**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

**NHL Playoff Team Popularity by Province**

**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

**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

**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?

**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

**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

**The Standard Normal Distribution** • A Normal distribution with mean 0and 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

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

**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

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

**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

**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%

**Example 3: Percentiles** • The kthpercentile 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 74th percentile – their mark is greater than 74% of the others

**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

**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.

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

**Hitting for the cycle** https://www.youtube.com/watch?v=ilWab_vyB4g 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

**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

**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 • 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

**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

**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

**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 5th 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.com • Click CHARTS Interactive • TECHNICAL INDICATORS Simple Moving Average (SMA) • Useful for showing long term trends

**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

**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/

**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

**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.

**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

**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).

**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

**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