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The Meaning and Application of Norms

A person’s score on any psychological measure is not very informative in itself. A score
of 90/100 or 100% may mean different things depending on the makeup of the test.

The strengths and weaknesses of a person, school, or district are more clearly illustrated
when performance is compared to a larger representative group.

Norms refer to information regarding the performance of a particular reference group on
a particular measure for which a person’s score can be compared to.

The size of the reference group used for norming is not as important as to how
representative that group is of the relevant population. Norming groups are selected to be
correspond to the total population in terms of geographic region, sex, age, urban vs. rural,
community size, parent’s occupational and educational level, and ethnicity.

Norms are NOT standards. A norm is not necessarily representative of what is desired,
but rather a measure of what is. However, sometimes norms are used to define standards.
The Normal Distribution

The normal distribution is a theoretical symmetrical distribution for continuous
variables that follows a bell shaped curve with many observations near the middle and
fewer scores at the extreme. The shape of the normal curve is defined by:
Y
1
2 
( X  )2
e
2 2
where:
Y represents the height of the curve above a given X,
 3.1416 and e 2.718,
= the standard deviation of the distribution of X- scores and
 = the mean of the distribution of X- scores

The scores on most standardized measures are approximately normally
distributed. This is true for cognitive tests, unless they are too easy or too
difficult

Test scores are often discussed in terms of the proportion of cases that are likely
to fall above or below a particular score, assuming a normal distribution. These
probabilities are known as percentile ranks.

For any normal distribution, mean = median = mode. Furthermore, approximately
68% of scores lie within 1 standard deviation of the mean and 95% of
observations lie within 2 standard deviations of the mean
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Example:
Plot a normal curve that has a mean of 100 and a standard deviation of 10. What score
corresponds to a percentile rank of 50? What score corresponds to a percentile rank of
98? What percentile rank does a score of 110 correspond to?
The Standard Normal Distribution and Standard Scores

The standard normal distribution is a normal distribution with a mean of 0 and a
standard deviation of 1. Variables that come from this distribution are referred to as zscores or standard scores and it is possible to convert any normal distribution to a
standard normal distribution by using the following transformation:
z
X 

We can then use standard normal tables to find the probability that a particular score falls
above or below a particular interval.

Z-scores are not widely used when reporting test results. Oftentimes, T-scores are
reported to avoid reporting negative standard scores. T-scores are simply standard scores
with a mean of 50 and a standard deviation of 10. To transform a raw score into a t-score
the raw score must first be converted into a z-score. Then
T –score = 50 + 10 (z-score)
In reality, z-scores can be converted into any standard score scale, each defined by a
mean and a standard deviation using the following formula:
Standard score = New mean + New standard deviation (z-score)

Standard scores (either expressed as z-scores or t-scores, or any other standard score)
from different test administrations can be directly compared since the mean and standard
deviation for all tests will be the same.

Stanine scores were developed and used by the air force in WWII. These standard scores
have a mean of 5 and a standard deviation of approximately 2.0. This scoring system
forces all raw scores into a discrete number from 1 – 10. These scores lack precision due
to the convenience of having only ten score categories.

NCE scores are standards scores with a mean of 50 and a standard deviation of
approximately 21. These scores are easily confused with percentiles, and therefore are
not recommended.

Grade equivalency scores are not typically standard scores! These type of scores are
computed by simply finding the mean or median scale score for students at different
grade levels (e.x. 4, 5, and 6) and interpolating between grade levels by dividing the
difference in scores from two consecutive grade levels into 10 different categories.
However, since it is possible to transform onto ANY scale it is possible to create a scale
with a mean that represents a particular grade level and a standard deviation of one.