2. 1 2.4 Describing Distributions Numerically – cont. Describing Symmetric Data

3. 2 Symmetric Data Body temp. of 93 adults

4. 3 Recall: 2 characteristics of a data set to measure n center measures where the “middle” of the data is located n variability measures how “spread out” the data is

5. 4 Measure of Center When Data Approx. Symmetric n mean (arithmetic mean) n notation

6. 5

7. 6 Connection Between Mean and Histogram n A histogram balances when supported at the mean. Mean x = 140.6

8. 7 Mean: balance point Median: 50% area each half right histo: mean 55.26 yrs, median 57.7yrs

9. 8 Properties of Mean, Median 1.The mean and median are unique; that is, a data set has only 1 mean and 1 median (the mean and median are not necessarily equal). 2.The mean uses the value of every number in the data set; the median does not.

10. 9 Think about mean and median n Six people in a room have a median age of 45 years and mean age of 45 years. n One person who is 40 years old leaves the room. n Questions: 1.What is the median age of the 5 people remaining in the room? 2.What is the mean age of the 5 people remaining in the room? Can’t answer 46 45  6=270; 270-40=230; 230/5=46

11. 10 Example: class pulse rates n 53 64 67 67 70 76 77 77 78 83 84 85 85 89 90 90 90 90 91 96 98 103 140

12. 11 2010, 2014 baseball salaries n 2010 n = 845  = $3,297,828 median = $1,330,000 max = $33,000,000 n 2014 n = 848  = $3,932,912 median = $1,456,250 max = $28,000,000

13. 12 Disadvantage of the mean n Can be greatly influenced by just a few observations that are much greater or much smaller than the rest of the data

14. 13 Mean, Median, Maximum Baseball Salaries 1985 - 2014

15. 14 Skewness: comparing the mean, and median n Skewed to the right (positively skewed) n mean>median

16. 15 Skewed to the left; negatively skewed n Mean < median n mean=78; median=87;

17. 16 Symmetric data n mean, median approx. equal

19. 18 Describing Symmetric Data (cont.) n Measure of center for symmetric data: n Measure of variability for symmetric data?

20. 19 Example n 2 data sets: x 1 =49, x 2 =51 x=50 y 1 =0, y 2 =100 y=50

21. 20 On average, they’re both comfortable 0 100 49 51

22. 21 Ways to measure variability 1. range=largest-smallest ok sometimes; in general, too crude; sensitive to one large or small obs.

23. 22 Previous Example

24. 23 The Sample Standard Deviation, a measure of spread around the mean n Square the deviation of each observation from the mean; find the square root of the “average” of these squared deviations

25. 24 Calculations … Mean = 63.4 Sum of squared deviations from mean = 85.2 (n − 1) = 13; (n − 1) is called degrees freedom (df) s 2 = variance = 85.2/13 = 6.55 inches squared s = standard deviation = √6.55 = 2.56 inches Women height (inches)

26. 25 1. First calculate the variance s 2. 2. Then take the square root to get the standard deviation s. Mean ± 1 s.d. We’ll never calculate these by hand, so make sure to know how to get the standard deviation using your calculator, Excel, or other software.

27. 26 Population Standard Deviation

28. 27 Remarks 1. The standard deviation of a set of measurements is an estimate of the likely size of the chance error in a single measurement

29. 28 Remarks (cont.) 2. Note that s and  are always greater than or equal to zero. 3. The larger the value of s (or  ), the greater the spread of the data. When does s=0? When does  =0?

30. 29 Remarks (cont.) 4. The standard deviation is the most commonly used measure of risk in finance and business –Stocks, Mutual Funds, etc. 5. Variance  s 2 sample variance   2 population variance  Units are squared units of the original data  square $, square gallons ??

31. 30 Remarks 6):Why divide by n-1 instead of n? n degrees of freedom n each observation has 1 degree of freedom however, when estimate unknown population parameter like , you lose 1 degree of freedom

32. 31 Remarks 6) (cont.):Why divide by n-1 instead of n? Example n Suppose we have 3 numbers whose average is 9 nx1=x2=nx1=x2= n then x 3 must be n once we selected x 1 and x 2, x 3 was determined since the average was 9 n 3 numbers but only 2 “degrees of freedom”

33. 32 Computational Example

34. 33 class pulse rates

35. 34 Example #1#2#3#4 32333837 41353942 44453945 47504046 50525647 53545748 56585850 59596167 68646268 n x50505050 n s10.610.610.610.6 n m50525647

36. 35 Boxplots: same mean, standard deviation

37. 36 More Boxplots of the 4 data sets

38. 37 Review: Properties of s and  s and  are always greater than or equal to 0 when does s = 0?  = 0? The larger the value of s (or  ), the greater the spread of the data n the standard deviation of a set of measurements is an estimate of the likely size of the chance error in a single measurement

39. 38 Summary of Notation