Educational Research

1. Educational Research Chapter 12 Descriptive Statistics Gay, Mills, and Airasian 10th Edition

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

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

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

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

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

9. Descriptive Statistics • Purpose – to describe or summarize data in a manner that is both understandable and short • Four types • Central tendency • Variability • Relative position • Relationships

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

11. Quiz 1 Results

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

13. Central Tendency • Mode • The most frequently occurring score • Appropriate for nominal data • Look for the most frequent number • 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 all scores • Look for the middle # (if 2 are in contention, get the mean of these 2 numbers.

14. Central Tendency • Mean • The arithmetic average of all scores • Characteristics • Advantageous statistical properties • Affected by outlying scores • Most frequently used measure of central tendency • Add all of the scores together and divide by the number of Ss

15. S1 = 10 S2 = 12 S3 = 14 S4 = 10 S5 = 14 S6 = 12 S7 = 12 S8 = 12 ??= ??? Mode Median Mean Calculate for the following data points:

16. You know the central score, do you need anything else? • What is the mean of the following: • 10, 20, 200, 10, 20 • What is the mean of the following: • 51, 52, 53, 52, 52 • Is there more we want to know about the data than just what is the middle point?

17. Quiz 1: Central Tendency • Count: 25 • Average/ Mean: 79.7 • Median: 83.5

18. Variability • Purpose – to measure the extent to which scores are spread apart • Four measures • Range • Variance • Standard deviation • (there are others, but these are the only ones we are going to talk about)

19. Variability • Range • The difference between the highest and lowest score in a data set • Characteristics • Unstable measure of variability • Rough, quick estimate • Calculate • What is the range of the following: • 10, 20, 200, 10, 20 • What is the range of the following: • 51, 52, 53, 52, 52

20. Quiz 1 • Count: 25 • Average: 79.7 • Median: 83.5 • Maximum: 93.4 • Minimum: 0.0

21. Variability • Variance • The average squared deviation of all scores around the mean • Characteristics • Many important statistical properties • Difficult to interpret due to “squared” metric • Used mostly to calculate standard deviation • Formula

22. 10 - 52 = -42 20 - 52 = -32 200-52 = 148 10 - 52 = -42 20 - 52 = -32 Variance • 51 - 52 = -1 • 52 - 52 = 0 • 53 - 52 = 1 • 52 - 52 = 0 • 52 - 52 = 0

23. 10 - 52 = -422 = 1764 20 - 52 = -322 = 1024 200-52 = 1482 =21904 10 - 52 = -422 = 1764 20 - 52 = -322 = 1024 Variance • 51 - 52 = -12 = 1 • 52 - 52 = 02 = 0 • 53 - 52 = 12 = 1 • 52 - 52 = 02 = 0 • 52 - 52 = 02 = 0

24. 10 - 52 = -422 = 1764 20 - 52 = -322 = 1024 200-52 = 1482 =21904 10 - 52 = -422 = 1764 20 - 52 = -322 = 1024 27480 27480/5 = 5496 Variance = 5496 Variance • 51 - 52 = -12 = 1 • 52 - 52 = 02 = 0 • 53 - 52 = 12 = 1 • 52 - 52 = 02 = 0 • 52 - 52 = 02 = 0 2 2/5=.4 Variance = .4

25. Variability • Standard deviation • The square root of the variance • Characteristics • Many important statistical properties • Relationship to properties of the normal curve • Easily interpreted • Formula

26. 10 - 52 = -422 = 1764 20 - 52 = -322 = 1024 200-52 = 1482 =21904 10 - 52 = -422 = 1764 20 - 52 = -322 = 1024 27480 27480/5= 5496= Variance ____ √5496 = 74.13 = SD Standard Deviation • 51 - 52 = -12 = 1 • 52 - 52 = 02 = 0 • 53 - 52 = 12 = 1 • 52 - 52 = 02 = 0 • 52 - 52 = 02 = 0 2 2/5=.4; Variance = .4 __ • √.4 = .63 = SD

27. So now you know middle # and spreadoutedness • How can you use that information to standardize all of the scores to have the same meaning. • First set of scores has a mean of 52 and a SD of .63; second set has a mean of 52 and a SD of 74.13. How do we compare an individual score on first to an individual score on second?

28. Quiz 1: Variance • Count: 25 • Average: 79.7 • Median: 83.5 • Maximum: 93.4 • Minimum: 0.0 • Standard Deviation: 18.44

29. The Normal Curve • A bell shaped curve reflecting the distribution of many variables of interest to educators • Gives a visual way of identifying where one person’s scores fit in with the rest of the people.

30. Normal Curve

31. 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.

32. The Normal Curve • Properties • Proportions under the curve • ±1 SD = 68% • ±1.96 SD = 95% • ±2.58 SD = 99%

33. Skewed Distributions • None - even • Positive – many low scores and few high scores • Negative – few low scores and many high scores

34. Skewed Distribution • Which direction are the following scores skewed: • 12,4,5,13,4,4,1,3,1,3,1,3,1,5 • Step 1: Reorder from lowest to highest • 1,1,1,3,3,3,4,4,4,5,5,12,13 • Step 2: Graph these numbers • Step 3: Compare the graph to the pictures we showed above (tail goes toward the direction… tail to the right, positive; tail to the left, negative)

35. Skewed Distribution Example 1 3 4 1 3 4 5 1 3 4 5 12 13

36. Skewed Distribution Example

37. 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

38. 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

39. Measures of Relative Position • z score • The deviation of a score from the mean in standard deviation units • 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

40. Measures of Relative Position • T score – a transformation of a z score • Characteristics • Mean = 50 • Standard deviation = 10 • No negative scores

41. Measures of Relative Position • Stanine – a transformation of a z score • Characteristics • Nine groups with 1 the lowest and 9 the highest

42. Measures of Relationship: Correlations • 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

43. Measures of Relationship • Interpretation – correlation does not mean causation • Formula see page 316 in your text book to discuss the formula for the Pearson r correlation coefficient.

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