1. Chapter 11 Descriptive Statistics Gay, Mills, and Airasian
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 Types of items
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 Coding
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 Median 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
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

35. Formula for the Mean

36. Formula for Variance

37. Formula for Standard Deviation

38. Formula for Pearson Correlation

39. Formula for z Score