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Foundations of Inferential Statistics: z-Scores

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Has Anyone Else Been Bored to Tears by Descriptive Statistics? Descriptives are very important Descriptives are very important They help us understand and summarize the data we have They help us understand and summarize the data we have But statistics, as a field, is much more than descriptives But statistics, as a field, is much more than descriptives 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!

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Taking a Look at z-Scores

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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. 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. The mean of this distribution is always made to equal 0 through this transformation (the means of the deviations are always zero) 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 The standard deviation of this distribution is always made to equal 1 through this transformation

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What Are z-Scores? Z-Scores are transformations of the raw scores Z-Scores are transformations of the raw scores What do z-scores tell us? What do z-scores tell us? They tell us exactly where a score falls relative to the other scores in the distribution 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 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 scoresIn other words they give us a standard way of looking at raw scores

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The Standard Distribution and z-Scores

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Yet Another Visual!

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About z-Scores What might the sign tell us? The sign tells us the direction. What might the Magnitude tell us? The magnitude tells us how far from the mean the score is in units of s.d.

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How Do We Calculate a z-Score? We must make the mean equal to zero What have we looked at that has a mean of zero? Deviations from the mean (X - μ) What is the other important property of z- Scores? The are in units of s.d. How do we standardize the scores in this way? Divide by σ Therefore z = (X - μ) / σ

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Example In Excel In Excel

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Standardizing a Distribution We might wish to look at a distribution with a different μ and σ We might wish to look at a distribution with a different μ and σ Say we wanted our μ to be 100 and our σ to be 10 Say we wanted our μ to be 100 and our σ to be 10 Lets look at the example Lets look at the example

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Example 8 83.3 11.314.6 4.71.4 10010) 100110120 90 80

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Samples Versus Populations s vs. σ s vs. σ s 2 vs. σ 2 s 2 vs. σ 2 As always M vs. μ As always M vs. μ N versus n-1 N versus n-1 This increases the size of the average deviant and makes it a more accurate, unbiased estimator of the population score 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 This is in essence a penalty for sampling Another way to think about it is because of the degrees of freedom Another way to think about it is because of the degrees of freedom

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