Foundations of Inferential Foundations of Inferential Statistics: z-ScoresStatistics: z-Scores

Has Anyone Else Been Bored to Has Anyone Else Been Bored to Tears by Descriptive Statistics?Tears by Descriptive Statistics?

Descriptives are very importantDescriptives are very important They help us understand and summarize the They help us understand and summarize the

data we havedata we have But statistics, as a field, is much more than But statistics, as a field, is much more than

descriptivesdescriptives What would we like to be able to do?What would we like to be able to do?

MAKE INFERENCES!MAKE INFERENCES! TEST HYPOTHESES!TEST HYPOTHESES! EXPLORE DATA AND RELATIONSHIPS!EXPLORE DATA AND RELATIONSHIPS!

Taking a Look at z-ScoresTaking a Look at z-Scores

What is a Standard Distribution?What is a Standard Distribution?

A A standard distributionstandard distribution is composed of scores is composed of scores that have been transformed to create that have been transformed to create predetermined values for μ and σ. Standardized predetermined values for μ and σ. Standardized distributions are used to make dissimilar distributions are used to make dissimilar distributions comparable.distributions comparable. The mean of this distribution is always made to equal The mean of this distribution is always made to equal

0 through this transformation (the means of the 0 through this transformation (the means of the deviations are always zero)deviations are always zero)

The standard deviation of this distribution is always The standard deviation of this distribution is always made to equal 1 through this transformationmade to equal 1 through this transformation

What Are z-Scores?What Are z-Scores?

Z-Scores are transformations of the raw Z-Scores are transformations of the raw scoresscores

What do z-scores tell us?What do z-scores tell us? They tell us exactly where a score falls They tell us exactly where a score falls

relative to the other scores in the distributionrelative to the other scores in the distribution They tell us how scores on one distribution They tell us how scores on one distribution

relate to scores on a totally different relate to scores on a totally different distributiondistribution• In other words they give us a standard way of In other words they give us a standard way of

looking at raw scoreslooking at raw scores

The Standard Distribution andThe Standard Distribution andz-Scoresz-Scores

Yet Another Visual!Yet Another Visual!

About z-ScoresAbout z-Scores

What might the sign tell us?What might the sign tell us? The sign tells us the direction.The sign tells us the direction.

What might the Magnitude tell us?What might the Magnitude tell us? The magnitude tells us how far from the mean The magnitude tells us how far from the mean

the score is in units of s.d.the score is in units of s.d.

How Do We Calculate a z-Score?How Do We Calculate a z-Score?

We must make the mean equal to zeroWe must make the mean equal to zero What have we looked at that has a mean of zero?What have we looked at that has a mean of zero?

• Deviations from the meanDeviations from the mean• (X - μ)(X - μ)

What is the other important property of z-What is the other important property of z-Scores?Scores? The are in units of s.d.The are in units of s.d. How do we standardize the scores in this way?How do we standardize the scores in this way? Divide by Divide by σσ

ThereforeTherefore z = (X - μ) / σz = (X - μ) / σ

ExampleExample

In ExcelIn Excel

Standardizing a DistributionStandardizing a Distribution

We might wish to look at a distribution with We might wish to look at a distribution with a different a different μμ and and σσ Say we wanted our Say we wanted our μμ to be 100 and our to be 100 and our σσ to to

be 10be 10 Lets look at the exampleLets look at the example

ExampleExample

88 3.3

11.3 14.64.71.4

100 10)

100 110 1209080

Samples Versus PopulationsSamples Versus Populations

s vs. s vs. σσ ss22 vs. vs. σσ22

As always M vs. As always M vs. μμ N versus n-1N versus n-1

This increases the size of the average deviant and This increases the size of the average deviant and makes it a more accurate, unbiased estimator of the makes it a more accurate, unbiased estimator of the population scorepopulation score

This is in essence a penalty for samplingThis is in essence a penalty for sampling Another way to think about it is because of the Another way to think about it is because of the

degrees of freedomdegrees of freedom