Chapter 6 Chapter06 Presentation 1213 The Standard Deviation as a Ruler and the Normal Model Copyright 2009 Pearson Education Inc 1 Tallest 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 1213 Copyright 2009 Pearson Education Inc 2 According to 2011 Edition of Guiness World Records the tallest living man is Sultan K sen 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 Chapter06 Presentation 1213 Sultan is taller than Yao but is his height more unusual Copyright 2009 Pearson Education Inc 3 The 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 1213 Copyright 2009 Pearson Education Inc 4 Standardizing with z scores We compare individual data values to their mean relative to their standard deviation using the following formula y y z s z tells us how many standard deviations the value y is away from the mean Chapter06 Presentation 1213 Copyright 2009 Pearson Education Inc 5 Standardizing with z scores Chapter06 Presentation 1213 Copyright 2009 Pearson Education Inc 6 Standardizing with z scores cont Calculate z scores for Sultan y 99 0 and Yao y 92 0 Recall for our Height example Men Women Mean 69 63 5 Std Dev 3 2 5 Chapter06 Presentation 1213 Copyright 2009 Pearson Education Inc 7 Different Units of Measure and z Scores C 5 F 32 9 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 Z Z Chapter06 Presentation 1213 Copyright 2009 Pearson Education Inc 8 When 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 1213 Copyright 2009 Pearson Education Inc 9 When 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 1213 Copyright 2009 Pearson Education Inc 10 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 1213 Copyright 2009 Pearson Education Inc 11 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 1213 Copyright 2009 Pearson Education Inc 12 When Is a z score Big cont There is a Normal model for every possible combination of mean and standard deviation We write N to represent a Normal model with a mean of and a standard deviation of When we standardize Normal data we still call the standardized value a z score and we write z Chapter06 Presentation 1213 y Copyright 2009 Pearson Education Inc 13 When 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 1213 Copyright 2009 Pearson Education Inc 14 The 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 1213 Copyright 2009 Pearson Education Inc 15 The 68 95 99 7 Rule cont The following shows what the 68 95 99 7 Rule tells us Chapter06 Presentation 1213 Copyright 2009 Pearson Education Inc 16 In 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 1213 Copyright 2009 Pearson Education Inc 17 In 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 USA Chapter06 Presentation 1213 Copyright 2009 Pearson Education Inc 18 So 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 1213 Copyright 2009 Pearson Education Inc 19 Working 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 1213 Copyright 2009 Pearson Education Inc 20 Beyond the 68 95 99 7 Rule What if we have an observation that is 1 5 SDs from 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 1213 Copyright 2009 Pearson Education Inc 21 Finding 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 Warning Use the Normal Distribution calculator found at http davidmlane com hyperstat z table html Google Normal Curve Calculator on your laptop tablet or smartphone and it should be the first link shown Chapter06 Presentation 1213 Copyright 2009 Pearson Education Inc 22 Finding Normal Percentages Using Technology Suppose SAT scores are normally distributed with a mean of 500 and a standard deviation of 100 You scored a 680 which is z 1 80 standard deviations above average What percent of people have SAT scores below yours
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