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- 1. Week 6 Part A: Standard Scores 1
- 2. Standard Scores In order to compare scores from different distributions and in different units of measurement, we need a common scale or common unit of measurement. Convert scores from each distribution to standard scores on a common scale. 2
- 3. Standard Scores Standard scores are transformed raw scores. Allow us to determine the exact position of raw scores in the distribution. 3
- 4. Z-Scores The most common or the basis of all standard scores is a z-score. Can be used as descriptive statistics and as inferential statistics. Descriptive: describes exactly where each individual is located. Inferential: determines whether a specific sample is representative of its population or is extreme and unrepresentative. 4
- 5. How does it tell us this? If our observation X is from a population with mean μ and standard deviation σ, then If the observation X is from a sample with mean and standard deviation s, then 5
- 6. z-standard (unit) normal distribution The mathematics of z-score transformation converts every observation in a distribution to its z- score. With this transformation, the mean of the new (z) distribution becomes 0 and the standard deviation becomes 1. The transformed distribution is called z-standard (unit) normal distribution. 6
- 7. z-standard (unit) normal distribution 7
- 8. Revisiting properties of the normal distribution 8
- 9. Interpretation of z-scores A z-score shows the distance of a score from the mean in terms of standard deviation. A z-score of .05 means that a score is half a deviation above the mean. A z-score of -.05 means that a score is .05 standard deviations below the mean. 9
- 10. What does a z-score tell us? Answers the question: “How many standard deviations away from the mean is this observation in a normal distribution.” 10
- 11. When is the Z score useful? The z score transformation is useful when we seek to compare the relative standings of observations from distributions with different means and/or different standard deviations. 11
- 12. For example: Last semester, Matt scored 70 in Ms. Lauren’s math class. The average score of the class was 60 and the standard deviation was 15. This year, Matt is in Ms. Molly’s class. He scored 88. The mean score was 90 and the standard deviation was 4. In which class did Matt perform better? 12
- 13. In notation: Write down the information given and the information you need: X1= 70 X2= 88 μ1 = 60 μ2 = 90 σ1 =15 σ2 = 4 13
- 14. Part B: Normal Distribution 14
- 15. Normal Distribution The normal distribution is not a single distribution but a family of distributions, each which is determined by its mean and standard deviation. Properties: Unimodal Continuous Asymptotic Theoretical! 15
- 16. z-standard (unit) normal distribution When all scores are converted to z- scores and plotted, they form a z- distribution. A z-score distribution is a normal distribution with the fixed mean of 0 and the standard deviation of 1. Any set of scores can be transformed into z-scores and plotted on a this distribution. 16
- 17. Area under the normal curve In every normal distribution, the distance between the mean and a given Z score cuts off a fixed proportion of the total area under the curve. Statisticians have provided us with tables indicating the value of these proportions for each possible Z score. 17
- 18. See page 634 of text: Mean to Z is the percentage between the mean and z score or sd. ‘Area beyond Z’ represents the smaller portion. 18
- 19. Smaller Portion Larger portion Smaller portion So, if we have a Z of +1, the smaller portion would be to the right. The larger portion would be to the left. 19
- 20. Smaller Portion Larger portion Smaller portion So, if we have a Z of -2, the smaller portion would be to the left. The larger portion would be to the right. 20
- 21. Probability versus Percentile Probability of an event is the proportion of times the event would happened if we could repeat the operation a great many times. Always between 0 (never happen) and 1 (always happen). Percentile is the point which a specified observations falls. 21
- 22. Practice interpreting table: What is the probability of selecting a score that falls beyond 1 Z? What is the probability of selecting a score that fall below – 2Z? What is the percentile rank of someone who has a Z score of 2? What is the percentile rank of someone who has a Z score of 1? 22
- 23. New score = σ(z)+μ How can we convert a particular score to a distribution with a different mean and standard deviation? Compute a “new score.” 23
- 24. Example of newscore[1]: There are several IQ tests, each consisting of different number of items. Yet, all the IQ test results are reported on the same scale with the mean of 100 and a deviation of 15. How can we convert any score from any test to this common scale that everybody can understand? 24
- 25. Example of newscore[2]: 1. Convert the given X into a Z-score. 2. Using the new score formula, we obtain: New score = σnew distribution (z) + μnew distribution New IQ score = 15 (z) + 100 New distribution New distribution standard deviation mean 25
- 26. Other standard scores Definitions Z-score T-score IQ GRE Mean 0 50 100 500 Standard 1 10 15 100 Deviation 26
- 27. Gre 200 300 400 500 600 700 800 If you scored a 600, how many standard deviations away from the mean would you be? What percentage of people did worse than you? If you scored a 300? 27
- 28. T Score What if we want to convert Matt’s Z-score of .67 to a T score. New score = σ (z) + μ Tscore = 10 (.67) + 50 = 56.70 Standard deviation Matt’s raw score of Mean of T of T distribution 70 converted to a distribution Z-score 28
- 29. Part C: Sampling Distribution of the Means 29
- 30. Is the Sample Representative of the Population? Often make conclusions/inferences about the population from the sample under study. How do we know if a sample is representative of the population when every sample is different? How can we transform a population distribution of individuals to a population distribution of sample means? 30
- 31. Sampling error Every sample is different from the population. Sampling error is the discrepancy/error between the sample and the population. Random sampling is used to minimize this error so that it occurs randomly 31
- 32. Distribution of Sample Means Randomly group people into similar sized samples. Calculate the sample means. Place them into a distribution. Result in a normal curve which is the distribution of sample means. 32
- 33. Sampling Distribution Any distribution that is of sample statistics and NOT individual observations/scores. 33
- 34. Properties of the distribution of sample means Approaches a normal distribution as sample size increases. Mean of the distribution is equal to the population mean of individuals. Standard deviation of this distribution is called the standard error of the sample mean. 34
- 35. Standard Error of the Sample Mean Standard error (σx) = σ √n Measures the standard distance between the sample mean and the population mean. A measure of how good an estimate will have for population mean. As sample size increases, the standard error decreases. 35
- 36. Probability and the Distribution of Sample Means: What does it tell us? What is the probability of obtaining a specific sample mean from the population of samples? 36
- 37. Note cards: • Z score formula • New score formula • T score values (sd = 10, mean = 50) In-Class Activity Creating a Sampling Distribution! 37

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