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Reminders • Homework due tomorrow • Quiz tomorrow 1 Warm Up - ACT Math Scores Density Distribution of ACT Math Scores 0 5 10 15 20 25 30 35 scores What percent of scores are between 12 and 24? Options: 38%, 50%, 53%, 68%, 77%, 86%, 90% , 95%, 99% 2 Warm Up - ACT Math Scores Density Distribution of ACT Math Scores 0 5 10 15 20 25 30 35 scores What percent of scores are between 6 and 30? Options: 38%, 50%, 53%, 68%, 77%, 86%, 90% , 95%, 99% 3 Warm Up - ACT Math Scores Density Distribution of ACT Math Scores 0 5 10 15 20 25 30 35 scores What can we say about the relationship between the mean and the median? 4 Chapter 13: Normal Distributions Aaron Zimmerman STAT 220 - Summer 2014 Department of Statistics University of Washington - Seattle 5 Density Curves • I’ve drawn a smooth Distribution of ACT Math Scores Density density curve over the histogram • Curves show proportions of observations in any region by areas under the curve • The entire area under the curve must be 1 and the curve must be non-negative. 0 5 10 15 20 25 30 35 scores 6 Strategies for exploring data • We have already discussed three “first steps” when you get new data on a single quantitative variable (1) Make a histogram (2) Look for the overall pattern (shape, center, spread) and for striking deviations such as outliers (3) Choose either the five-number summary or the mean and standard deviation to briefly describe the center and spread in numbers • Today, we are going to add one more strategy (4) Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve The smooth curve we will discuss is a Normal curve 7 Motivating example Distribution of ACT Math Scores ACT Math scores is symmetric with no clear outliers Density • The distribution of • This is the type of distribution that is well-described by a Normal curve 0 5 10 15 20 25 30 35 scores 8 Properties of Normal curves • You can get new Normal curves by sliding and stretching other Normal curves • A specific Normal curve is completely described by giving its mean and standard deviation • The mean determines the center of the distribution. It is located at the center of symmetry of the curve ?The mean ACT Math score is 18 Distribution of ACT Scores 0 5 10 15 20 25 30 35 x 9 Properties of Normal curves • The mean determines the center of the distribution. It is located at the center of symmetry of the curve ?The mean ACT Math score is 18 • The standard deviation determines the shape of the curve. It is the distance from the mean to the change-of-curvature point on either side ?The standard deviation of ACT Math scores is 6 Distribution of ACT Scores 0 5 10 15 20 25 30 35 x 10 Some cautions before we continue! • There are many types of data that are Normally distributed ?Human birth weight ?Heights of trees (within species) ?Sample means and proportions calculated from repeated random samples from the same population • However, there are many, many types of data that are not even close to Normal (and we have seen many of these already) ?Always remember to check a histogram of your data before performing any Normal calculations! 11 The 68-95-99.7 Rule In any Normal distribution • 68% of the observations fall within one SD of the mean • 95% of the observations fall within two SDs of the mean • 99.7% of the observations fall within three SDs of the mean Mean = 0, SD=1 12 The 68-95-99.7 Rule Distribution of ACT Scores ACT math scores are normally distributed with a mean of 18 and a standard deviation of 6 • So, 68% of ACT math scores are between 12 (one SD below the mean) and 24 (one SD above the mean) 0 5 10 15 20 25 30 35 x Mean = 18, SD=6 13 The 68-95-99.7 Rule Distribution of ACT Scores ACT math scores are normally distributed with a mean of 18 and a standard deviation of 6 • 95% of ACT math scores are between 6 (two SDs below the mean) and 30 (two SDs above the mean) 0 5 10 15 20 25 30 35 x Mean = 18, SD=6 14 The 68-95-99.7 Rule Distribution of ACT Scores ACT math scores are normally distributed with a mean of 18 and a standard deviation of 6 • Note: we already calculated these first two approximations using the histogram on the warm up! 0 5 10 15 20 25 30 35 x Mean = 18, SD=6 15 The 68-95-99.7 Rule Distribution of ACT Scores ACT math scores are normally distributed with a mean of 18 and a standard deviation of 6 • 99.7% of ACT math scores are between 0 (three SDs below the mean) and 36 (three SDs above the mean) 0 5 10 15 20 25 30 35 x Mean = 18, SD=6 16 Birth Weight Distribution of Birth Weights Human birth weight is Normally distributed with a mean of 3300 grams and a standard deviation of 300 grams. Approximately what percent of babies have a birth weight between 3000 grams and 3600 grams? 2500 3000 3500 4000 4500 x Mean = 3300, SD=300 17 Cherry Tree Heights Distribution of Cherry Tree Heights The heights of cherry trees are Normally distributed with a mean of 10 feet and a standard deviation of 2 feet. The heights of approximately 95% of all cherry trees lie between what two values? 5 10 15 x Mean = 10, SD=2 18 Cherry Tree Heights Distribution of Cherry Tree Heights Approximately what percentage of cherry trees have heights between 6 and 12 feet? 5 10 15 x Mean = 10, SD=2 19 Standard scores • The 68-95-99.7 rule is convenient for quick calculations, but we can calculate more precise percentages using standard scores ?To calculate the standard score for any observation in a Normal distribution, subtract the mean and then divide by the standard deviation standard score = observation − mean standard deviation Key idea #1: The standard score tells you how many standard deviations above or below the mean the observation lies 20 Standard scores Suppose that Grainne scored a 27 on the ACT Math test, while Karthik scored a 15. Remember, the mean is 18 and the SD is 6. • To calculate Grainne’s standard score: standard score = observation − mean 27 − 18 = = 1.5 standard deviation 6 ? Conclusion : Grainne scored 1.5 SDs above the mean • To calculate Karthik’s standard score: standard score = observation − mean 15 − 18 = = −0.5 standard deviation 6 ? Conclusion : Karthik scored 0.5 SDs below the mean 21 Why do standard scores work? Distribution of ACT Scores We start back with the distribution of ACT math scores 0 5 10 15 20 25 30 35 x Mean = 18, SD=6 22 Why do standard scores work? When we subtract the mean from every observation, the mean of the distribution becomes zero Distribution of ACT Scores Distribution of ACT Scores 0 5 10 15 20 25 x Mean = 18, SD=6 30 35 −15 −10 −5 0 5 10 15 x Mean = 0, SD=6 23 Why do standard scores work? When we divide every observation by the SD, the SD of the distribution becomes one Distribution of ACT Scores Distribution of ACT Scores −15 −10 −5 0 5 10 x Mean = 0, SD=6 15 −3 −2 −1 0 1 2 3 x Mean = 0, SD=1 24 Why do standard scores work? ? When we subtract the mean from every observation, the mean of the distribution becomes zero ? When we divide every observation by the SD, the SD of the distribution becomes one Distribution of ACT Scores Distribution of ACT Scores +/− 1 SD, Area=68% −20 −10 0 10 x 20 Mean=18, SD=6 30 Distribution of ACT Scores +/− 1 SD, Area=68% −20 −10 0 10 20 x Mean=0, SD=6 30 +/− 1 SD, Area=68% −20 −10 0 10 20 30 x Mean=0, SD=1 ? All of the blue areas are the same size 25 Why do standard scores work? Grainne’s score is blue while Karthik’s is green Distribution of ACT Scores 0 5 10 15 20 25 30 35 x 26 Why do standard scores work? Grainne’s score is blue while Karthik’s is green ?Subtract the mean Distribution of ACT Scores −15 −10 −5 0 5 10 15 x 27 Why do standard scores work? Grainne’s score is blue while Karthik’s is green ?Subtract the mean ?Divide by the standard deviation Distribution of ACT Scores −3 −2 −1 0 1 2 3 28 Why do standard scores work? • The idea here is that although we’ve stretched and shifted the Normal curve, we’ve done it to all points in the same way • Specifically, we’ve done it so areas under the curve are the same • As a result, once we have standard scores, we can use a standard normal distribution to learn something about the original distribution 29 Birth Weight Distribution of Birth Weights Human birth weight is Normally distributed with a mean of 3300 grams and a standard deviation of 300 grams. If Aaron was 3450 grams when he was born, what was his standard score? 2500 3000 3500 4000 4500 x 30 Cherry Tree Heights Distribution of Cherry Tree Heights The heights of cherry trees are Normally distributed with a mean of 10 feet and a standard deviation of 2 feet. What is the standard score for a tree that is 7.5 feet tall? 5 10 15 x 31 Homework • Read Chapter 13 • Do problems 13.1, 13.2, 13.7 (use information at bottom of page 280), 13.10a, 13.13, 13.14 (pay close attention to the mean and SD for each age group), 13.16b 32