Statistics Used In Special Education National Association Of
Special Education Teachers
Slide 2
Definition Statistics-Mathematical procedures used to describe
and summarize samples of data in a meaningful fashion
Slide 3
Basic Statistics You Need to Understand Measures of Central
Tendency Frequency Distributions Range Standard Deviation Normal
Curve Percentile Ranks Standard Scores Scaled Scores T Scores
Stanines
Slide 4
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
Slide 5
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
Slide 6
Example Mean Problem In the distribution: 8, 10, 8, 14, and 40,
What is the Mean?
Slide 7
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 The Mean is 16.
Slide 8
Problem with the Mean Score 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
Slide 9
Median Definition: The Median is the Midpoint in the
Distribution It is the MIDDLE score Half the scores fall ABOVE the
Median and half the scores fall BELOW the Median
Slide 10
Calculate the Median In the distribution of scores: 8, 10, 8,
14, 40 Calculate the Median
Slide 11
Remember the Rule for Median Score **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
Slide 12
Answer to Median Problem 8, 8, 10, 14, 40 Cross of the low then
cross off the high (in our example 8 & 40) 8, 8, 10, 14, 40
Repeat until a Middle Number Obtained 8, 8, 10, 14, 40 The Median
is 10
Slide 13
What if There are Two Middle Scores? Suppose our distribution
was 8, 10, 8, 14, 40 and 12. When you put the scores in order you
get 8, 8, 10, 12, 14, 40 After crossing off the low and high
scores, 8, 8, 10, 12, 14, 40 This leaves you with 10 and 12. What
would you do?
Slide 14
Rule: When You Have Two Middle Scores, Find Their MEAN 8, 8,
10, 12, 14, 40 Middle Numbers are 10 and 12 Find the Mean: 10 + 12
= 22 22/2 = 11 The Median is 11
Slide 15
Mode Definition: The Mode is the score that occurs most
frequently in the distribution What is the mode in the distribution
of 8, 10, 8, 14, 40?
Slide 16
Frequency Distribution Score Tally Frequency 40 I 1 14 I 1 10 I
1 8 I I2 Frequency Distribution-a listing of scores from lowest to
highest with the number of times each score appears in a
sample
Slide 17
Answer to Mode Problem In our distribution of 8, 10, 8, 14, 40,
the score 8 appears twice. All other scores appear once Score Tally
Frequency 40 I 1 14 I 1 10 I 1 8 I I 2 The Mode is 8
Slide 18
What if Two or More Scores Appear the Same Number of Times?
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
Answer to Both Mode Problems 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
Slide 21
Calculate the Measures of Central Tendency STUDENT NAMEIQ SCORE
1. Billy105 2. Juan125 3. Carmela 70 4. Fred115 5. Yvonne 85 6. Amy
105 7. Carol 95 8. Sarah100
Slide 22
Answer to Measures of Central Tendency Question Mean = 100
800/8 = 100 Median = 102.5 100, 125, 70, 115, 85, 105, 95, 100 70,
85, 95, 100, 105, 105, 115, 125, M = 205/2 = 102.5 Median is 102.5
Mode = 105 70, 85, 95, 100, 105, 105, 115, 125,
Slide 23
Range 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?
Slide 24
Answer to Range Problem The Range is 32 High score is 4 Low
score is 8 40 8 = 32
Slide 25
Problem with the Range The range tells you nothing about the
scores in between the high and low scores. 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.
Slide 26
Standard Deviation Lets look at the following two distributions
of scores on a 50-question spelling test (each score represents the
number of words correctly spelled) Scores for 5 students in Group
A: 28, 29, 30, 31, 32 Scores for 5 students in Group B: 10, 20, 30,
40, 50 Calculate the MEAN for Groups A and B
Slide 27
Standard Deviation Mean of Group A = 30 Mean of Group B = 30
The means of both groups are 30. Now, if you knew nothing about
these two groups other than their mean scores, you might think they
looked similar. 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).
Slide 28
Standard Deviation 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.
Slide 29
Normal Curve 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.
Slide 30
Normal Curve 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).
Slide 31
Percentages Under the Normal Curve 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%).
Slide 32
Normal Curve 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. Approximately 95% of the scores
lie within two standard deviations of the mean (13.5% + 34% + 34% +
13.5% = 95%)
Slide 33
The Importance of the Normal Curve Now, how does this help you?
Well, lets 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
Slide 34
Gifted Programs 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)? By looking at the normal curve you may have
figured it outthe minimum is normally an IQ of 130 for entrance.
Why? 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.
Slide 35
Mental Retardation How about mental retardation? On the
Wechsler Scales, the classification of mental retardation is
determined if a child receives an IQ score of below 70. Why 70?
This score was not just randomly chosen.
Slide 36
Why 70? 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 childs 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).
Slide 37
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?
Slide 38
Answer to Practice Problem between 65 and 85? 68% (34 + 34)
above 85? 16% (13.5 + 2.5%) between 55 and 95? 95% (13.5% + 34% +
34% + 13.5%) above 95? 2.5%