STAT 110 - Section 5 Lecture 16

1. STAT 110 - Section 5 Lecture 16 Professor Hao Wang University of South Carolina Spring 2012 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

2. 4 5 6 7 8 9 7 5 2 3 6 6 7 4 8 8 9 0 2 3 Last time: histogram and stemplot

3. Chapter 12 – Describing Distributions with Numbers Percent of College Graduates by State Year AL AK FL GA KY LA MS NC SC TN % 22.3 18.8 26.0 27.6 21.0 22.4 20.1 23.4 24.9 24.3 18192021222324 25 26 27 8 1 03 443 9 0 6

4. Median median (M) – the midpoint of a distribution, the number such that at least half the observations are less than or equal to it and at least half are greater than or equal to it

5. Median

6. To find the median of a distribution: • 1. Arrange all observations in order from smallest to largest. • 2. Is the number of observations odd or even?

7. Median • Percent of College Graduates by State • 22.3 18.8 26.0 27.6 21.0 22.4 20.1 23.4 24.9 24.3 • Order the observations.18.8 20.1 21.0 22.3 22.4 23.4 24.3 24.9 26.0 27.6 • Is the number of observations even or odd? • So, what’s the median?

8. Median • If the number of observations n is even, the median is the average of the two center observations in the ordered list. • - Count (n + 1)/2 observations up from the bottom of the list. • - Average this number with the one above it. • If the number of observations n is odd, the median is the center observation in the ordered list. • - Count (n + 1)/2 observations up from the bottom of the list.

9. Median • For States in the Midwest • IL IN IA MI MN OH WI • 27.4 21.1 24.3 24.4 32.5 24.6 25.6 • Order the observations.21.1 24.3 24.4 24.6 25.6 27.4 32.5 A - 24.4 D – 25.1 B – 24.5 E – 25.6 C – 24.6

10. Quartiles Q1 – the point that is one-quarter of the way up the ordered list of observations Q3 – the point that is three-quarters of the way up the ordered list of observations • Quartiles help to measure spread. • How do you think Q2 would be defined? • What’s another name for Q2?

11. Quartiles To calculate the quartiles: 1. Arrange the observations in increasing order. 2. Locate the median. 3. Q1 is the median of the observations whose position in the ordered list is to the left of the location of the overall median. 4. Q3 is the median of the observations whose position in the ordered list is to the right of the location of the overall median.

12. Quartiles Percent of College Graduates in the Southeast 22.3 18.8 26.0 27.6 21.0 22.4 20.1 23.4 24.9 24.3 Order the observations.18.8 20.1 21.0 22.3 22.4 23.4 24.3 24.9 26.0 27.6 • We said the median was 22.9. So, what’s Q1? 18.8 20.1 21.0 22.3 22.4  n is odd

13. Quartiles Order the observations.18.8 20.1 21.0 22.3 22.4 23.4 24.3 24.9 26.0 27.6 • What is Q3? • A) 21.0 • B) 24.6 • 24.9 • 25.45 • 26.0

14. Technically Speaking **not in text • The median and quartiles are examples of percentiles. The pth percentile is the value such that: • At least p% of the data are less than or equal to that value • At least (1-p)% of the data are greater than or equal to that value • So, for example Q1 is actually the value such that: • At least 25% of the data are less than or equal to that value • At least 75% of the data are greater than or equal to that value

15. Median and quartile illustration

16. What percent of the observations are between Q1 and Q3? A 25% B 50% C 75%

17. Five-Number Summary five-number summary – describes center and spread of a distribution • minimum • Q1 • median • Q3 • maximum

18. Example: Career home runs Find the quartiles for Mark McGwire’s Home Run data 49 32 33 39 22 42 9 9 39 52 58 70 65 Order the observations. 9 9 22 32 33 39 39 42 49 52 58 65 70 What are the five-number summaries for McGwire?

19. Example: Career home runs • Find the quartiles for Roger Maris’ Home Run data 14 28 16 39 61 33 23 26 8 13 Order the observations. 8 13 14 16 23 26 28 33 39 61 What are the five-number summaries for Maris?

20. Boxplot boxplot – a graph of the five-number summary • A central box spans the quartiles. • A line in the box marks the median. • Lines extend from the box out to the smallest and largest observations (usually….). • Boxplots can be drawn horizontally or vertically. • A numerical scale should be included.

21. Example: Career home runs

23. The median is around • 0.5 • 2.0 • C) 3.3 • D) 5.1

24. mean – ( ) the average of a set of observations Mean • To find the mean of n observations: 1. Add the values. 2. Divide the sum by n.

25. Mean USC’s Points Scored 70 70 75 77 78 79 79 82 82 83 86 93 Sum = 954 so the mean is 954 / 12 = 79.5 Opponent’s Points Scored 59 68 69 71 73 75 76 77 78 82 96 97 Sum = 921 so the mean is 921 / 12= 76.75

26. Consider the data set 3 7 9 6 1 10 The mean is A) 3.5 D) 7 B) 6 E) 7.5 C) 6.5

27. Mean versus Median USC’s Points Scored70 70 75 77 78 79 79 82 82 83 86 93 = 79.5 M = 79 What if we really turned it on one game… 70 70 75 77 78 79 79 82 82 83 86 129 = 82.5 M = 79

28. Mean, Median, and Skewness The mean is typically pulled in the direction of the skewness or outlier.

29. Mean pulled by skewness and outliers

30. For this data set A) Mean < Median B) Mean = Median C) Mean > Median