Download Z-scores & Review

Z-scores
& Review
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The Standard Normal Distribution
• Z-scores
– A descriptive statistic
that represents the
distance between an
observed score and
the mean relative to
the standard deviation
xx
z 
s
X 
z

Standard Normal Distribution
• Z-scores
– Converts distribution to:
• Have a mean = 0
• Have standard deviation = 1
– However, if the parent distribution is not
normal the calculated z-scores will not be
normally distributed.
Why do we calculate z-scores?
• To compare two different measures
– e.g., Math score to reading score, weight to
height.
– Area under the curve
• Can be used to calculate what proportion of scores
are between different scores or to calculate what
proportion of scores are greater than or less than a
particular score.
Class practice
How much do you weigh? _____
132, 149, 144,143, 113
Calculate z-scores for 120 & 133
What percentage of scores are less than
120?
What percentage are less than 133?
What percentage are between 120 and 133?
Z-scores to raw scores
• If we want to know
what the raw score of
a score at a specific
%tile is we calculate
the raw using this
formula.
• Using previous data
– What are the weights
of individuals at the
20%tile & the 33%tile?
x  z ( s)  x
Transformation scores
• We can transform scores to
have a mean and standard
deviation of our choice.
• Why might we want to do this?
Let’s say we have a set of spelling
scores with a mean of 15 and
a standard deviation of 5. We
want to transform them to have
a mean of 50 and a standard
deviation of 10.
What would be the transformed
scores for 12 and 18?
T  z ( s)  x
IQ scores
• We want:
– Mean = 100
– s = 15
• Transform:
– Z scores of:
• -1.23
• 1.56
• 1.32
x  z ( s)  x