1 / 22

Applying the Normal Distribution: Z-Scores

Applying the Normal Distribution: Z-Scores. Chapter 3.5 – Tools for Analyzing Data Mathematics of Data Management (Nelson) MDM 4U. Comparing Data. Consider the following two students: Student 1 MDM 4U, Mr. Lieff, Semester 1, 2004-2005 Mark = 84%, Student 2

deva
Télécharger la présentation

Applying the Normal Distribution: Z-Scores

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. 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 Semester1, 2004-05 Semester 2, 2005-06 90 50 58 82 98 74 66 40.6 50.4 60.2 70 79.8 89.6 99.4

  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(0,12) 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… 99.7% 95% 34% 34% 13.5% 13.5% 2.35% 2.35% 6 8 10 12 14 16 7 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 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

  12. Example 4: Ranges • Now find the percent of data between a mark of 60 and 80 • For 60: • z = (60 – 68)/10.9 = -0.73 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

  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 • Interval Data

  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 Underweight Below 18.5 Normal 18.5 - 24.9 Overweight 25.0 - 29.9 Obese 30.0 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. 1B Miguel Cabrera, Detroit Tigers http://sports.yahoo.com/mlb/players/7163 • 2008 Statistics: 616 AB, 180 H, 36 2B, 2 3B, 37 HR SLG = (H + 2B + 2*3B + 3*HR) / AB = (180 + 36 + 2*2 + 3*37) / 616 = 331 / 616 = 0.537 (3 decimal places) • This means Miggy attained 0.537 bases per AB

  20. Example 3: 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 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  Technical chart • n-Day Moving Average

  21. 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 2008), 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 • Wikipedia (2004). Online Encyclopedia. Retrieved September 1, 2004 from http://en.wikipedia.org/wiki/Main_Page

More Related