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Technical Information on the Development of Boxplots and Effect Sizes |
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The use of boxplots was
first advocated by Tukey (1978) as a method to graphically
represent differences in performance. The boxplot provides
information on the distribution of the data, the interquartile
spread of scores (25th percentile to 75th
Percentile) and the mean. The graphical nature allows most
people to readily identify differences in performance between
two groups represented on the boxplot. |
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ALERT Boxplots
and Effect Sizes |
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The software package PHP
used to generate the boxplots does not allow for the reporting
of the mean and the median. However, to aide in interpretation,
we have elected to use the mean to represent the middle bar. |
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1 -
Represents the 90th Percentile
2 - Represents the 75th Percentile
3 - Represents the Mean
4 - Represents the 25th Percentile
5 - Represents the 10th Percentile
6 - Represents the Interquartile Spread
(25th to 75th Percentile)
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The goal of the
boxplots is to represent many facets of the scores in the
subject area, while also allowing you to make comparisons
between the specific group. For example
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Mobile Students
|
Non-Mobile Students |
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| Performance Values: |
| |
Mobile
Students |
Non-Mobile Students |
| 1: |
255 |
280 |
| 2: |
208 |
233 |
| 3: |
184 |
197 |
| 4: |
160 |
181 |
| 5: |
147 |
165 |
| 6: |
48 |
47 |
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Practical Example and
Development of Boxplots
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| The boxplots used in
the ALERT series was developed following the models provided by
Tukey (1978), with a few minor variations. |
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| Calculation of
Mean Literacy Scaled Scores |
All the Literacy
scaled scores from the Arkansas Benchmark 4th, 6th, and 8th
grade exams are combined with the End-of-Course (EOC) Literacy
Exam results to produce a composite score for the district.
|
| Example 1: |
Fourth Grade Benchmark
Exam |
n = 128 |
| |
Sixth Grade Benchmark
Exam |
n = 138 |
| |
Eighth Grade Benchmark
Exam |
n = 125 |
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EOC Literacy Exam |
n = 129 |
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Total
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N = 520 |
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| Further, let us
assume a the following values: |
| Mobility |
N |
Mean Literacy Scaled
Score |
Standard Deviation |
| |
|
|
|
| Mobile |
287 |
184 |
37.5 |
| Non-Mobile |
233 |
197 |
32.6 |
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| All computations for
the Mobile/non-Mobile comparison would be generated from this
pool of 520 literacy scaled scores. First, a mean scores, then
standard deviation scores are generated. Next, a pooled variance
score is computed using: |
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Next, the effects
sizes (δ) are computed using:
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Interpretation
Thus, the achievement gap between Mobile and Non-Mobile student
is δ = .367 or representative of a moderate effect size
according to Cohen (1995).
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