1. 3-1 Stats Unit 3 Summary Statistics (Descriptive Statistics) FPP Chapter 4 For one variable - - Center of distribution "central value", "typical value" - Spread of distribution How variable are the values in a set of data? - Measure how many / what proportion of observations are above / below a given value. W.01

2. 3-2 Stats Summary Statistics Purposes: compact reporting easy comparison Important considerations: interpretable stable We will discuss: how the statistics are defined when each is (in)appropriate how to interpret them how to compute them "guesstimation" techniques

3. 3-3 Stats Example: Hospital Charges Total charge (in dollars) of the hospital stay for 29 normal deliveries of babies Charges 1,905 2,324 2,048 2,888 2,907 2,840 2,607 2,823 2,310 2,953 2,138 3,418 4,903 3,729 3,709 5,063 3,932 3,392 3,287 3,819 4,248 2,640 2,921 2,785 2,804 2,955 2,219 2,184 2,681 14,898

4. 3-4 Stats Definitions 1500 2500 3500 4500 5500 Hospital Charges (in Dollars) 10 8 6 4 2 freq. 12 mode = most frequently occurring value = _______________ median = "middle value" = __________________ = mean = sum / # measurements in the data set = = __________/___________ = _________ = another way to compute the mean:

5. 3-5 Stats Locating These Summary Statistics on a Histogram mode: median: mean: comparing mean & median: For skewed histograms, the mean could be deceiving. 1500 2500 3500 4500 5500 Hospital Charges (in Dollars) 10 8 6 4 2 freq. 12

6. 3-6 Stats

7. 3-7 Stats Event Day Abnormal Returns (ref. "Marketing Science", Fall 1987, vol 6, no 4, pages 320- 335, "Does It Pay to Change Your Company's Name?") -1.84-0.31 0.02 0.30 0.53 1.09 -1.38 -0.24 0.06 0.34 0.55 1.12 -1.00 -0.24 0.09 0.36 0.58 1.23 -0.59 -0.20 0.10 0.39 0.78 1.43 -0.57 -0.16 0.13 0.40 0.81 1.50 -0.56 -0.06 0.21 0.41 0.96 1.60 -0.51 -0.05 0.23 0.43 0.98 1.64 -0.44 -0.02 0.24 0.45 0.99 1.79 -0.39 -0.02 0.25 0.48 1.00 -0.33 -0.01 0.29 0.50 1.03

8. 3-8 Stats mode = most frequently occurring value =______ median = "middle value" = __________ mean = "average" = (sum of values in list)/(# values in list) = _____ / _____ = _____ p th percentile = the value with p percent of the list less than (or equal to it) and 100-p percent greater than it 10 th percentile = _____ 25 th percentile = _____ 80 th percentile = _____

9. 3-9 Stats Histogram for Abnormal Returns 0.4 5 10 15 20 RETURNS 0.3 0.2 0.1 -2.0 -0.5 1.0 2.5 4.0

10. 3-10 Stats Does This Statistic Make Sense? Some summary statistics make sense only for certain types of data. mean: median: mode:

11. 3-11 Stats Water Watch

12. 3-12 Stats Aug 1-22 the average consumption was 223.7 million gallons per day. Aug 1-25 the average consumption was 224.4 million gallons per day. Q1: Was the average consumption higher Aug 1-22 or Aug 23-25? Q2: What was the total amount of water consumed Aug 23-25? Q3: What was the average daily consumption Aug 23-25?

13. 3-13 Stats Baseball Batting Averages Suppose batting average = (# hits / # at bats) x 1000 Before the game starts, a player has batting average = 250. - first at bat, strikes out - new batting average = 200 Q1: How many times has this batter been up? Another player starts the game with batting average 500. After his first at bat, his new batting average is 524. Q2: Did he get a hit? Q3: How many times has this batter been up?

14. 3-14 Stats

15. 3-15 Stats Measures of Location & Spread of a Data Set LOCATION mean median mode SPREAD standard deviation (SD) range variance

16. 3-16 Stats Range RANGE: (largest measurement) - (smallest measurement) example:

17. 3-17 Stats Deviation from Average definition: deviation from average = data value - average note: A deviation can be zero. 1 2 5 7 10 data value

18. 3-18 Stats Standard Deviation of a list of numbers definition: standard deviation = SD = rms size of the deviations from average =

19. 3-19 Stats rms (root mean square) size of a list of numbers root-mean-square (rms) operation 1 2 5 7 10 data value deviation

20. 3-20 Stats Standard Deviation Try another list of numbers. Find the standard deviation (rms size of the deviations from average) for this list of numbers. 2, - 6, 12, 4, 6 I.Find the average of this list of numbers. II.Find the deviation of each value from this average. III.Find the rms size of the list of deviations. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 data

21. 3-21 Stats Standard Deviation The STANDARD DEVIATION (SD) OF A DATA SET measures how far away numbers are from their average. Most entries on the list will be somewhere around one SD away from the average. Very few will be more than two or three SDs away.

22. 3-22 Stats Interpreting the Standard Deviation * Roughly 68% of the entries on a list (roughly 2/3 of the entries) are within one SD of the average. * The other 32% (approximately 1/3) are further away. ** Roughly 95% (19 out of 20) are within two SDs of the average. ** The other 5% are further away. The 2/3 rule is true for most data sets. The 95% rule is true for many data sets, but not all.

23. 3-23 Stats Delivery Times Example Class Limits Tallies Frequency 25-34|1 35-44|||3 45-54||||4 55-64|1 65-74|||| |||8 75-84|||| ||||10 85-94|||| |||| 9 95-104|||3 105-114||||4 115-124||||5 125-134||2 TIME IN DAYS 27687991107 43718091108 43718193108 44718394116 47738494120 49738494120 50748497122 54758697123 587688103127 657788106128

24. 3-24 Stats Delivery Times Continued 25 45 65 85 105 125 days Elapsed Time to Delivery.20.16.12.08.04 rel. freq. Days Elapsed Between Order Date and Delivery Date for 50 Orders average (mean) = median = SD =

25. 3-25 Stats Delivery Times - 3 “The 2/3 Rule” says that Roughly 2/3 or 68% of the entries on a list are within one SD of the average. 108.0 days Actually, 49 out of 50 deliveries took between 35.1 and 132.3 days. 49/50 = 0.98 = 98% Actually, in this data set, 34 out of 50 deliveries took between 59.4 and 108.0 days. 34/50 = 0.68 = 68% “The 95% Rule” says that Roughly 95% of the entries on a list are within two SD’s of the average. 108.0 days

26. 3-26 Stats

27. 3-27 Stats Guesstimating the SD Middle 2/3 Rule 1. Locate the middle 2/3 of the data. 2. The range of the middle 2/3 of the data is approximately 2 SD's. So, 1/2 of this range is approximately 1 SD.

28. 3-28 Stats Variance The variance of a list of numbers is the SD squared. That is, the SD is the square root of the variance.

29. 3-29 Stats z-score The z-score says how many SD's above (+) or below (-) the average a value is. The sample z-score for a measurement is z = The population z-score for a measurement is z = example:

30. 3-30 Stats Interpreting z-scores Interpretation of z-Scores for "Mound-Shaped" Distributions of Data 1. Approximately 68% of the measurements will have a z-score between -1 and +1. 2. Approximately 95% of the measurements will have a z-score between -2 and +2. 3. All or almost all of the measurements will have a z-score between -3 and +3.

31. 3-31 Stats Wonderlic Scores

32. 3-32 Stats USC had average team score 20.3. What is their z- score? Is this value extreme among NCAA Division I teams? How about Michigan State whose average team score is 16.6? Find their z-score and interpret it. How about Stanford whose average team score is 28.2? Find their z-score and interpret it..