Transcript Educational Research

Educational Research
Chapter 12
Descriptive Statistics
Gay, Mills, and Airasian
10th Edition
Topics Discussed in this Chapter
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Preparing data for analysis
Types of descriptive statistics
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Central tendency
Variation
Relative position
Relationships
Calculating descriptive statistics
Preparing Data for Analysis
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Issues
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Scoring procedures
Tabulation and coding
Use of computers
Scoring Procedures
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Instructions
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Standardized tests detail scoring instructions
Teacher-made tests require the delineation of
scoring criteria and specific procedures
Types of items
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Selected response items - easily and objectively
scored
Open-ended items - difficult to score objectively
with a single number as the result
Tabulation and Coding
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Tabulation is organizing data
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Identifying all information relevant to the analysis
Separating groups and individuals within groups
Listing data in columns
Coding
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Assigning names to variables
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EX1 for pretest scores
SEX for gender
EX2 for posttest scores
Tabulation and Coding
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Reliability
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Concerns with scoring by hand and
entering data
Machine scoring
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Advantages
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Reliable scoring, tabulation, and analysis
Disadvantages
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Use of selected response items, answering on
scantrons
Tabulation and Coding
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Coding
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Assigning identification numbers to
subjects
Assigning codes to the values of nonnumerical or categorical variables
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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
Computerized Analysis
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Need to learn how to calculate descriptive
statistics by hand
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Creates a conceptual base for understanding the
nature of each statistic
Exemplifies the relationships among statistical
elements of various procedures
Use of computerized software
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SPSS-Windows
Other software packages
Descriptive Statistics
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Purpose – to describe or summarize
data in a manner that is both
understandable and short
Four types
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Central tendency
Variability
Relative position
Relationships
Descriptive Statistics
Graphing data – a
frequency polygon
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SCORE
5
Vertical axis
represents the
frequency with which
a score occurs
Horizontal axis
represents the
scores themselves
Frequency
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4
3
2
1
Std. Dev = 1.63
Mean = 6.0
N = 16.00
0
3.0
SCORE
4.0
5.0
6.0
7.0
8.0
9.0
Quiz 1 Results
Central Tendency
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Purpose – to represent the typical score
attained by subjects
Three common measures
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Mode
Median
Mean
Central Tendency
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Mode
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The most frequently occurring score
Appropriate for nominal data
Look for the most frequent number
Median
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The score above and below which 50% of all
scores lie (i.e., the mid-point)
Characteristics
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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.
Central Tendency
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Mean
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The arithmetic average of all scores
Characteristics
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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
Calculate for the following
data points:
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S1 = 10
S2 = 12
S3 = 14
S4 = 10
S5 = 14
S6 = 12
S7 = 12
S8 = 12
??= ???
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Mode
Median
Mean
You know the central score,
do you need anything else?
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What is the mean of the following:
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What is the mean of the following:
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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?
Quiz 1: Central Tendency
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Count: 25
Average/ Mean: 79.7
Median:
83.5
Variability
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Purpose – to measure the extent to
which scores are spread apart
Four measures
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Range
Variance
Standard deviation
(there are others, but these are the only
ones we are going to talk about)
Variability
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Range
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The difference between the highest and lowest
score in a data set
Characteristics
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Unstable measure of variability
Rough, quick estimate
Calculate
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What is the range of the following:
 10, 20, 200, 10, 20
What is the range of the following:
 51, 52, 53, 52, 52
Quiz 1
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Count: 25
Average:
Median:
Maximum:
Minimum:
79.7
83.5
93.4
0.0
Variability
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Variance
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The average squared deviation of all scores
around the mean
Characteristics
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Many important statistical properties
Difficult to interpret due to “squared” metric
Used mostly to calculate standard deviation
Formula
Variance
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51
52
53
52
52
-
52
52
52
52
52
=
=
=
=
=
-1
0
1
0
0
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10 - 52 = -42
20 - 52 = -32
200-52 = 148
10 - 52 = -42
20 - 52 = -32
Variance
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51
52
53
52
52
-
52
52
52
52
52
=
=
=
=
=
-12 = 1
02 = 0
12 = 1
02 = 0
02 = 0
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10 - 52 = -422 = 1764
20 - 52 = -322 = 1024
200-52 = 1482 =21904
10 - 52 = -422 = 1764
20 - 52 = -322 = 1024
Variance
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51
52
53
52
52
-
52
52
52
52
52
=
=
=
=
=
-12 = 1
02 = 0
12 = 1
02 = 0
02 = 0
2
2/5=.4
Variance = .4
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10 - 52 = -422 = 1764
20 - 52 = -322 = 1024
200-52 = 1482 =21904
10 - 52 = -422 = 1764
20 - 52 = -322 = 1024
27480
27480/5 = 5496
Variance = 5496
Variability
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Standard deviation
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The square root of the variance
Characteristics
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Many important statistical properties
Relationship to properties of the normal curve
Easily interpreted
Formula
Standard Deviation
-12 = 1
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02 = 0
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12 = 1
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02 = 0
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02 = 0
2
2/5=.4; Variance = .4
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 √.4 = .63 = SD
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51
52
53
52
52
-
52
52
52
52
52
=
=
=
=
=
10 - 52 = -422 = 1764
2
 20 - 52 = -32 = 1024
 200-52 = 1482 =21904
 10 - 52 = -422 = 1764
 20 - 52 = -322 = 1024
27480
27480/5= 5496= Variance
____
√5496 = 74.13 = SD
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So now you know middle #
and spreadoutedness
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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 74.13. How do we
compare an individual score on first to
an individual score on second?
Quiz 1: Variance
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Count: 25
Average:
79.7
Median:
83.5
Maximum: 93.4
Minimum:
0.0
Standard Deviation:
18.44
The Normal Curve
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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.
Normal Curve
The Normal Curve
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Characteristics
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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.
The Normal Curve
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Properties
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Proportions under the curve
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±1 SD = 68%
±1.96 SD = 95%
±2.58 SD = 99%
Skewed Distributions
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None - even
Positive –
many low
scores and few
high scores
Negative – few
low scores and
many high
scores
Skewed Distribution
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Which direction are the following scores
skewed:
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Step 1: Reorder from lowest to highest
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12,4,5,13,4,4,1,3,1,3,1,3,1,5
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)
Skewed Distribution Example
1
1
1
3
3
3
4
4
4
5
5
12
13
Skewed Distribution Example
3
2.5
2
1.5
1st Qtr
1
0.5
0
1
3
4
5
12
13
Measures of Relative Position
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Purpose – indicates where a score is in
relation to all other scores in the
distribution
Characteristics
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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
Measures of Relative Position
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Types
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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
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z score
T score
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Stanine
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Measures of Relative Position
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z score
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The deviation of a score from the mean in
standard deviation units
Characteristics
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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
Measures of Relative Position
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T score – a transformation of a z score
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Characteristics
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Mean = 50
Standard deviation = 10
No negative scores
Measures of Relative Position
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Stanine – a transformation of a z score
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Characteristics
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Nine groups with 1 the lowest and 9 the
highest
Measures of Relationship:
Correlations
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Purpose – to provide an indication of the relationship
between two variables
Characteristics of correlation coefficients
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Strength or magnitude – 0 to 1
Direction – positive (+) or negative (-)
Types of correlation coefficients – dependent on the
scales of measurement of the variables
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Spearman rho – ranked data
Pearson r – interval or ratio data
Measures of Relationship
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Interpretation – correlation does not
mean causation
Formula see page 316 in your text
book to discuss the formula for the
Pearson r correlation coefficient.
Calculating Descriptive Statistics
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Using SPSS Windows
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Means, standard deviations, and standard
scores
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The DESCRIPTIVE procedures
Correlations
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The CORRELATION procedure
Objectives 10.1, 10.2, 10.3, & 10.4