EDCI 6312 Educational Measurement Measures of Variability

1
EDCI 6312 - Educational MeasurementMeasures of
Variability

• Dr. Reynaldo Ramirez, Jr
• Associate Professor for Secondary and Science
  Education

2
PERCENTILES AND NORMS

• Age-Grade Norms
• Percentile Norms
• Others
• T
• Z
• NCE
• Stanine

3
AGE-GRADE NORMS

• A test is given to a representative sample of
  students of various ages and in various grades.
• Table is constructed of the raw score, the
  equivalent age, and equivalent grade
  corresponding to the raw score.
• How could we interpret the score of a 13 year old
  girl at the beginning of seventh grade who made a
  score of 49 on an arithmetic achievement test
  using the data on Table 13?

4
AGE-GRADE NORMS

• AGE
• GRADE
• SCORE

How could we interpet the score of this
individual student?
5
AGE-GRADE NORMS

• AGE
• GRADE
• SCORE

How could we interpet the score of this
individual student?
REALLY?
6
TEST YOUR KNOWLEDGE

• Ricardo has just finished the third grade and
  scores a 19 on the arithmetic achievement test.
• What can you say about his performance?

7
TEST YOUR KNOWLEDGE

• Sylvia scores 27 on the same arithmetic test. If
  she is 7 years and 11 months old, why is it
  incorrect to say that she has the same
  proficiency as the average 9 year-old?

8
PERCENTILE NORMS

• A percentile is the value of a variable below
  which a certain percent of observations fall.
• It is a way of expressing the location of a
  particular raw score in a distribution.
• Percentiles are noted by the symbol Pp, where
  P is a percentile and p is the percentage of
  the cases below that point.

9
PERCENTILE NORMS

• Distribution of College Entrance Exam
• Raw score 23
• 64th percentile
• 64 of the original group scored lower than 23.
• What is the percentile of a raw score of 21?

10
PERCENTILE NORMS

• Percentile scores help us to determine the
  relative strengths or weaknesses of an
  individual performance on different tests.
• Why wouldnt the raw scores be able to give us
  the same information?

11
Grouped Data TechniquesCalculation of
Percentiles, Grouped Data
In this case, we want to know what score
corresponds to the 65th ile.
Scores Frequency (f) Calculations 85-89
1 (a) 65 of 50 32.5 80-89 3 (b) 2 3
8 12 25 75-79 6 (c) R 32.5 25
7.5 70-74 15 (d) Fp 15 65-69
12 (e) L 69.5 60-64 8 55-59 3 50-54
2 N 50
12
Grouped Data TechniquesCalculation of
Percentiles, Grouped Data
In this case, we want to know what score
corresponds to the 65th ile.
R Fp
69.5 2.5
Pp L i
7.5 15
69.5 5
P65 72
37.5 15
69.5 5
13
Group PracticeCalculation of Percentiles,
Grouped DataCalculate P30, P64, and P90
Scores Frequency (f) 70-79 2 60-69
7 50-59 14 40-49
25 30-39 13 20-29 7 10-19 9
0-9 3 N 80
14
Individual PracticeCalculation of Percentiles,
Grouped DataCalculate P50, P90, and P22
Scores Frequency (f) 85-89 1 80-89
3 75-79 6 70-74 15 65-69
12 60-64 8 55-59 3 50-54 2 N
50
15
MEASURES OF VARIABILITY

• The fluctuation of scores about a central
  tendency is called variability.
• We use measures of variability to compare two
  sets of scores.
• Although the means may be the same, the
  distribution may be different.

16
MEASURES OF VARIABILITY

• Range
• Standard Deviation

17
RANGE

• Defined The distance between two extreme
  scores.
• It informs us about the dispersion of our
  distribution.
• The larger the range the larger the dispersion
  from the mean value.
• Although the mean of the scores of two
  distributions can be identical their ranges may
  be different.

18
DRAWBACKS TO THE RANGE

• Good preliminary measure, but one single extreme
  value can influence the range significantly.
• The calculation of the range is derived from the
  highest and lowest values and doesnt tell us
  anything about the variability of the different
  values.

19
TWO DISTRIBUTIONS WITH EQUAL MEANS BUT DIFFERENT
RANGES

• Distribution 1
  Distribution II
• X X
• 10 12
• 9 12
• 8 12
• 8 11
• 7 10
• 7 5
• 6 4
• 6 2
• 5 1
• 4 1
• 70 Sum of X1 70 Sum of X2
• N 10 N 10

M1 7
M2 7
20
TWO DISTRIBUTIONS WITH EQUAL RANGES BUT
DISSIMILAR PATTERNS OF DISPERSION

• Distribution ! Distribution II
• X X
• 17 17
• 15 17
• 15 17
• 15 17
• 14 16
• 14 10
• 14 10
• 14 10
• 13 10
• 10 10
• 142 Sum of X1 142 Sum of X2
• N 10 N 10

M1 14.2
M2 14.0
R1 17-10 7
R2 17-10 7
21
THE STANDARD DEVIATION

• Each score in a distribution varies from the mean
  by a greater or lesser amount, except when the
  score is the same as the mean.
• Deviations from the mean can be noted as either
  positive or negative deviations from the mean.
• It would be impossible to find the average of
  these deviations because the mean would equal
  zero.

22
CALCULATION OF THE STANDARD DEVIATION (DEVIATION
METHOD)

• X x x2
• 15 5 25
• 14 4 16
• 12 2 4
• 12 2 4
• 11 1 1
• 10 0 0
• 9 -1 1
• 8 -2 4
• 7 -3 9
• 2 -8 64
• 100 Sum of X 128 Sum of
  X2
• N 10

SD 3.58
23
CALCULATION OF THE STANDARD DEVIATION (WHOLE
SCORE METHOD)

• X x2
• 12 144
• 12 144
• 14 196
• 17 289
• 14 196
• 13 169
• 16 256
• 15 225
• 15 225
• 15 225
• 143 2069 Sum of X2
  

N 10
143 10
14.3
M
M2 (14.3)2 204.5
The Standard Deviation is equal to the square
root of the sum of the score squared divided by
the number of scores less the square of the mean.
SD 1.55
24
COMPARISON OF THE STANDARD DEVIATION FOR TWO
DISTRIBUTIONS
25
No Child Left Behind

• Read the article on NCLB
• What is it?
• What are the reasons for it?
• Where are the dangers facing educators and
  children?

26
Next Week

• Chapter 1 Educational Testing and Assessment
  Context, Issues, and Trends
• Standard Deviations and the Normal Curve
• Calculating the z-score and T-score
• Damn-Calculating SD from Group Data