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UT Knoxville STAT 201 - Chapter 6

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1Chapter06 Presentation 1213Copyright © 2009 Pearson Education, Inc.Chapter 6 The Standard Deviation as a Ruler and the Normal ModelChapter06 Presentation 1213Copyright © 2009 Pearson Education, Inc.2Tallest Living Humans Recent research estimates that the average height of men is 5’ 9” (69 inches) with a standard deviation of 3.0 inches. For women, the average is 5’ 3.5” (63.5 inches) with a standard deviation of 2.5 inches.Chapter06 Presentation 1213Copyright © 2009 Pearson Education, Inc.3 According to 2011 Edition of Guiness World Records, the tallest living man is Sultan Kösen of Turkey. He stands 8’ 3.0” (99.0 inches). The tallest living woman is Yao Defen of China. She stands 7’ 8.0” (92.0 inches). Sultan is taller than Yao, but is his height more unusual?Chapter06 Presentation 1213Copyright © 2009 Pearson Education, Inc.4The Standard Deviation as a Ruler The standard deviation is the most common measure of variation. The trick in comparing very different-looking values is to use standard deviations as our rulers.Chapter06 Presentation 1213Copyright © 2009 Pearson Education, Inc.5Standardizing with z-scores We compare individual data values to their mean, relative to their standard deviation using the following formula: z tells us how many standard deviations the value y is away from the mean.z y ysChapter06 Presentation 1213Copyright © 2009 Pearson Education, Inc.6Standardizing with z-scoresChapter06 Presentation 1213Copyright © 2009 Pearson Education, Inc.7Standardizing with z-scores (cont.) Calculate z-scores for Sultan (y=99.0”) and Yao (y=92.0”). Recall, for our Height example:Men WomenMean 69” 63.5”Std. Dev.3” 2.5”Chapter06 Presentation 1213Copyright © 2009 Pearson Education, Inc.8Different Units of Measure and z-Scores Daily high temperature in Knoxville, TN from 05/01/08 to 05/31/08 in Fahrenheit (F°) and Celsius (C°). Calculate the Z-scores for the maximum values:C° = 5(F° - 32)9Z = Z =Chapter06 Presentation 1213Copyright © 2009 Pearson Education, Inc.9When Is a z-score BIG? A z-score gives us an indication of how unusual a value is because it tells us how far it is from the mean. The z-scores for Sultan and Yao are at least 10, but these are world records (very unusual)! Where is the cutoff between typical and unusual values?Chapter06 Presentation 1213Copyright © 2009 Pearson Education, Inc.10When Is a z-score Big? (cont.) There is no universal standard for z-scores, but there is a model that shows up over and over in Statistics. This model is called the Normal Model. Normal models are appropriate for distributions whose shapes are unimodal and symmetric.Chapter06 Presentation 1213Copyright © 2009 Pearson Education, Inc.11 This is a histogram of the heights (in inches) of 1500 women (mean 63.5”, standard deviation 2.5”) with a Normal model drawn on top of it.Chapter06 Presentation 1213Copyright © 2009 Pearson Education, Inc.12 This is the same histogram, showing where Yao Defen’s height falls (recall, her height was z=11.4 standard deviations above the mean).92.0”Chapter06 Presentation 1213Copyright © 2009 Pearson Education, Inc.13When Is a z-score Big? (cont.) There is a Normal model for every possible combination of mean and standard deviation.  We write N(μ,σ) to represent a Normal model with a mean of μ and a standard deviation of σ. When we standardize Normal data, we still call the standardized value a z-score, and we write yzChapter06 Presentation 1213Copyright © 2009 Pearson Education, Inc.14When is a z-score Big? (cont.) Once we have standardized, we need only one Normal model:  The N(0,1) model is called the Standard Normal Model (or the Standard Normal Distribution).Chapter06 Presentation 1213Copyright © 2009 Pearson Education, Inc.15The 68-95-99.7 Rule We will be more precise in the near future, but until then we will use a simple rule that tells us a lot about the Normal model. On exams, be very familiar with this rule!Chapter06 Presentation 1213Copyright © 2009 Pearson Education, Inc.16The 68-95-99.7 Rule (cont.) The following shows what the 68-95-99.7 Rule tells us:Chapter06 Presentation 1213Copyright © 2009 Pearson Education, Inc.17In-Class Activity – Groups of 2 (or 3) You will need a calculator for this activity. Please do not do any internet searches to help you answer these questions. With your teammate(s), use your knowledge of normal models to guess plausible standard deviations for the 2 variables on the next page. You can base your estimates on a general knowledge that the distributions of these 2 variables are likely to be unimodal and symmetric. Then, estimate that the range is probably about ±3 standard deviations wide, or about 6 standard deviations total.Chapter06 Presentation 1213Copyright © 2009 Pearson Education, Inc.18In-Class Activity (cont.) Height (in inches) of a sample of one hundred 19 year old girls in the USA Weight (in pounds) of a sample of one hundred 19 year old boys in USAChapter06 Presentation 1213Copyright © 2009 Pearson Education, Inc.19So, When Is a z-score Big?  z-scores bigger than 3 (in absolute value) are considered “big” or unusual values. Sometimes values beyond 3 standard deviations from the mean are called outliers. Sultan and Vao’s z-scores were at least 10: these are extreme outliers.Chapter06 Presentation 1213Copyright © 2009 Pearson Education, Inc.20Working with Normal Models When we use the Normal model, we are assuming the data we are working with is Normal. No real data set is perfectly Normal, so we check the following condition: Nearly Normal Condition: The shape of the data’s distribution is unimodal and symmetric. One way to check this condition it to make a histogram.Chapter06 Presentation 1213Copyright © 2009 Pearson Education, Inc.21Beyond the 68-95-99.7 Rule What if we have an observation that is 1.5 SDsfrom the mean? What percent of observations should be within 1.5 SDs of the mean? Would it be (68%+95%)/2 = 81.5%?Chapter06 Presentation 1213Copyright © 2009 Pearson Education, Inc.22Finding Normal Percentages Using Technology Many calculators and statistics programs have the ability to find normal percentages for us. Your textbook mentions that the ActivStats Multimedia Assistant offers such technology tools.


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UT Knoxville STAT 201 - Chapter 6

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