Notes 2: Normal Distribution and Standard Scores
1. Normal Distributions
ā¢ bell shaped: symmetrical, unimodal, and smooth (meaning that there is an unlimited number of
observations so a polygon or histogram will show a smooth distribution)
ā¢ central tendency: mean, median, and mode are identical (note that a normal distribution may have
any mean and any variance, e.g., the population distribution of IQs supposedly has Āµ = 100 and Ļ =
15, whereas the population of verbal SAT scores supposedly has Āµ = 500 and Ļ = 100 and both IQ
and SAT could be normally distributed)
ā¢ standard normal, unit normal curve, z-distribution: a normal distribution with Āµ = 0.00 and Ļ = Ļ
2
=
1.00; any normally distributed data set with scores transformed to z scores will have a standard
normal distribution; similarly, any non-normally distributed data with scores transformed to z scores
will retain their non-normal shape (z scores do not change shape of distribution)
ā¢ area: the area under the normal distribution can be found in a table of z scores, such as the one
provided on the next page
2. Relative Position and Standard Scores
ā¢ relative position: relative position or relative standing refers to the position of a particular score in
the distribution relative to the other scores in the distribution, e.g., percentile ranks, PR; relative
position is similar to the concept of norm-referenced interpretations found with many common tests
ā¢ standard scores: a standard score in one in which the mean and standard deviation of the distribution
will have pre-defined values, e.g., z scores have a M of 0.00 and s of 1.00; note that PR are not
standard scores
ā¢ z score: z scores provide a measure of the relative standing for a score within a distribution based
upon two things, the mean and standard deviation; z scores are standard scores since M = 0.00 and s
= 1.00; the formula for a sample is:
z =
SD
XX ā =
s
XX ā
ā¢ interpretation of z scores: since zās are expressed in standard deviations unit from the mean, a z of 1
implies that the raw score is 1 standard deviation above the mean of the distribution; if z is -2, then
the corresponding raw score is 2 standard deviations below the group mean; a z of -1.34 indicates
that the raw score is 1.34 standard deviations below the mean of the group
ā¢ mean, variance of z: the mean of z scores is 0.00, and the standard deviation (and variance) of z
scores is 1.00.
ā¢ shape of original distribution: if one transforms a set of raw scores into z scores, the transformation
does not change the shape of the original distribution; if the original distribution was normally
distributed, then the z score distribution will remain normally distributed; if the original distribution
was bimodal, then the z score distribution will be bimodalātransforming Xs into z scores changes
only the mean and variance of the distribution, with the transformed mean equal to 0.00 and the
transformed variance equal to 1.00
