Textbook Credits
• Textbook
Shavelson, R.J. (1996). Statistical reasoning for the behavioral
sciences (3rd Ed.). Boston: Allyn & Bacon.
• Supplemental Material
Ruiz-Primo, M.A., Mitchell, M., & Shavelson, R.J. (1996).
Student guide for Shavelson statistical reasoning for the
behavioral sciences (3rd Ed.). Boston: Allyn & Bacon.
2
Overview
• Example data on Teacher Expectancy Research
• Frequency Distributions
• Measures of Central Tendency & Variability
• The Normal Distribution
3
Teacher Expectancy Data
4
Research on Teacher Expectancy Study Design
• A 2 x 6 x 2 (treatment x grade level x test occasions) randomized experiment
Schematic of design:
Occasion 1
Grade
Treatment
Occasion 2
1
Experimental X
(“Bloomers”)
2
IQ Pretest
3
4
Control
5
Randomly Assigned
6
5
IQ Posttest
Teacher Expectancy Data Matrix
• Another convenient way to easily depict the data
6
Teacher Expectancy Frequency Distribution
• Posttest scores from the treatment group
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Sum = 30
Frequency Distribution Using Class Intervals
• Treatment group posttest scores divided into 11 class intervals
• Each class interval size is 3 (score values 123, 124, 125, …)
• Clear patterns emerge. Look @ interval 114-116 with highest f
8
Frequency Distribution Using Class Intervals
•
•
•
•
Use 11 intervals
Number of Class Intervals: Highest – Lowest score: 125-95 = 30
Class intervals size (i) = H–L / # of class intervals: 30/11 = 2.7(round to 3)
Rule: Lowest interval score must be divisible by interval class size. Lowest score 95 is not
divisible by 3 so subtract 1. 94 is still not divisible by 3 so subtract 1. 93 is divisible by 3 so
lowest class interval score begins with 93.
9
Teacher Expectancy: Histogram
• Histogram showing the class interval posttest scores on the abscissa and frequency
on the ordinate
• Lower and upper limit scores with zero values are shown.
10
Teacher Expectancy: Polygon
• Polygon showing the class interval posttest scores on the abscissa and frequency
on the ordinate
• Lower and upper midpoint values with zero f are shown.
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Teacher Expectancy: Polygon
• Polygon showing the class interval posttest scores on the abscissa and frequency
on the ordinate
• Lower and upper midpoint values with zero f are shown.
12
Teacher Expectancy: Polygon
13
Teacher Expectancy: Stem-and-leaf plot
• Stem-and-leaf plot containing the data matrix posttest scores in increments of 5’s
14
Common Frequency Distribution Shapes
Normal Distribution(bell shape)
Unimodal distribution : 1 peak
Positively Skewed
Symmetric about the mean
Bimodal Distribution: 2 peaks
Multimodal Distribution: > 2 peaks
Negatively Skewed
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Rectangular Distribution(no peaks)
Kurtosis (peakedness): platykurtic
Symmetric about the mean
Kurtosis (peakedness): leptokurtic
The Relative Frequency (Probability) Distribution
• Score frequencies are shown as a proportion of the total number of frequencies in
the sample: RF = f / total # of subjects
Sum = 10
Sum = 20
Total Sum = 30
16
The Relative Frequency Polygon
• Relative frequency polygons are constructed as the frequency polygons except the
relative frequency is listed in the ordinate
17
The Cumulative Frequency Distribution
• The cumulative frequency distribution shows the number of scores falling below a
certain point on the scale of scores
• The cf of a score is defined as the number of cases falling below the upper real limit
of the class interval
18
The Cumulative Frequency Polygon
• The cumulative frequency polygon uses upper real limits and cumulative frequency
19
Cumulative Proportions and Percentiles
• 80% of the subjects in the experimental group of the expectancy study received a
posttest score below 119.5
• CP = CF / total number of subjects
• C% = CP x 100
20
Percentile Scores
• Example: A raw score of 113.5 has a percentile rank of 57
21
Measures of Central Tendency
The central tendency of the set of measurements - that is, the tendency of the
data to cluster, or center, about certain numerical values.
Central Tendency
(Location)
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Measures of Central Tendency
The variability of the set of measurements–that is, the spread of the data.
Variation (Dispersion)
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Standard Notation
Measure
Sample
m
Mean
Size
Population
n
N
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Mean
• Most common measure of central tendency
• Acts as ‘balance point’
• Affected by extreme values (‘outliers’)
• Denoted as
where
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Mean Example
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Median
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Median Example: Odd Size Sample
• Raw Data: 24.1 22.6 21.5 23.7 22.6
• Ordered: 21.5 22.6 22.6 23.7 24.1
• Position:
1
2
3
4
5
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Median Example: Even Size Sample
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Mode
• Measure of central tendency
• Value that occurs most often
• Not affected by extreme values
• May be no mode or several modes
• May be used for quantitative or qualitative data
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Mode Example
• No Mode
Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7
• One Mode
Raw Data: 6.3 4.9 8.9
6.3 4.9 4.9
• More Than 1 Mode
Raw Data: 21 28
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28
31
43
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Measures of Variability
• Measure of dispersion
• Difference between largest & smallest
observations
Range = xlargest – xsmallest
• Ignores how data are distributed
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Standard Notation
33
Sample & Population Variance
The variance is the average of the squares of the distance
each value is from the mean.
n
s2 
 x
i
 x
2
i1
n 1
x1  x   x2  x 


2
2
L  xn  x 
2
n 1
x  m 


2

2
N
N in the
denominator!
n – 1 in the
denominator!
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Standard Deviation
The standard deviation is the square root of the variance.
s  s2
n

2
x

x


 i
  
2
i1
n 1
x1  x   x2  x  L  xn  x 


n 1
2
2
2
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2


x

m

N
Sample Standard Deviation Formula
S
=
= 2.523
36
Overview of the Normal Distribution
• Serves as a reasonable good model of many natural
phenomena
• Provides a good model for the frequency distribution of scores
• Of particular importance is in inferential statistics as a
probability distribution
• There exists a close connection between the sample size and
the distribution of means calculated for many samples of
subjects drawn from the same population
• As the sample size increases, the distribution of scores
becomes normal
• May provide a good approximation to probabilities of other
distributions that are more difficult to work with
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Properties of the Normal Distribution
• It is unimodal, observing the value of X and the mean
• It is symmetric about the mean; ½ the scores fall
below the mean and ½ the scores fall above the mean
• The mean, mode, and median are all equal
• It is asymptotic (never touches the abscissa)
• It is continuous for all values of X from - ∞ to +∞
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Properties of the Normal Distribution
•
•
•
•
Unimodal
Symmetric
Mode=median=mean
Asymptotic
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Empirical Rule of the Normal Distribution Areas
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Interpretation of z-Scores Example
• Approximately 68% of the measurements will have a
z-score between –1 and 1.
• Approximately 95% of the measurements will have a
z-score between –2 and 2.
• Approximately 99.7% of the measurements will have
a z-score between –3 and 3.
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Empirical Rule Example
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Computing the z-Score
43
z-Score Example 1
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Z-score Example 1
Find the area between the mean and a given raw score
• z score: Mean 0, s = 1
– Distance between a score (X) and the mean of a distribution in standard
deviation (s) units
– Used to display and interpret areas of the normal distribution
Assume score is 9, mean = 8, s = 2
z = (x - mean) / s
z = (9-8)/2 = 0.5
• Next, find the area between
the mean and z = 0.5
From Table B Column 2 in
Appendix II we find: 0.1915
0.1915 or 19.15% of the cases
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Z-score Example 2
Find the area below a given raw score
• z score: Mean 0, s = 1
– Distance between a score (X) and the mean of a distribution in
standard deviation (s) units
– Used to display and interpret areas of the normal distribution
Assume score is 9, mean = 8, s = 2
z = (x - mean) / s
z = (9-8)/2 = 0.5
Below!
• Mark off area in the ND
• Find area below z = 0.5
• From Table B column 3 we find:
0.6915 or 69.15% of the cases
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Z-score Example 3
Find the area above a given raw score
• z score: Mean 0, s = 1
– Distance between a score (X) and the mean of a distribution in standard
deviation (s) units
– Used to display and interpret areas of the normal distribution
Assume score is 9, mean = 8, s = 2
z = (x - mean) / s
z = (9-8)/2 = 0.5
• Mark off area in the ND
• Find area above z = 0.5
Above!
• From Table B column 3 we find:
0.3085 or 30.85% of the cases
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Z-score Example 3
Find the area between two given raw scores
• z score: Mean 0, s = 1
– Distance between a score (X) and the mean of a distribution in standard deviation (s)
units
– Used to display and interpret areas of the normal distribution
Assume 1st score is 9, 2nd score is 5.8, mean = 8, s = 2
z = (x - mean) / s
z = (9-8)/2 = 0.5
•
•
z = (x - mean) / s
z = (5.8-8)/2 = -1.1
Find area between mean and z = 0.5
Find area between mean and z = -1.1
zX=5.8 = -1.1
Between!
zX=9 = .5
• Total = 0.1915 + 0.3643 = 0.5558
0.5558 or 55.58% of the cases
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Practice Exercises
1. Select a hypothetical product or a process and create some test data
of your choice (plausible, no more than 10) as shown in
textbook/class
2. Show your type of experimental approach
3. Create a detailed table of frequency distributions
4. Display your data with different types of graphs
5. Calculate the measures of central tendency and variability
6. Calculate the Z-score(s) and indicate the relative position in the
normal distribution.
7. Provide any other pertinent information as a result
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Questions ?
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