2. Numerical Summaries of Quantitative Data. Means, Standard Deviations, z-scores

3. Warmup 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?

4. 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. Measure of the “middle”

6. Recall: Warmup 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

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

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

9. Mean, Median, Maximum Baseball Salaries 1985 - 2014

10. DESCRIBING VARIABILITY OF QUANTITATIVE DATA

11. 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

12. 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)

13. 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 or software.

14. Population Standard Deviation

15. 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

16. 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? When all data values are the same.

17. 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 ??

18. 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

19. 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” Since the average (mean) is 9, x 1 + x 2 + x 3 must equal 9*3 = 27, so x 3 = 27 – (x 1 + x 2 ) Choose ANY values for x 1 and x 2

20. class pulse rates

21. 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

22. Summary of Notation

23. Using the Mean and Standard Deviation Together. Z-scores: Standardized Data Values Measures the distance of a number from the mean in units of the standard deviation

24. z-score corresponding to y

25. n Exam 1: y 1 = 88, s 1 = 6; exam 1 score: 91 Exam 2: y 2 = 88, s 2 = 10; exam 2 score: 92 Which score is better?

26. Comparing SAT and ACT Scores n SAT Math: Eleanor’s score 680 SAT mean =500 sd=100 n ACT Math: Gerald’s score 27 ACT mean=18 sd=6 n Eleanor’s z-score: z=(680-500)/100=1.8 n Gerald’s z-score: z=(27-18)/6=1.5 n Eleanor’s score is better.

27. Z-scores add to zero Student/Institutional Support to Athletic Depts For the 9 Public ACC Schools: 2013 ($ millions) SchoolSupporty - ybarZ-score Maryland15.56.41.79 UVA13.14.01.12 Louisville10.91.80.50 UNC9.20.10.03 VaTech7.9-1.2-0.34 FSU7.9-1.2-0.34 GaTech7.1-2.0-0.56 NCSU6.5-2.6-0.73 Clemson3.8-5.3-1.47 Mean=9.1000, s=3.5697 Sum = 0

28. In a recent year the mean tuition at 4-yr public colleges/universities in the U.S. was $6185 with a standard deviation of $1804. In NC the tuition was $4320. What is NC’s z-score? 1. 1.03 2. -1.03 3. 2.39 4. 1865 5. -1865

29. End of Numerical Summaries