Most important Chapter 5 The Standard Deviation as a Ruler and the Normal Model Chapter05 Presentation 0115 Copyright 2014 2012 2009 Pearson Education Inc 1 5 1 Standardizing with z Scores Chapter05 Presentation 0115 Copyright 2014 2012 2009 Pearson Education Inc 2 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 Chapter05 Presentation 0115 Copyright 2014 2012 2009 Pearson Education Inc 3 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 Chapter05 Presentation 0115 Sultan is taller than Yao but is his height more unusual Copyright 2014 2012 2009 Pearson Education Inc 4 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 Chapter05 Presentation 0115 Copyright 2014 2012 2009 Pearson Education Inc 5 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 Chapter05 Presentation 0115 Copyright 2014 2012 2009 Pearson Education Inc 6 Standardizing with z scores Chapter05 Presentation 0115 Copyright 2014 2012 2009 Pearson Education Inc 7 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 Chapter05 Presentation 0115 Copyright 2014 2012 2009 Pearson Education Inc 8 5 2 Shifting and Scaling Chapter05 Presentation 0115 Copyright 2014 2012 2009 Pearson Education Inc 9 Units of Measure and z Scores If you convert an entire data set into z scores This is known as standardizing the data The z scores are unitless numbers This shifts the mean to 0 and This rescales the standard deviation to 1 Shifting and rescaling a data set does not change the shape of the distribution This applies to converting data into z scores or converting data from one unit of measure to another Chapter05 Presentation 0115 Copyright 2014 2012 2009 Pearson Education Inc 10 Different Units of Measure and z Scores 5 F 32 C 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 Chapter05 Presentation 0115 Copyright 2014 2012 2009 Pearson Education Inc 87 77 22 5 879 1 6767 1 6767 11 5 3 Normal Models Chapter05 Presentation 0115 Copyright 2014 2012 2009 Pearson Education Inc 12 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 Chapter05 Presentation 0115 Copyright 2014 2012 2009 Pearson Education Inc 13 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 Chapter05 Presentation 0115 Copyright 2014 2012 2009 Pearson Education Inc 14 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 Chapter05 Presentation 0115 Copyright 2014 2012 2009 Pearson Education Inc 15 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 Chapter05 Presentation 0115 Copyright 2014 2012 2009 Pearson Education Inc 16 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 y m z s Chapter05 Presentation 0115 Copyright 2014 2012 2009 Pearson Education Inc 17 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 Chapter05 Presentation 0115 Copyright 2014 2012 2009 Pearson Education Inc 18 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 Chapter05 Presentation 0115 Copyright 2014 2012 2009 Pearson Education Inc 19 The 68 95 99 7 Rule cont The following shows what the 68 95 99 7 Rule tells us when unimodal and symmetric m Chapter05 Presentation 0115 m m m Copyright 2014 2012 2009 Pearson Education Inc m m m 20 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 Chapter05 Presentation 0115 Copyright 2014 2012 2009 Pearson Education Inc 21 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 Chapter05 Presentation 0115 Copyright 2014 2012 2009 Pearson Education Inc 22 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 Yao s z scores were at least 10 these are extreme outliers Chapter05 Presentation 0115 Copyright 2014 2012 2009 Pearson Education Inc 23 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
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