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# Standard Scores

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### Standard Scores

1. 1. Week 6 Part A: Standard Scores 1
2. 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. 3. Standard Scores  Standard scores are transformed raw scores.  Allow us to determine the exact position of raw scores in the distribution. 3
4. 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. 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. 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. 7. z-standard (unit) normal distribution 7
8. 8. Revisiting properties of the normal distribution 8
9. 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. 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. 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. 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. 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. 14. Part B: Normal Distribution 14
15. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 26. Other standard scores Definitions Z-score T-score IQ GRE Mean 0 50 100 500 Standard 1 10 15 100 Deviation 26
27. 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. 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. 29. Part C: Sampling Distribution of the Means 29
30. 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. 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. 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. 33. Sampling Distribution  Any distribution that is of sample statistics and NOT individual observations/scores. 33
34. 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. 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. 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. 37. Note cards: • Z score formula • New score formula • T score values (sd = 10, mean = 50) In-Class Activity Creating a Sampling Distribution! 37