2. TOPICS DISCUSSED IN THIS
CHAPTER
• Preparing data for analysis
• Types of descriptive statistics
– Central tendency
– Variation
– Relative position
– Relationships
• Calculating descriptive statistics
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=Rahul, 002=Rajani, 003= Rita, … 256=Harish
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 parsimonious
manner
• 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
SCORE
9.0
8.0
7.0
6.0
5.0
4.0
3.0
SCORE
Frequency
5
4
3
2
1
0
Std. Dev = 1.63
Mean = 6.0
N = 16.00
11. CENTRAL TENDENCY
• Purpose – to represent the typical score attained by subjects
• Three common measures
– Mode
– Median
– Mean
12. 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
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
14. VARIABILITY
• Purpose – to measure the extent to which scores are spread
apart
• Four measures
– Range
– Quartile deviation
– Variance
– Standard deviation
15. VARIABILITY
• Range
– The difference between the highest and lowest score in a data set
– Characteristics
• Unstable measure of variability
• Rough, quick estimate
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
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
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
19. THE NORMAL CURVE
• A bell shaped curve reflecting the distribution of many variables
of interest to educators
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.
21. THE NORMAL CURVE
• Properties
– Proportions under the curve
• ±1 SD = 68%
• ±1.96 SD = 95%
• ±2.58 SD = 99%
– Cumulative proportions and percentiles
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
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
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
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
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
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
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
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
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
32. CALCULATING DESCRIPTIVE
STATISTICS
• Using SPSS Windows
– Means, standard deviations, and standard scores
• The DESCRIPTIVE procedures
• Interpreting output
– Correlations
• The CORRELATION procedure
• Interpreting output
