1. Educational Research: Data analysis and interpretation – 1Descriptive statistics EDU 8603 Educational Research Richard M. Jacobs, OSA, Ph.D.

2. Statistics... • A set of mathematical procedures for describing, synthesizing, analyzing, and interpreting quantitative data …the selection of an appropriate statistical technique is determined by the research design, hypothesis, and the data collected

3. Preparing data for analysis... Data must be accurately scored and systematically organized to facilitate data analysis: scoring: assigning a total to each participant’s instrument tabulating: organizing the data in a systematic manner coding: assigning numerals (e.g., ID) to data

4. descriptive statistics... …permit the researcher to describe many pieces of data with a few indices

5. statistics... …indices calculated by the researcher for a sample drawn from a population

6. parameters... …indices calculated by the researcher for an entire population

7. Types of descriptive statistics… 1. graphs 2. measures of central tendency 3. measures of variability

8. graphs... …representations of data enabling the researcher to see what the distribution of scores look like

9. 1. Graphs… • frequency polygon • pie chart • boxplot • stem-and-leaf chart

10. measures of central tendency... …indices enabling the researcher to determine the typical or average score of a group of scores

11. 2. Measures of central tendency… • mode • median • mean

12. mode... …the score attained by more participants than any other score

13. median... …the point in a distribution above and below which are 50% of the scores

14. mean... …the arithmetic average of the scores

15. measures of variability... …indices enabling the researcher to indicate how spread out a group of scores are

16. 3. Measures of variability… • range • quartile deviation • variance • standard deviation

17. range... …the difference between the highest and lowest score in a distribution

18. quartile deviation... …one half of the difference between the upper quartile (the 75%’ile) and the lower quartile (the 25%’ile) in a distribution

19. variance... …a summary statistic indicating the degree of variability among participants for a given variable

20. standard deviation... …the square root of variance providing an index of variability in the distribution of scores

21. Normal distributions of data(the normal curve)... A bell-shaped distribution of scores having four identifiable properties… …50% of the scores fall above the mean and 50% of the scores fall below the mean …the mean, median, and mode are the same value

22. …most scores are near the mean and, the farther from the mean a score is, the fewer the number of participants who attained that score …the same number, or percentage, of scores is between the mean and plus one standard deviation as is between the mean and minus one standard deviation

23. Non-normal distributions of data (skewed distributions)... A non-bell-shaped distribution of scores where… …mean < median < mode (a “negatively skewed distribution”) …mean > median > mode (a “positively skewed distribution”)

24. measures of relative position... …indices enabling the researcher to describe a participant’s performance compared to the performance of all other participants

25. 4. Measures of relative position… • percentile ranks • standard scores

26. percentile rank... …indicates the percentage of scores that fall at or below a given score

27. standard score... …a measure of relative position

28. Types of standard scores... …z score …T score …stanines

29. z score... …a statistic expressing how far a score is from the mean in terms of standard deviation units

30. T score... …a transformed z score that voids negative numbers and decimals by multiplying the z score by 10 and adding 50

31. stanines... …a standard score that divides a distribution into nine parts

32. measures of relationship... …indices enabling the researcher to indicate the degree to which two sets of scores are related

33. 5. Measures of relationship… • Spearman Rho • Pearson r

34. correlations …determines whether and to what degree a relationship exists between two or more quantifiable variables …the degree of the relationship is expressed as a coefficient of correlation

35. …the presence of a correlation does not indicate a cause-effect relationship primarily because of the possibility of multiple confounding factors

36. Correlation coefficient… -1.00 0.00 +1.00 strong positive strong negative no relationship

37. Spearman Rho... …a measure of correlation used for rank and ordinal data

38. Pearson r... …a measure of correlation used for data of interval or ratio scales …assumes that the relationship between the variables being correlated is linear

39. Mini-Quiz… • True and false… …the analysis of the data is as important as any other component of the research process True

40. True and false… …descriptive statistics are normally computed separately for each group in a research study True

41. True and false… …every instrument administered must always be scored accurately and consistently, using the same procedures and criteria True

42. True and false… …tentative scoring procedures must always be tried out beforehand by administering the instrument to the study participants False

43. True and false… …a computer should not be used to perform an analysis that a researcher has never completed by hand or, at least, studied extensively True

44. True and false… …the first step in data analysis is to describe, or summarize, the data using descriptive statistics True

45. True and false… …the number resulting from the computation of a measure of central tendency represents the typical score attained by a group of participants True

46. True and false… …the mean is the most precise, stable index of typical performance that is especially useful in situations in which there are extreme scores False

47. True and false… …unless a correlation coefficient is used to compute the reliability of an instrument in a causal-comparative or experimental study, a correlation coefficient is only computed in a correlation study True

48. True and false… …plus and/or minus two standard deviations includes more the 99% of the scores False

49. True and false… …standard scores are rarely used in research studies True

50. True and false… …to test a hypothesis adequately, more than descriptive statistics are normally needed True