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Foundations of Inferential Statistics: z-Scores

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

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Foundations of Inferential Statistics: z-Scores

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  1. Foundations of Inferential Statistics: z-Scores

  2. Has Anyone Else Been Bored to Tears by Descriptive Statistics? • Descriptives are very important • They help us understand and summarize the data we have • But statistics, as a field, is much more than descriptives • What would we like to be able to do? • MAKE INFERENCES! • TEST HYPOTHESES! • EXPLORE DATA AND RELATIONSHIPS!

  3. Taking a Look at z-Scores

  4. 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. • 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

  5. What Are z-Scores? • Z-Scores are transformations of the raw scores • 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 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

  6. The Standard Distribution andz-Scores

  7. Yet Another Visual!

  8. 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.

  9. 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

  10. Example • In Excel

  11. Standardizing a Distribution • 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 • Lets look at the example

  12. 8 3.3 1.4 4.7 8 11.3 14.6 100 10) 90 80 100 110 120 Example

  13. Samples Versus Populations • s vs. σ • s2 vs. σ2 • As always M vs. μ • 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 is in essence a penalty for sampling • Another way to think about it is because of the degrees of freedom

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