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

2. Comparing Data Consider the following two students: Student 1  MDM 4U, Mr. Lieff, Semester 1, 2004-2005  Mark = 84%, Student 2 MDM 4U, Mr. Lieff, Semester 2, 2005-2006  Mark = 83%, Can we compare the two students fairly when the mark distributions are different?

3. Mark Distributions for Each Class Semester 1, 2004-05 Semester 2, 2005-06 74 66 585082 90 99.489.679.87060.250.440.6 98

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

5. The Standard Normal Distribution A distribution with a mean of zero and a standard deviation of one X~N(0,1²) Each element of any normal distribution can be translated to the same place on a Standard Normal Distribution using the z-score of the element the z-score is the number of standard deviations the piece of data is below or above the mean If the z-score is positive, the data lies above the mean, if negative, below

6. Standardizing The process of reducing the normal distribution to a standard normal distribution N(0,1 2 ) is called standardizing Remember that a standardized normal distribution has a mean of 0 and a standard deviation of 1

7. 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

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

9. Standard Deviation A recent math quiz offered the following data The z-scores offer a way to compare scores among members of the class, find out how many had a mark greater than yours, indicate position in the class, etc. mean = 68.0 standard deviation = 10.9

10. Example 2: Suppose your mark was 64 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

11. 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 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

12. 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

13. Back to the two students... Student 1 Student 2 Student 2 has the lower mark, but a higher z- score!

14. Exercises read through the examples on pages 180-185 try page 186 #2-5, 7, 8, 10

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

16. What is an Index? An index is an arbitrarily defined number that provides a measure of scale These are used to indicate a value, but do not actually represent some actual measurement or quantity so that we can make comparisons

17. 1) BMI – Body Mass Index A mathematical formula created to determine whether a person’s mass puts them at risk for health problems BMI =m = mass(kg), h = height(m) Standard / Metric BMI Calculator http://nhlbisupport.com/bmi/bmicalc.htm http://nhlbisupport.com/bmi/bmicalc.htm UnderweightBelow 18.5 Normal18.5 - 24.9 Overweight25.0 - 29.9 Obese30.0 and Above

18. 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) 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 and 1B = singles, 2B = doubles, 3B = triples, HR = homeruns

19. Slugging Percentage Example e.g. DH Frank Thomas, Toronto Blue Jays http://sports.espn.go.com/mlb/players/stats?playerId=2370 2006 Statistics: 466 AB, 126 H, 11 2B, 0 3B, 39 HR SLG = (H + 2B + 2*3B + 3*HR) / AB = (126 + 11 + 2*0 + 3*39) / 466 = 254 / 466 = 0.545 (3 decimal places)

20. Moving Average Used when time-series data show a great deal of fluctuation (e.g. long term trend of a stock) takes the 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 http://ca.finance.yahoo.com http://ca.finance.yahoo.com Click Charts  Technical chart n-Day Moving Average

21. Exercises read pp. 189-192 1a (odd), 2-3 ac, 4 (alt: calculate SLG for 3 players on your favourite team for 2007), 8, 9, 11

22. 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