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1. Educational Research Chapter 11 Descriptive Statistics Gay, Mills, and Airasian

2. Topics Discussed in this Chapter • Preparing data for analysis • Types of descriptive statistics • Central tendency • Variation • Relative position • Relationships • Calculating descriptive statistics

3. Preparing Data for Analysis • Issues • Scoring procedures • Tabulation and coding • Use of computers

4. Scoring Procedures • Instructions • Standardized tests detail scoring instructions • Teacher-made tests require the delineation of scoring criteria and specific procedures • Types of items • Selected response items - easily and objectively scored • Open-ended items - difficult to score objectively with a single number as the result Objectives 1.1 & 1.2

5. Tabulation and Coding • Tabulation is organizing data • Identifying all information relevant to the analysis • Separating groups and individuals within groups • Listing data in columns • Coding • Assigning names to variables • EX1 for pretest scores • SEX for gender • EX2 for posttest scores Objectives 2.1, 2.2, & 2.3

6. Tabulation and Coding • Reliability • Concerns with scoring by hand and entering data • Machine scoring • Advantages • Reliable scoring, tabulation, and analysis • Disadvantages • Use of selected response items, answering on scantrons Objectives 1.4 & 1.5

7. Tabulation and Coding • Coding • Assigning identification numbers to subjects • Assigning codes to the values of non-numerical or categorical variables • Gender: 1=Female and 2=Male • Subjects: 1=English, 2=Math, 3=Science, etc. • Names: 001=John Adams, 002=Sally Andrews, 003=Susan Bolton, … 256=John Zeringue Objectives 2.2 & 2.3

8. Computerized Analysis • Need to learn how to calculate descriptive statistics by hand • Creates a conceptual base for understanding the nature of each statistic • Exemplifies the relationships among statistical elements of various procedures • Use of computerized software • SPSS-Windows • Other software packages Objective 2.4

9. Descriptive Statistics • Purpose – to describe or summarize data in a parsimonious manner • Four types • Central tendency • Variability • Relative position • Relationships Objective 2.4

10. Descriptive Statistics • Graphing data – a frequency polygon • Vertical axis represents the frequency with which a score occurs • Horizontal axis represents the scores themselves Objectives 3.1 & 3.2

11. Central Tendency • Purpose – to represent the typical score attained by subjects • Three common measures • Mode • Median • Mean Objective 4.1

12. Central Tendency • Mode • The most frequently occurring score • Appropriate for nominal data • Median • The score above and below which 50% of all scores lie (i.e., the mid-point) • Characteristics • Appropriate for ordinal scales • Doesn’t take into account the value of each and every score in the data Objectives 4.2, 4.3, & 4.4

13. Central Tendency • Mean • The arithmetic average of all scores • Characteristics • Advantageous statistical properties • Affected by outlying scores • Most frequently used measure of central tendency • Formula Objectives 4.2, 4.3, & 4.4

14. Variability • Purpose – to measure the extent to which scores are spread apart • Four measures • Range • Quartile deviation • Variance • Standard deviation Objective 5.1

15. Variability • Range • The difference between the highest and lowest score in a data set • Characteristics • Unstable measure of variability • Rough, quick estimate Objectives 5.2 & 5.3

16. Variability • Quartile deviation • One-half the difference between the upper and lower quartiles in a distribution • Characteristic - appropriate when the median is being used Objectives 5.2 & 5.3

17. Variability • Variance • The average squared deviation of all scores around the mean • Characteristics • Many important statistical properties • Difficult to interpret due to “squared” metric • Formula Objectives 5.2 & 5.3

18. Variability • Standard deviation • The square root of the variance • Characteristics • Many important statistical properties • Relationship to properties of the normal curve • Easily interpreted • Formula Objectives 5.2 & 5.3

19. The Normal Curve • A bell shaped curve reflecting the distribution of many variables of interest to educators • See Figure 14.2 • See the attached slide Objective 6.1

20. The Normal Curve • Characteristics • Fifty-percent of the scores fall above the mean and fifty-percent fall below the mean • The mean, median, and mode are the same values • Most participants score near the mean; the further a score is from the mean the fewer the number of participants who attained that score • Specific numbers or percentages of scores fall between ±1 SD, ±2 SD, etc. Objectives 6.1, 6.2, & 6.3

21. The Normal Curve • Properties • Proportions under the curve • ±1 SD = 68% • ±1.96 SD = 95% • ±2.58 SD = 99% • Cumulative proportions and percentiles Objectives 6.3 & 6.4

22. Skewed Distributions • Positive – many low scores and few high scores • Negative – few low scores and many high scores • Relationships between the mean, median, and mode • Positively skewed – mode is lowest, median is in the middle, and mean is highest • Negatively skewed – mean is lowest, median is in the middle, and mode is highest Objectives 7.1 & 7.2

23. Measures of Relative Position • Purpose – indicates where a score is in relation to all other scores in the distribution • Characteristics • Clear estimates of relative positions • Possible to compare students’ performances across two or more different tests provided the scores are based on the same group Objectives 7.1 & 7.2

24. Measures of Relative Position • Types • Percentile ranks – the percentage of scores that fall at or above a given score • Standard scores – a derived score based on how far a raw score is from a reference point in terms of standard deviation units • z score • T score • Stanine Objectives 9.3 & 9.4

25. Measures of Relative Position • z score • The deviation of a score from the mean in standard deviation units • The basic standard score from which all other standard scores are calculated • Characteristics • Mean = 0 • Standard deviation = 1 • Positive if the score is above the mean and negative if it is below the mean • Relationship with the area under the normal curve Objective 9.5

26. Measures of Relative Position • z score (continued) • Possible to calculate relative standings like the percent better than a score, the percent falling between two scores, the percent falling between the mean and a score, etc. • Formula Objective 9.5

27. Measures of Relative Position • T score – a transformation of a z score where T = 10(z) + 50 • Characteristics • Mean = 50 • Standard deviation = 10 • No negative scores Objective 9.6

28. Measures of Relative Position • Stanine – a transformation of a z score where the stanine = 2(z) + 5 rounded to the nearest whole number • Characteristics • Nine groups with 1 the lowest and 9 the highest • Categorical interpretation • Frequently used in norming tables Objective 9.7

29. Measures of Relationship • Purpose – to provide an indication of the relationship between two variables • Characteristics of correlation coefficients • Strength or magnitude – 0 to 1 • Direction – positive (+) or negative (-) • Types of correlation coefficients – dependent on the scales of measurement of the variables • Spearman rho – ranked data • Pearson r – interval or ratio data Objectives 8.1, 8.2, & 8.3

30. Measures of Relationship • Interpretation – correlation does not mean causation • Formula for Pearson r Objective 8.2

31. Calculating Descriptive Statistics • Symbols used in statistical analysis • General rules for calculating by hand • Make the columns required by the formula • Label the sum of each column • Write the formula • Write the arithmetic equivalent of the problem • Solve the arithmetic problem Objectives 10.1, 10.2, 10.3, & 10.4

32. Calculating Descriptive Statistics • Using SPSS Windows • Means, standard deviations, and standard scores • The DESCRIPTIVE procedures • Interpreting output • Correlations • The CORRELATION procedure • Interpreting output Objectives 10.1, 10.2, 10.3, & 10.4

33. Calculating Descriptive Statistics • See the Statistical Analysis of Data module on the web site for problems related to descriptive statistics

34. Formula for the Mean

35. Formula for Variance

36. Formula for Standard Deviation

37. Formula for Pearson Correlation

38. Formula for z Score