Download Understanding Statistics Used In Special Education

Statistics Used In
Special Education
National Association
Of
Special Education Teachers
Definition

Statistics-Mathematical
procedures used to describe
and summarize samples of
data in a meaningful
fashion
Basic Statistics You Need to
Understand
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Measures of Central
Tendency
Frequency
Distributions
Range
Standard Deviation
Normal Curve
Percentile Ranks
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Standard Scores
Scaled Scores
T Scores
Stanines
Measures of Central
Tendency
Measures of Central Tendency-A single
number that tells you the overall
characteristics of a set of data
Mean
 Median
 Mode

Mean

Definition: The Mean is the
Mathematical Average

It is defined as the summation
(addition) of all the scores in your
distribution divided by the total
number of scores
Statistically, it is represented by M
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Example Mean Problem
In the distribution:
8, 10, 8, 14, and 40,
What is the Mean?
Answer to Mean Problem

Add up the scores: 8 +10+8+14+40 = 80
Adding the scores up gives you a total of
80.
There are 5 scores
80/5 is 16 M = 16
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The Mean is
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16.
Problem with the Mean Score
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Extreme Scores can greatly affect
the Mean
In our example, the mean is 16 but
there is only one score that is greater
than 16 (The 40)
So, extreme scores (whether high or
low) can affect the Mean
Median
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Definition: The Median is the Midpoint
in the Distribution
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It is the MIDDLE score
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Half the scores fall ABOVE the Median
and half the scores fall BELOW the
Median
Calculate the Median
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In the distribution of scores:
8, 10, 8, 14, 40

Calculate the Median
Remember the Rule for
Median Score
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**RULE: In order to calculate the
Median, you must first put the scores
in order from lowest to highest

For our example, this would be
8, 8, 10, 14, 40
Answer to Median Problem
8, 8, 10, 14, 40
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Cross of the low then cross off the high (in our example 8 & 40)
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8, 8, 10, 14, 40
Repeat until a Middle Number Obtained
 8, 8, 10, 14, 40
The Median is 10
What if There are
Two Middle Scores?
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Suppose our distribution was
8, 10, 8, 14, 40 and 12.
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When you put the scores in order you get
 8, 8, 10, 12, 14, 40
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After crossing off the low and high scores,
 8, 8, 10, 12, 14, 40
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This leaves you with 10 and 12. What would you do?
Rule: When You Have Two
Middle Scores, Find Their MEAN
8, 8, 10, 12, 14, 40
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Middle Numbers are 10 and 12
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Find the Mean:
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The Median is 11
10 + 12 = 22
22/2 = 11
Mode
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Definition: The Mode is the score that
occurs most frequently in the distribution
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What is the mode in the distribution of
8, 10, 8, 14, 40?
Frequency Distribution
Score
Tally
Frequency
40
I
1
14
I
1
10
I
1
8
II
2
Frequency Distribution-a listing of scores
from lowest to highest with the number of
times each score appears in a sample
Answer to Mode Problem

In our distribution of 8, 10, 8, 14, 40, the score 8
appears twice. All other scores appear once
Score
40
14
10
8
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The Mode is
8
Tally
I
I
I
I I
Frequency
1
1
1
2
What if Two or More Scores Appear
the Same Number of Times?
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When two scores appear the same number of
times, both scores are considered modes
When you have two modes, it is a bimodal
distribution
When you have three or more modes, it is a
multimodal distribution
When all scores appear the same number of
times, there is “No Mode”
Calculate the Mode (s)
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1. 8, 10, 8, 10,
14, 40
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2.
8, 9, 10, 12,
14, 40, 14,
40, 12, 10,
9, 8
Answer to Both Mode Problems
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1. There are two modes-It is a bimodal
distribution.
The modes are 8 and 10
2. Since all scores appear twice, there is
no mode
Calculate the Measures of
Central Tendency
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STUDENT NAME
IQ SCORE
1.
Billy
Juan
Carmela
Fred
Yvonne
Amy
Carol
Sarah
105
125
70
115
85
105
95
100
2.
3.
4.
5.
6.
7.
8.
Answer to Measures of Central
Tendency Question
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Mean = 100
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Median = 102.5
800/8 = 100
100, 125, 70, 115, 85, 105, 95, 100
70, 85, 95, 100, 105, 105, 115, 125, M = 205/2 = 102.5
Median is 102.5
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Mode = 105
70, 85, 95, 100, 105, 105, 115, 125,
Range
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Definition: The Range is the difference
between the highest and lowest score in
the distribution.
To calculate the range, simply take the
highest score and subtract the lowest
score.
In the distribution 8,10,8,14, 40, what is
the range?
Answer to Range Problem
The Range is
32
High score is 4
Low score is 8
40 – 8 =
32
Problem with the Range
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The range tells you nothing about the scores in
between the high and low scores.
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Extreme scores can greatly affect the range.
e.g., Suppose the distribution was 8, 9, 8, 9, 8,
and 1,000. The range would be 992 (1,000 – 8 =
992). Yet, only one score is even close to 992, the
1,000.
Standard Deviation
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Let’s look at the following two distributions of scores
on a 50-question spelling test (each score represents the
number of words correctly spelled)
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Scores for 5 students in Group A: 28, 29, 30, 31, 32
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Scores for 5 students in Group B: 10, 20, 30, 40, 50
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Calculate the MEAN for Groups A and B
Standard Deviation
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Mean of Group A = 30
Mean of Group B = 30
The means of both groups are 30.
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Now, if you knew nothing about these two groups other
than their mean scores, you might think they looked
similar.
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However, the spread of scores around the mean in
Group A (28 to 32) is much smaller than the spread of
scores around the mean Group B (10 to 50).
Standard Deviation
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There is a statistic that describes for us
the spread of scores around the mean
Definition: The standard deviation is the
spread of scores around the mean.
It is an extremely important statistical
concept to understand when doing
assessment in special education.
Normal Curve
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A normal distribution hypothetically
represents the way test scores would
fall if a particular test is given to every
single student of the same age or grade
in the population for whom the test was
designed.
Normal Curve
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The normal curve (also referred to as the Bell
Curve) tells us many important facts about test
scores and the population.
The beauty of the normal curve is that it never
changes.
As students, this is great for you because once
you memorize it, it will never change on you
(and, yes, you do have to memorize it at some
point in your academic or professional career).
Percentages Under the
Normal Curve
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34% of the scores lie between the mean and
1 standard deviation above the mean.
An equal proportion of scores (34%) lie
between the mean and 1 standard deviation
below the mean.
Approximately 68% of the scores lie within
one standard deviation of the mean (34% +
34% = 68%).
Normal Curve
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13.5% of the scores lie between one and two
standard deviations above the mean, and
between one and two standard deviations below
the mean.
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Approximately 95% of the scores lie within two
standard deviations of the mean (13.5% + 34%
+ 34% + 13.5% = 95%)
The Importance of the
Normal Curve
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Now, how does this help you?
Well, let’s take an example that you will
come across numerous times in special
education: IQ.
The mean IQ score on many IQ tests is
100 and the standard deviation is 15
Gifted Programs
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Do you know what the requirements are for most gifted
programs regarding minimum IQ scores (that have a
mean of 100 and SD of 15)?
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By looking at the normal curve you may have figured it
out—the minimum is normally an IQ of 130 for
entrance. Why?
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Gifted programs will take only students who are 2
standard deviations or more above the mean. In a
sense, they want only those whose IQs are better than
97.5% of the population.
Mental Retardation
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How about mental retardation?
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On the Wechsler Scales, the classification
of mental retardation is determined if a
child receives an IQ score of below 70.
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Why 70? This score was not just
randomly chosen.
Why 70?
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What we are saying is that in order to receive
this classification you normally have to be 2 or
more standard deviations below the mean.
In a sense, the child’s IQ is only as high as
2.5% (or even lower) of the normal population
(or, in other words, 97.5% or more of the
population has a higher IQ than this child).
Practice Problem

In School district XYZ, the mean score on an exam was
75. The standard deviation was 10. Draw the normal
curve for this distribution.
Based on the normal curve, what percentage of students
scored:
 between 65 and 85?
 above 85?
 between 55 and 95?
 above 95?
Answer to Practice Problem
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between 65 and 85?
68%
(34 + 34)
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above 85?
16%
(13.5 + 2.5%)
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between 55 and 95?
95%
13.5%)
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above 95?
2.5%
(13.5% + 34% + 34% +