Transcript 5 z-scores - Joaquin Roca

Foundations of Inferential
Statistics: z-Scores
Has Anyone Else Been Bored to
Tears by Descriptive Statistics?
 Descriptives
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
They help us understand and summarize the
data we have
But statistics, as a field, is much more than
descriptives
 What
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are very important
would we like to be able to do?
MAKE INFERENCES!
TEST HYPOTHESES!
EXPLORE DATA AND RELATIONSHIPS!
Taking a Look at z-Scores
What is a Standard Distribution?

A standard distribution is composed of scores
that have been transformed to create
predetermined values for μ and σ. Standardized
distributions are used to make dissimilar
distributions comparable.
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The mean of this distribution is always made to equal
0 through this transformation (the means of the
deviations are always zero)
The standard deviation of this distribution is always
made to equal 1 through this transformation
What Are z-Scores?
 Z-Scores
are transformations of the raw
scores
 What do z-scores tell us?
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They tell us exactly where a score falls
relative to the other scores in the distribution
They tell us how scores on one distribution
relate to scores on a totally different
distribution
• In other words they give us a standard way of
looking at raw scores
The Standard Distribution and
z-Scores
Yet Another Visual!
About z-Scores
 What
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The sign tells us the direction.
 What
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might the sign tell us?
might the Magnitude tell us?
The magnitude tells us how far from the mean
the score is in units of s.d.
How Do We Calculate a z-Score?
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We must make the mean equal to zero
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What have we looked at that has a mean of zero?
• Deviations from the mean
• (X - μ)
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What is the other important property of zScores?
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The are in units of s.d.
How do we standardize the scores in this way?
Divide by σ
Therefore
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z = (X - μ) / σ
Example
 In
Excel
Standardizing a Distribution
 We
might wish to look at a distribution with
a different μ and σ
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Say we wanted our μ to be 100 and our σ to
be 10
Lets look at the example
Example
1.4
4.7
8
8
11.3
3.3
14.6
100
80
90
100
110
120
10)
Samples Versus Populations
s vs. σ
 s2 vs. σ2
 As always M vs. μ
 N versus n-1
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This increases the size of the average deviant and
makes it a more accurate, unbiased estimator of the
population score
This is in essence a penalty for sampling
Another way to think about it is because of the
degrees of freedom