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Fact Sheet 54
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Reliabilities and Confidence Ranges for Lucid CoPS
What are confidence ranges and why are they used?
When reporting a standard score, it is good practice also to report the confidence range
attached to that score. The reason for this is that all psychological and educational tests
scores give only estimates of ability, based on a sample of behaviour at a given point in time.
If you were to assess a student on several occasions you would not expect them to obtain
exactly the same score each time – there would be a spread of scores and somewhere within
that spread we would expect the (hypothetical) true score to lie. The amount of spread or
variation of actual scores obtained by an individual is dependent on the reliability of the test.
The confidence range is the zone around the standard score in which we are reasonably
confident the true score lies. Different confidence ranges may be set: for Lucid CoPS we have
set a confidence level of 90%, which means that there is a 90% probability that the true
standard score lies within the stated confidence interval. Put another way, if the student was
retested 100 times, on 90 out of 100 occasions the score would lie within the stated
confidence interval.
How are confidence intervals calculated?
Confidence ranges are calculated on the basis of the Standard Error of Measurement (SEM) of
a test which, in turn, is determined by the reliability of the test and the standard deviation of
test scores. Table 1 shows the reliabilities (Cronbach’s alpha or split-half reliability) for each
of the Lucid CoPS tests. Note that these are all high (mostly around 0.8 – 0.9) except for the
Zoid’s Letter Names, where the reliability is 0.74. This is because Zoid’s Letter Names
comprises fewer test items as a result of necessary reductions in the overall length of the test.
A compromise normally has to be reached between test length and overall reliability.
Table 1. Reliability measures and 90% confidence ranges for the tests in Lucid CoPS.
Reliability
Confidence
range for z
scores(±)
Confidence
range for
standard
scores(±)
Rabbits
0.82
0.68
10
Zoid's Friends
0.80
0.72
11
Toybox
0.95
0.36
5
Zoid's Letters
0.91
0.48
7
Zoid's Letter
Names
0.74
0.82
12
Races
0.89
0.53
8
Rhymes
0.79
0.73
11
Wock
0.90
0.51
8
Test
Copyright © 2012 Lucid Research Ltd
F54 v01 March 12
Fact Sheet 54
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How to apply confidence ranges
Table 1 also shows the confidence ranges for each test. These are given for both z scores and
standard scores. (Note that confidence ranges do not apply to centile scores because they are
not distributed in a normal curve.) Applying a confidence band for any of the tests requires
the following two steps:
1. Obtain the child’s z score or standard score* for that test.
* If required, a z score can be converted into a standard score by multiplying the z
score by the standard deviation (15), and then subtracting it from the mean (100) if
the z score was negative, or adding it to the mean if the z score was positive. For
example, a z score of -0.5 means ‘half a standard deviation below the mean’, or 100 –
(15 x 0.5) = standard score 92.5; similarly, a z score of 1.25 means ‘one-and-a-quarter
standard deviations above the mean’, or 100 + (15 x 1.25) = 118.5 standardised
score.
2. Determine the appropriate confidence range by consulting Table 1 and then (a)
subtracting that figure from the score, and (b) adding that figure from the score. For
example, if a child obtained a z score of 0.40 on Zoid’s Letters, for which the z score
confidence range is ±0.48, the 90% confidence limits for the test are
0.40 ± 0.48 = –0.92 (lower limit) to 0.88 (upper limit). This would mean that you can
be sure within a 90% degree of confidence that the child’s true score lies between
–0.92 and 0.88. Since a z score of 0.40 is equivalent to a standard score of 100 + (15
x 0.40) = 106, and the standard score confidence range is ±7, the 90% confidence
limits for the test are 106 ± 7 = 99 (lower limit) to 113 (upper limit).
For more information about Lucid or the developments or research please visit the Lucid
web site www.lucid-research.com. The Lucid staff can be contacted by email [email protected], telephone +44 (0)1482 882121 or fax +44 (0)1482 882911.
Please note that the information contained in this document is correct at time of going to
press.
Copyright © 2012 Lucid Research Ltd
F54 v01 March 12