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. NHL Playoff Team Popularity by Province

3. 3.5 Applying the Normal Distribution: Z-Scores Chapter 3 – Tools for Analyzing Data Learning goal: use z-scores to calculate the % of data between 2 values in a Normal Distribution Due now: p. 176 #1, 3b, 6, 8-10 MSIP / Home Learning: p. 186 #2-5, 7, 8, 10

4. AGENDA Comparing Data in different Normal Distributions The Standard Normal Distribution Ex. 1: z-scores Ex. 2: Percentage of data below/above Ex. 3: Percentiles Ex. 4: Ranges

5. Comparing Data Consider the following two students: Student 1  MDM 4U Semester 1  Mark = 84%, Student 2  MDM4U Semester 2  2  Mark = 83%, How can we compare the two students when the class mark distributions are different?

6. Mark Distributions for Each Class Semester 1 Semester 2 74 66 585082 90 99.489.679.87060.250.440.6 98

7. 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

8. The Standard Normal Distribution A Normal distribution with mean 0 and std.dev. 1  X~N(0, 1²) A z-score translates from a Normal distribution to the Standard Normal Distribution The z-score is the number of standard deviations a data point lies below or above the mean  Positive z-score  data lies above the mean  Negative z-score  data lies below the mean

9. Standard Normal Distribution 34% 13.5% 2.35% 95% 99.7% 012-23-3 68%

10. 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

11. Example continued… 34% 13.5% 2.35% 95% 99.7% 10121486 7 16 18.5

12. 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 in the class, etc. mean = 68.0 standard deviation = 10.9

13. Example 2 If your mark was 64, what % of the class scored lower?  Calculate your z-score  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%

14. Example 3: Percentiles The k th 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  0.7389 From the table on page 398 we get 0.7389 or 73.9%, so the student is in the 74 th percentile – their mark is greater than 74% of the others

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

16. 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.

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

18. 3.6 Mathematical Indices Chapter 3 – Tools for Analyzing Data Learning goal: Calculate mathematical indices and draw conclusions Questions? p. 186 #2-5, 7, 8, 10 MSIP/Home Learning: pp. 193-195 #1a (odd), 2-3 ac, 4 (look up recent stats if desired), 8, 9, 11 Hitting for the cycle https://www.youtube.com/watch?v=ilWab_vyB4g

19. What is a Mathematical Index? An arbitrarily defined number Most are based on a formula Used to make cross-sectional and/or longitudinal comparisons Does not always represent an actual measurement or quantity

20. 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://www.nhlbi.nih.gov/guidelines/obesity/BMI/bmicalc.htm  UnderweightBelow 18.5  Normal18.5 - 24.9  Overweight25.0 - 29.9  Obese30.0 and Above NOTE: BMI is not accurate for athletes and the elderly

21. 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 AVG = (hits) ÷ (at bats)  probability Slugging percentage (SLG) 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) 1B = singles, 2B = doubles, 3B = triples, HR = homeruns

22. Slugging Percentage Example e.g. 1B Adam Lind, Toronto Blue Jays 2013 Statistics: 465 AB, 134 H, 26 2B, 1 3B, 23 HR NOTE: H (Hits) includes 1B, 2B, 3B and HR So  1B = H – (2B + 3B + HR)  = 134 – (26 + 1 + 23)  = 84 SLG = (1B + 2×2B + 3×3B+ 4×HR) ÷ AB  = (84 + 2×26 + 3×1 + 4×23) ÷ 465  = 231 ÷ 465  = 0.497 This means Adam attained 0.497 bases per AB

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

24. Other examples: Big Mac Index A Big Mac costs:  $5.26 USD in Canada  $3.09 USD in Latvia Which currency has MORE purchasing power? The Big Mac Index uses the cost of a Big Mac to compare the purchasing power of different currencies

25. Christmas Price Index Totals the cost of the items in “Twelve Days of Christmas” Measures inflation from year-to-year Created by PNC Bank http://www.pncchristmaspriceindex.com/

26. Other Examples: Fan Cost Index Which NHL cities do you think are the most expensive to take a family of 4 to a hockey game? 5. Chicago 4. New York 3. Boston 2. Vancouver 1. Toronto Compares the prices of: 4 average-price tickets 2 small draft beers 4 small soft drinks 4 regular-size hot dogs 1 parking pass 2 game programs 2 least-expensive, adult-size adjustable caps http://www.fancostexperience.com/pages/fcx/blog_pdfs/entry0000020_pdf001.pdf

27. Fan Cost Index cont’d 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. Consumer Price Index (CPI) Managed by Statistics Canada An indicator of changes in Canadian consumer prices Compares the cost of a fixed basket of commodities (600 items) over time Expressed as a % of the base year (2002). http://www.statcan.gc.ca/tables-tableaux/sum-som/l01/cst01/cpis01g-eng.htm

29. 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).

30. MSIP / Home Learning 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 2013), 8, 9, 11 Ch3 Review: p. 199 #1a, 3a, 4-6 You will be provided with:  Formulas in Back Of Book  z-score table on p. 398-9

31. References Halls, S. (2004). Body Mass Index Calculator. Retrieved October 12, 2004 from http://www.halls.md/body-mass-index/av.htm 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