1. Descriptive Statistics
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 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

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

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 Median 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. Calculate for the following data points:
??= ???
Mode
Median
Mean

16. You know the central score, do you need anything else?
What is the mean of the following:
10, 20, 200, 10, 20
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
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. Variance
= -1
= 0
= 1
= -42
= -32
= 148

23. Variance
= -12 = 1
= 02 = 0
= 12 = 1
= -422 = 1764
= -322 = 1024
= 1482 =21904

24. Variance
= -12 = 1
= 02 = 0
= 12 = 1 2
2/5=.4
Variance = .4
= -422 = 1764
= -322 = 1024
= 1482 =21904
27480
27480/5 = 5496
Variance = 5496

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

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%

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

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