence that there really is a difference.  To test the assumption of no difference, a cutoff point is selected before analysis.

15. 15 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Alpha (  )  Risk of making a type I error  The threshold at which statistical significance is reached

16. 16 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Cutoff Point  Referred to as level of significance or alpha (α)  Point at which the results of statistical analysis are judged to indicate a statistically significant difference between groups  For most nursing studies, level of significance is 0.05.  Sometimes written as α = 0.05

17. 17 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Cutoff Point (cont’d)  The cutoff point is absolute.  If value obtained is only a fraction above the cutoff point, groups are from the same population.  No meaning can be attributed to differences between the groups.  Results that reveal a significant difference of 0.001 are not considered more significant than the cutoff point.

18. 18 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Levels of Acceptable Significance  0.05  0.01  0.005  0.001

19. 19 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Inference  A conclusion or judgment based on evidence  Judgments are made based on statistical results  Statistical inferences must be made cautiously and with great care  Decision theory rules were designed to increase the probability that inferences are accurate

20. 20 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Generalization  A generalization is the application of information that has been acquired from a specific instance to a general situation.  Generalizing requires making an inference.  Both inference and generalization require the use of inductive reasoning.

21. 21 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Generalization (cont’d)  An inference is made from a specific case and extended to a general truth, from a part to a whole, from the known to the unknown.  In research, an inference is made from the study findings to a more general population.

22. 22 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Normal Curve  A theoretical frequency distribution of all possible values in a population  No real distribution exactly fits the normal curve.  However, in most sets of data, the distribution is similar to the normal curve.  Levels of significance and probability are based on the logic of the normal curve.

23. 23 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Normal Curve (cont’d)

24. 24 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Tailedness  An extreme score can occur in either tail of the normal curve.  An extreme score is higher or lower than 95% of the population.  Mean scores of a population also can be extreme and occur in the tail of the normal curve.

25. 25 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Tailedness (cont’d)  If the mean score is an extreme value, the population is not likely to be the same as that represented by the normal curve; it is significantly different.  However, extreme values that are members of the population do occur. Thus there is always a risk of making an error in deciding that the groups are different.

26. 26 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Two-Tailed Test  Assumes that an extreme score can occur in either tail of the normal curve  Nondirectional hypothesis: tests for significance in either tail  Hypothesis: the extreme score is higher or lower than 95% of the population; thus sample with extreme score is not a member of the same population  A two-tailed test of significance is used.

27. 27 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Two-Tailed Test (cont’d)

28. 28 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. One-Tailed Test  Extreme values occur on a single tail of the curve.  The hypothesis is directional: one-tailed test of significance used  The 5% of statistical values considered significant will be in one tail rather than two.  Extreme values in the other tail are not considered significantly different.  One-tailed tests are more powerful than two- tailed tests.

29. 29 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. One-Tailed Test (cont’d)

30. 30 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Type I and Type II Errors  Type I error occurs when the researcher rejects the null hypothesis when it is true.  The results indicate that there is a significant difference, when in reality there is not.  Type II error occurs when the researcher regards the null hypothesis as true but it is false.  The results indicate there is no significant difference, when in reality there is a difference.

31. 31 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Data analysisIn reality, theIn reality, the indicates:null hypothesis null hypothesis is true:is false: Results significant—nullType I errorCorrect decision Results not significant—nullCorrect decisionType II error not rejected Occurrence of Type I and Type II Errors

32. 32 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Risk of Type I Error

33. 33 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Power and Risk for Type II Error  Power analysis = 0.80 minimum  Influenced by the sample size and the effect size

34. 34 Copyright © 2011 by Saunders, an imprint of Elsevier Inc.  You have six scores and the mean = 6. What is the value of score #6? Can the value of score #6 vary? Can the other five scores vary? The number of scores that can vary is your degree of freedom. 1 = 54 = 4 2 = 75 = 4 3 = 86 = ? Degrees of Freedom

35. 35 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Using Statistics to Describe  Descriptive statistics are also referred to as summary statistics.  In any study in which the data are numerical, data analysis begins with descriptive statistics.  In simple descriptive studies, analysis may be limited to descriptive statistics.

36. 36 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Types of Descriptive Statistics  Frequency distributions  Ungrouped frequency distributions  Grouped frequency distributions  Percentage distributions  Measures of central tendency  Measures of dispersion

37. 37 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Example of an Ungrouped Frequency Distribution  Data are presented in raw, counted form. 1: / 2: ///// 3: /// 4: / 5: //

38. 38 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Example of a Grouped Frequency Distribution  Data are pregrouped into categories. Ages 20 to 39: 14 Ages 40 to 59: 43 Ages 60 to 79: 26 Ages 80 to 100: 4

39. 39 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Example of Percentage Distribution  Salaries: 41.7%  Maintenance: 8.3%  Equipment: 16.7%  Fixed costs: 8.3%  Supplies: 25%

40. 40 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Commonly Used Graphic Displays of Frequency Distribution

41. 41 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Measures of Central Tendency  What is a typical score?

42. 42 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Mode  Is the numerical value or score that occurs with greatest frequency  Is expressed graphically  Is not always the center of distribution

43. 43 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Bimodal Distribution

44. 44 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Median  Is the value in exact center of ungrouped frequency distribution  Is obtained by rank ordering the values  When number of values is uneven, may not be an actual value in data set

45. 45 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Mean  Is the sum of values divided by the number of values being summed  Like the median, the mean may not be a data set value.

46. 46 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Measures of Dispersion  Range  Variance  Standard deviation  Standardized scores  Scatterplots

47. 47 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Range  Is obtained by subtracting lowest score from highest score  Uses only the two extreme scores  Very crude measure and sensitive to outliers

48. 48 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Difference Scores  The sum of all difference scores in a data set is zero, making it a useless measure.  Difference scores are the basis for many statistical analysis procedures.

49. 49 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Difference Scores (cont’d)  Are obtained by subtracting the mean from each score  Sometimes referred to as a deviation score because it indicates the extent to which a score deviates from the mean

50. 50 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Standard Deviation  Is the square root of the variance  Just as the mean is the “average” value, the standard deviation is the “average” difference score.

51. 51 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Standardized Scores  Raw scores that cannot be compared and are transformed into standardized scores  Common standardized score is a Z-score.  Provides a way to compare scores in a similar process

52. 52 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Scatterplots  Have two scales: horizontal axis (X) and vertical axis (Y)  Illustrates a relationship between two variables

53. 53 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Structure of a Plot

54. 54 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Example of a Scatterplot

55. 55 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Chi-Square Test of Independence  Used with nominal or ordinal data  Tests for differences between expected frequencies if groups are alike and frequencies actually observed in the data

56. 56 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Regular No Regular Exercise Exercise Total Male 3515 50 Female 1040 50 Total 4555 100 Example of Chi-Square Table

57. 57 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Chi-Square Results  Indicates that there is a significant difference between some of the cells in the table  The difference may be between only two of the cells, or there may be differences among all of the cells.  Chi-square results will not tell you which cells are different.

58. 58 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Example of Chi-Square Results   2 = 4.98, df = 2, p = 0.05

59. 59 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Pearson Product-Moment Correlation  Tests for the presence of a relationship between two variables  Called bivariate correlation  Types of correlation are available for all levels of data. Best results are obtained using interval data.

60. 60 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Correlation  Performed on data collected from a single sample  Measures of the two variables to be examined must be available for each subject in the data set.

61. 61 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Correlation (cont’d)  Results  Nature of the relationship (positive or negative)  Magnitude of the relationship (–1 to +1)  Testing the significance of a correlation coefficient  Does not identify direction of a relationship (one variable does not cause the other)  Are symmetrical

62. 62 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Correlation Results  r = 0.56(p = 0.03)  r = –0.13(p = 0.2)  r = 0.65(p < 0.002)  Which ones are significant?

63. 63 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Explained Variance  Definition: The R 2 is the variation between two variables expressed as a percentage.

64. 64 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Factor Analysis  Examines relationships among large numbers of variables  Disentangles those relationships to identify clusters of variables most closely linked  Sorts variables according to how closely related they are to the other variables  Closely related variables grouped into a factor

65. 65 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Factor Analysis (cont’d)  Several factors may be identified within a data set.  The researcher must explain why the analysis grouped the variables in a specific way.  Statistical results indicate the amount of variance in the data set that can be explained by each factor and the amount of variance in each factor that can be explained by a particular variable.

66. 66 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Usefulness of Factor Analysis  Aids in development of theoretical constructs  Aids in development of measurement scales

67. 67 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Regression Analysis  Used when one wishes to predict the value of one variable based on the value of one or more other variables  For example, one might wish to predict the possibility of passing the credentialing exam based on grade point average (GPA) from a graduate program.

68. 68 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Regression Analysis (cont’d)  Regression analysis could also be used to predict the length of stay in a neonatal unit based on the combined effect of multiple variables such as gestational age, birth weight, number of complications, and sucking strength.

69. 69 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Regression Analysis (cont’d)  The outcome of analysis is the regression coefficient R.  When R is squared, it indicates the amount of variance in the data that is explained by the equation.  The R 2 is also called the coefficient of multiple determination.

70. 70 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Regression Results  R 2 = 0.63  This result indicates that 63% of the variance in length of stay can be predicted by the combined effect of age, weight, complications, and sucking strength.

71. 71 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Overlay of Scatterplot and Best-Fit Line

72. 72 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. t-Test  Requires interval level measures  Tests for significant differences between two samples  Most commonly used test of differences

73. 73 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Example of t-Test Results  t = 4.169 (p < 0.05)

74. 74 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Analysis of Variance (ANOVA)  Tests for differences between means  More flexible than other analyses in that it can examine data from two or more groups

75. 75 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. ANOVA (cont’d)  Multiple versions of ANOVA are available that can be used in studies examining multiple outcome variables, or repeated measures of outcome variables across several time periods.  Can look at between-group variance, within- group variance, and total variance

76. 76 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Results of ANOVA  F = 9.75 (2, 95) (p = 0.002)  If there are more than two groups under study, it is not possible to determine where the significant differences are.  Post hoc tests are used to determine the location of differences.

77. 77 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Analysis of Covariance (ANCOVA)  Allows the researcher to examine the effect of a treatment apart from the effect of one or more potentially confounding variables  Potentially confounding variables that are commonly of concern include pretest scores, age, education, social class, and anxiety level.

78. 78 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. ANCOVA (cont’d)  The effects on study variables are statistically removed by performing regression analysis before performing ANOVA.  Allows the effect of the treatment to be examined more precisely

79. 79 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Information Needed for Algorithm 1.Determine whether the research question focuses on differences (I) or associations (relationships) (II). 2.Determine level of measurement (A, B, or C). 3.Select the design listed that most closely fits the study you are critiquing (1, 2, or 3). 4.Determine whether the study samples are independent (a), dependent (b), or mixed (c).

80. 80 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Algorithm for Choosing a Statistical Test

81. 81 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Judging Statistical Suitability  Factors that must be considered include:  Study purpose  Hypotheses, questions, or objectives  Design  Level of measurement

82. 82 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Judging Statistical Suitability (cont’d)  Requires you to be familiar with the statistical procedures used in the study  Requires you to compare the statistical procedures used with other statistics that could have been used to greater advantage  Are there dependent or independent groups?

83. 83 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Judging Statistical Suitability (cont’d)  You must judge whether the procedure was performed appropriately and the results were interpreted correctly.  Judgments required  Whether the data for analysis were treated as nominal, ordinal, or interval  The number of groups in the study  Whether the groups were dependent or independent

84. 84 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Types of Results  Significant and predicted results  Nonsignificant results  Mixed results  Unexpected results

85. 85 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Significant and Predicted Results  Are in keeping with those predicted by researcher and support logical links developed by researcher among the framework, questions, variables, and measurement tools

86. 86 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Nonsignificant Results  Also called negative or inconclusive results  Analysis showed no significant differences or relationships.  Could be a true reflection of reality. If so, the researcher or theory used by researcher to develop hypothesis is in error. In this case, negative findings are an important addition to the body of knowledge.

87. 87 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Nonsignificant Results (cont’d)  Results could stem from a type II error  Causes of type II error include:  Inappropriate methods  Biased or small sample  Internal validity problems

88. 88 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Nonsignificant Results (cont’d)  Inadequate measurement  Weak statistical measures  Faulty analysis

89. 89 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Significant and Unpredicted Results  Are opposite of those predicted  Indicate flaws in the logic of both the researcher and the theory being tested  If valid, are an important addition to the body of knowledge

90. 90 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Mixed Results  Most common outcome of studies  One variable may uphold predicted characteristics, whereas another does not.  Or two dependent measures of the same variable may show opposite results  May be caused by methodology problems  May indicate need to modify existing theory

91. 91 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Unexpected Results  Relationships between variables that were not hypothesized and not predicted from the framework being used  Can be useful in theory development, modification of existing theory, development of later studies

92. 92 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Unexpected Results (cont’d)  Serendipitous results are important as evidence in developing the implications of the study.  They must be evaluated carefully because the study was not designed to examine these results.

93. 93 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Findings  Results of the study that have been translated and interpreted  A consequence of evaluating evidence

94. 94 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Conclusions  A synthesis of the findings using:  Logical reasoning  Creative formation of meaningful whole from pieces of information obtained through data analysis and findings from previous studies  Receptivity to subtle clues in data  Alternative explanations of data

95. 95 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Conclusions (cont’d)  Risk in developing conclusions is going beyond the data  Forming conclusions not warranted by data  Occurs more frequently in published studies than one would like to believe

96. 96 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Implications  The meanings of conclusions for the body of nursing knowledge, theory, and practice  Based on, but more specific than, conclusions  Provide specific suggestions for implementing the findings

97. 97 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Significance of Findings  Associated with importance to the nursing body of knowledge  May be associated with:  Amount of variance explained  Control in the study design to eliminate unexplained variance  Detection of statistically significant differences

98. 98 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Clinical Significance  Findings can have statistical significance but not clinical significance.  Related to practical importance of the findings  No common agreement in nursing about how to judge clinical significance  Effect size?  Difference sufficiently important to warrant changing the patient’s care?

99. 99 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Clinical Significance (cont’d)  Who should judge clinical significance?  Patients and their families?  Clinician/researcher?  Society at large?  Clinical significance is ultimately a value judgment.

100. 100 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Generalizing the Findings  Extends the implications of the findings:  From the sample studied to a larger population  From the situation studied to a more general situation  How far can generalizations be made?

101. 101 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Empirical Generalizations  Are based on accumulated evidence from many studies  Are important for verification of theoretical statements or for development of new theory  Are the basis of a science  Contribute to scientific conceptualization

102. 102 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Suggesting Further Studies  Researcher gains knowledge and experience from conducting the study that can be used to design a better study next time.  Researcher often makes suggestions for future studies that logically emerge from the present study.

103. 103 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Suggesting Further Studies (cont’d)  Replications  Different design  Larger sample  Hypotheses emerging from findings  Strategies to further test framework in use

104. 104 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Critiquing Statistics in a Study  What statistics were used to describe the characteristics of the sample?  Are the data analysis procedures clearly described?  Did statistics address the purpose of the study?

105. 105 Copyright © 2011 by Saunders, an imprint of Elsevier Inc. Critiquing Statistics in a Study (cont’d)  Did the statistics address the objectives, questions, or hypotheses of the study?  Were the statistics appropriate for the level of measurement of each variable?