1 / 29

Warm up

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

harris
Download Presentation

Warm up

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 I can: use z-scores to calculate the % of data in any interval of a Normal Distribution MSIP / Home Learning: p. 186 #2-5, 7, 8, 10

  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 / Home Learning

  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 Learning goal: Calculate mathematical indices using formulas and compare data

  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 AVG = (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) 1B = singles, 2B = doubles, 3B = triples, HR = homeruns

  21. Slugging Percentage Example • e.g. 3B/OF Jose Bautista, Toronto Blue Jays • 2010 Statistics: 569 AB, 148 H, 35 2B, 3 3B, 54 HR • NOTE: H (Hits) includes 1B as well as 2B, 3B and HR • So • 1B = H – (2B + 3B + HR) • = 148 – (35 + 3 + 54) • = 56

  22. Slugging Percentage Example cont’d SLG = (1B + 2×2B + 3×3B+ 4×HR) / AB = (56 + 2×35 + 3×3 + 4×54) / 569 = 351 / 569 = 0.617 • This means Jose attained 0.617 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  Interactive • TECHNICAL INDICATORS  SMA • 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

More Related