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

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

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

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?

Mark Distributions for Each Class Semester 1 Semester

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

Standard Normal Distribution 34% 13.5% 2.35% 95% 99.7% %

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

Standardizing The process of mapping any normal distribution to the standard normal distribution N(0,1 2 )

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 = 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… 34% 13.5% 2.35% 95% 99.7%

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

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 = (using the z-score table on page 398) We get 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 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 = ( )/10.9 = 0.64 From the table on page 398 we get or 73.9%, so the student is in the 74 th 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 = -0.73gives 23.3% For 80:  z = (80 – 68)/10.9 = 1.10gives 86.4% 86.4% % = 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 Complete p. 186 #2-5, 7, 8, 10

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

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)

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 UnderweightBelow 18.5 Normal Overweight Obese30.0 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 (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

Slugging Percentage Example e.g. 1B Miguel Cabrera, Detroit Tigers 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 – ( )  = 105

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

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 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 e.g. Look up a stock symbol at Click Charts  Technical chart n-Day Moving Average Useful for showing long term trends

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. bin/imdb/p2SV.pl?Function=getSurvey&SDDS=2301&lang =en&db=imdb&adm=8&dis=2

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).

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.

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 $1CAD=$ USD.

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

References Halls, S. (2004). Body Mass Index Calculator. Retrieved October 12, 2004 from Wikipedia (2004). Online Encyclopedia. Retrieved September 1, 2004 from