Educational Research

1. Educational Research Chapter 14 Descriptive Statistics Gay 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

5. Tabulation and Coding • Tabulation is organizing data • Identifying all relevant information to the analysis • Separating groups and individuals within groups • Listing data in columns

6. 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.

7. 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

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

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

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

11. 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

12. 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

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

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

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

16. 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

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

18. 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

19. 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.

20. The Normal Curve • Properties • Proportions under the curve • 1 SD  68% • 1.96 SD  95% • 2.58 SD  99% • Cumulative proportions and percentiles

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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 correlations coefficients – dependent on the scales of measurement of the variables • Spearman Rho – ranked data • Pearson r – interval or ratio data

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

30. Calculating Descriptive Statistics • Symbols used in statistical analysis • General rules form 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

31. Calculating Descriptive Statistics • Using SPSS Windows • Means, standard deviations, and standard scores • The DESCRIPTIVES procedures • Interpreting output • Correlations • The CORRELATION procedure • Interpreting output

33. Formula for the Mean

34. Formula for Variance

35. Formula for Standard Deviation

36. Formula for Pearson Correlation

37. Formula for Z-Score