Warm up

1. Warm up • In a class where • State the interval containing the following % of marks: • a) 68% • b) 95% • c) 99.7% • Answers: • a) 66 – 82 • b) 58 – 90 • c) 50 – 98

2. Applying the Normal Distribution: Z-Scores Chapter 3.5 – Tools for Analyzing Data Mathematics of Data Management (Nelson) MDM 4U

3. AGENDA • Comparing Data • Standard Normal Distribution • Ex. 1: z-scores • Ex. 2: Percentage of data below/above • Ex. 3: Percentiles • Ex. 4: Ranges • MSIP / Homework

4. Comparing Data • Consider the following two students: • Student 1 • MDM 4U, Mr. Norbraten, Semester 1 • Mark = 84%, • Student 2 MDM 4U, Mr. Lieff, Semester 2 • Mark = 83%, • Can we compare the two students fairly when the mark distributions are different?

5. Mark Distributions for Each Class Semester1 Semester 2 90 50 58 82 98 74 66 40.6 50.4 60.2 70 79.8 89.6 99.4

6. Comparing Distributions • It is difficult to compare two distributions when they have different characteristics • For example, the two histograms have different means and standard deviations • z-scores allow us to make the comparison

7. Standard Normal Distribution 99.7% 95% 68% 34% 34% 13.5% 13.5% 2.35% 2.35% -3 -2 -1 0 1 2 3

8. The Standard Normal Distribution • A distribution with mean zero and standard deviation of one X~N(0,1²) • z-score translates from any Normal distribution to the Standard Normal Distribution • z-score is the number of standard deviations below or above the mean • Positive z-score  data lies above the mean • Negative below

9. Example 1 • For the distribution X~N(10,2²) determine the number of standard deviations each value lies above or below the mean: • a. x = 7 z = 7 – 10 2 z = -1.5 • 7 is 1.5 standard deviations below the mean • 18.5 is 4.25 standard deviations above the mean (anything beyond 3 is an outlier) • b.x = 18.5 z = 18.5 – 10 • 2 • z = 4.25

10. Example continued… 99.7% 95% 34% 34% 13.5% 13.5% 2.35% 2.35% 6 8 10 12 14 16 7 18.5

11. Standard Deviation • A recent math quiz offered the following data • z-scores offer a way to compare scores among members of the class, find out what % had a mark greater than yours, indicate position (percentile) in the class, etc. • mean = 68.0 • standard deviation = 10.9

12. Example 2: • If your mark was 64, what % of the class scored lower? • Compare your mark to the rest of the class • z = (64 – 68.0)/10.9 = -0.37 (using the z-score table on page 398) • We get 0.3557 or 35.6% • So 35.6% of the class has a mark less than or equal to yours • What % scored higher? • 100 – 35.6 = 64.4%

13. Example 3: Percentiles • The kth percentile is the data value that is greater than k% of the population • If another student has a mark of 75, what percentile is this student in? • z = (75 - 68)/10.9 = 0.64 • From the table on page 398 we get 0.7389 or 73.9%, so the student is in the 74th percentile – their mark is greater than 74% of the others

14. Example 4: Ranges • Now find the percent of data between a mark of 60 and 80 • For 60: • z = (60 – 68)/10.9 = -0.73 gives 23.3% • For 80: • z = (80 – 68)/10.9 = 1.10 gives 86.4% • 86.4% - 23.3% = 63.1% • So 63.1% of the class is between a mark of 60 and 80

15. Back to the two students... • Student 1 • Student 2 • Student 2 has the lower mark, but a higher z-score, so he/she did better compared to the rest of her class.

16. MSIP / Homework • Read through the examples on pages 180-185 • Complete p. 186 #2-5, 7, 8, 10

17. Mathematical Indices Chapter 3.6 – Tools for Analyzing Data Mathematics of Data Management (Nelson) MDM 4U

18. What is an Index? • An arbitrarily defined number that provides a measure of scale • Used to indicate a value so that we can make comparisons, but does not always represent an actual measurement or quantity • Interval Data (no meaningful starting point)

19. 1) BMI – Body Mass Index • A mathematical formula created to determine whether a person’s mass puts them at risk for health problems • BMI = where m = mass in kg, h = height in m • Standard / Metric BMI Calculator http://nhlbisupport.com/bmi/bmicalc.htm Underweight Below 18.5 Normal 18.5 - 24.9 Overweight 25.0 - 29.9 Obese 30.0 and Above NOTE: BMI is not accurate for athletes and the elderly

20. 2) Slugging Percentage • Baseball is the most statistically analyzed sport in the world • A number of indices are used to measure the value of a player • Batting Average (AVG) measures a player’s ability to get on base (hits / at bats)  probability • Slugging percentage (SLG) also takes into account the number of bases that a player earns (total bases / at bats) SLG = where TB = 1B + 2B×2 + 3B×3 + HR×4, OR TB = H + 2B + 3B×2 + HR×3 where 1B = singles, 2B = doubles, 3B = triples, HR = homeruns

21. Slugging Percentage Example • e.g. 1B Miguel Cabrera, Detroit Tigers http://sports.yahoo.com/mlb/players/7163 • 2008 Statistics: 616 AB, 180 H, 36 2B, 2 3B, 37 HR • NOTE: H (hits) includes 1B as well as 2B, 3B and HR • So • 1B = H – (2B + 3B + HR) • = 180 – (36 + 2 + 37) • = 105

22. Slugging Percentage Example cont’d SLG = (1B + 2×2B + 3×3B+ 4×HR) / AB = (105 + 2×36 + 3×2 + 4×37) / 616 = 331 / 616 = 0.537 (3 decimal places) • This means Miggy attained 0.537 bases per AB

23. Example 3: Moving Average • Used when time-series data show a great deal of fluctuation (e.g. stocks, currency exchange) • Average of the previous n values • e.g. 5-Day Moving Average • cannot calculate until the 5th day • value for Day 5 is the average of Days 1-5 • value for Day 6 is the average of Days 2-6 • e.g. Look up a stock symbol at http://ca.finance.yahoo.com • Click Charts  Technical chart • n-Day Moving Average • Useful for showing long term trends

24. Other Examples 1) Consumer Price Index (CPI) • An indicator of changes in Canadian consumer prices • Compares the cost of a fixed basket of commodities through time • Commodities are of unchanging or equivalent quantity and quality reflecting only pure price change. http://www.statcan.gc.ca/cgi-bin/imdb/p2SV.pl?Function=getSurvey&SDDS=2301&lang=en&db=imdb&adm=8&dis=2

25. What is included in the CPI? • 8 major categories • FOOD AND BEVERAGES (breakfast cereal, milk, coffee, chicken, wine, full service meals, snacks) • HOUSING (rent of primary residence, owners' equivalent rent, fuel oil, bedroom furniture) • APPAREL (men's shirts and sweaters, women's dresses, jewelry) • TRANSPORTATION (new vehicles, airline fares, gasoline, motor vehicle insurance) • MEDICAL CARE (prescription drugs and medical supplies, physicians' services, eyeglasses and eye care, hospital services) • RECREATION (televisions, toys, pets and pet products, sports equipment, admissions); • EDUCATION AND COMMUNICATION (college tuition, postage, telephone services, computer software and accessories); • OTHER GOODS AND SERVICES (tobacco and smoking products, haircuts and other personal services, funeral expenses).

26. Other Examples cont’d 2) NHL Fan Cost Index (FCI) • Comprises the prices of: • four (4) average-price tickets • two (2) small draft beers • four (4) small soft drinks • four (4) regular-size hot dogs • parking for one (1) car • two (2) game programs • two (2) least-expensive, adult-size adjustable caps.

27. Other Examples cont’d 2) NHL Fan Cost Index (FCI) Details • Average ticket price represents a weighted average of season ticket prices. • Costs were determined by telephone calls with representatives of the teams, venues and concessionaires. Identical questions were asked in all interviews. • All prices are converted to USD at the exchange rate of $1CAD=$.932418 USD.

28. MSIP / Homework • Read pp. 189-192 • Complete pp. 193-195 #1a (odd), 2-3 ac, 4 (alt: calculate SLG for 3 players on your favourite team for 2010), 8, 9, 11

29. References • Halls, S. (2004). Body Mass Index Calculator. Retrieved October 12, 2004 from http://www.halls.md/body-mass-index/av.htm • Wikipedia (2004). Online Encyclopedia. Retrieved September 1, 2004 from http://en.wikipedia.org/wiki/Main_Page