Reminders. Homework due tomorrow Quiz tomorrow

Transcription

1 Reminders Homework due tomorrow Quiz tomorrow 1

2 Warm Up - ACT Math Scores Distribution of ACT Math Scores Density scores What percent of scores are between 12 and 24? Options: 38%, 50%, 53%, 68%, 77%, 86%, 90%, 95%, 99% 2

3 Warm Up - ACT Math Scores Distribution of ACT Math Scores Density scores What percent of scores are between 6 and 30? Options: 38%, 50%, 53%, 68%, 77%, 86%, 90%, 95%, 99% 3

4 Warm Up - ACT Math Scores Distribution of ACT Math Scores Density scores What can we say about the relationship between the mean and the median? 4

5 Chapter 13: Normal Distributions Aaron Zimmerman STAT Summer 2014 Department of Statistics University of Washington - Seattle 5

6 Density Curves I ve drawn a smooth 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. Density Distribution of ACT Math Scores scores 6

7 Strategies for eploring 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

8 Motivating eample Distribution of ACT Math Scores The distribution of ACT Math scores is symmetric with no clear outliers Density This is the type of distribution that is well-described by a Normal curve scores 8

9 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

10 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

11 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

12 The 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

13 The Rule 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) Mean = 18, SD=6 13

14 The Rule 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) Mean = 18, SD=6 14

15 The Rule ACT math scores are normally distributed with a mean of 18 and a standard deviation of 6 Note: we already calculated these first two approimations using the histogram on the warm up! Mean = 18, SD=6 15

16 The Rule ACT math scores are normally distributed with a mean of 18 and a standard deviation of % of ACT math scores are between 0 (three SDs below the mean) and 36 (three SDs above the mean) Mean = 18, SD=6 16

17 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. Approimately what percent of babies have a birth weight between 3000 grams and 3600 grams? Mean = 3300, SD=300 17

18 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 approimately 95% of all cherry trees lie between what two values? Mean = 10, SD=2 18

19 Cherry Tree Heights Distribution of Cherry Tree Heights Approimately what percentage of cherry trees have heights between 6 and 12 feet? Mean = 10, SD=2 19

20 Standard scores The 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

21 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 standard deviation = = 1.5 Conclusion : Grainne scored 1.5 SDs above the mean To calculate Karthik s standard score: standard score = observation mean standard deviation = = 0.5 Conclusion : Karthik scored 0.5 SDs below the mean 21

22 Why do standard scores work? We start back with the distribution of ACT math scores Mean = 18, SD=6 22

23 Why do standard scores work? When we subtract the mean from every observation, the mean of the distribution becomes zero Mean = 18, SD= Mean = 0, SD=6 23

24 Why do standard scores work? When we divide every observation by the SD, the SD of the distribution becomes one Mean = 0, SD= Mean = 0, SD=1 24

25 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 +/ 1 SD, Area=68% +/ 1 SD, Area=68% +/ 1 SD, Area=68% Mean=18, SD=6 Mean=0, SD=6 Mean=0, SD=1 All of the blue areas are the same size 25

26 Why do standard scores work? Grainne s score is blue while Karthik s is green

27 Why do standard scores work? Grainne s score is blue while Karthik s is green Subtract the mean

28 Why do standard scores work? Grainne s score is blue while Karthik s is green Subtract the mean Divide by the standard deviation

29 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

30 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?

31 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?

32 Homework Read Chapter 13 Do problems 13.1, 13.2, 13.7 (use information at bottom of page 280), 13.10a, 13.13, (pay close attention to the mean and SD for each age group), 13.16b 32