PSYC32H3 Chapter Notes - Chapter 1: Percentile Rank, Truncated Distribution, Standard Score
Chapter 1 Psychometrics in Neuropsychological Assessment
The Normal Curve
Basis of many commonly used statistical and psychometric models and is the assumed
distribution for many psychological variables.
Definition and Characteristics
Unimodal, symmetrical, asymptotic at the tails.
The ordinate (height of the curve at any point along the x-axis) is the proportion of
persons within the sample who obtained a given score.
Normal curve can also be referred to as a probability distribution.
Relevance for Assessment
As a frequency distribution, the area under any given segment of the normal curve
indicates the frequency of observations or cases within that interval.
•This provides psychologists with an estimate of the normality/abnormality of any
given test score or range of scores
oNormality – score falls in the center of the bell shape, where most of the
scores are located
oAbnormality – score falls at the ends of the bell shape, where there are
Z Scores and Percentiles
Percentile: the percentage of scores that fall at or below a given test score
Converting scores to percentiles – raw scores are ‘standardized’. Usually to Z scores
z = (x – X)/SD
x= measurement value (test score)
X= the mean of the test score distribution
SD= the standard deviation of the test score distribution
Resulting distribution of Z scores has a mean of 0 and a SD of 1.
Interpretation of Percentiles
The relationship between raw or Z scores and percentiles is not linear.
•A constant difference between raw or Z scores will be associated with a variable
difference in percentile scores, as a function of the distance of the two scores
from the mean.
•This is due to the fact that there are proportionally more observations (scores)
near the mean than there are farther from the mean
oOtherwise, the distribution would be rectangular or non-normal
Linear Transformations of Z Scores: T Scores and Other Standard Scores
Linear transformation can be used to produce other standardized scores.
T scores, Z scores, standard scores, and percentile equivalents are derived from
samples. They are often treated as population values, any limitations of generalizability
due to reference sample composition or testing circumstances must be taken into
consideration when standardized scores are interpreted.
The Meaning of Standardized Test Scores: Score Interpretation
When comparing scores, it should be done when the distributions for tests that are
being compared are approximately normal in the population. If standardized scores are
to be compared, they should be derived from similar samples or (more ideally) from the
Also when comparing scores, the reliability of the two measures must be considered
and they intercorrelation before determining if a significance exists.
•Relatively large disparities between standard scores may not actually reflect
reliable differences and therefore may not be clinically meaningful.
When test scores are not normally distributed, standardized scores may not accurately
reflect actual population rank.
Comparability across tests does not imply equality in meaning and relative importance
Interpreting Extreme Scores
In clinical practice, one may encounter standard scores that are either extremely low or
high. The meaning/comparability of the scores depends on the characteristics of the
normative sample from which they derive.
Whenever extreme scores are being interpreted, examiners should verify that an
examinee’s score falls within the range of raw scores in the normative sample.
•If the normative sample size is substantially smaller than the estimated
prevalence size and the examinee’s score falls outside the sample range, then
considerable caution may be indicated in interpreting the percentile associated
with the standardized score.
When interpreting extreme scores, it depends on the properties of the normal samples
The Normal Curve and Test Construction
A test with a normal distribution in the general population may show extreme skew or
other divergence from normality when administered to a population that differs
considerably from the average individual.
Whether a test produces a normal distribution is also an important aspect of evaluating
tests for bias across different populations.
Depending on the characteristics of the construct being measured and the purpose for
which a test is being designed, a normal distribution of scores may not be obtainable or
desirable. For example:
•The population distribution of the construct being measured may not be normally
•One may want only to identify and/or discriminate between persons at only one
end of a continuum of abilities
oThe characteristics of only one side of the sample score distribution are
critical while the characteristics on the other side of the distribution are not
oThe measure may even be deliberately designed to have floor or ceiling
It is not unusual for test score distributions to be markedly non-normal, even with large
The degree to which a given distribution approximates the underlying population
distribution increases as the number of observations (N) increases and becomes less
accurate as N decreases.
•Larger sample will produce a more normal distribution only if the underlying
population from which the sample is obtained is normal.