Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 16 Mathematics of Normal Distributions 16.1Approximately Normal Distributions of Data 16.2Normal Curves and Normal Distributions 16.3Standardizing Normal Data 16.4The Rule 16.5Normal Curves as Models of Real- Life Data Sets 16.6Distribution of Random Events 16.7Statistical Inference

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The reason we like to idealize a real-life, approximately normal data set by means of a normal distribution is that we can use many of the properties we just learned about normal distributions to draw useful conclusions about our data. For example, the rule for normal curves can be reinterpreted in the context of an approximately normal data set as follows. Real-Life and The Rule

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 1. In an approximately normal data set, about 68% of the data values fall within (plus or minus) one standard deviation of the mean. 2. In an approximately normal data set, about 95% of the data values fall within (plus or minus) two standard deviations of the mean. THE RULE FOR APPROXIMATELY NORMAL DATA

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 3. In an approximately normal data set, about 99.7%, or practically 100%, of the data values fall within (plus or minus) three standard deviations of the mean. THE RULE FOR APPROXIMATELY NORMAL DATA

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. We are now going to use what we learned so far to analyze the 2007 SAT mathematics scores. As you may recall, there were N = 1,494,531 scores, distributed in a nice, approximately normal distribution. The two new pieces of information that we are going to use now are that the mean score was  = 515 points and the standard deviation was  = 114 points. Example SAT Math Scores: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Just knowing the mean and the standard deviation (and that the distribution of test scores is approximately normal) allows us to draw a lot of useful conclusions: Example SAT Math Scores: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Median. In an approximately normal distribution, the mean and the median should be about the same. Given that the mean score was  = 515, we can expect the median score to be close to 515. Moreover, the median has to be an actual test score when N is odd (which it is in this example), and SAT scores come in multiples of 10, so a reasonable guess for the median would be either 510 or 520 points. Example SAT Math Scores: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. First Quartile. Recall that the first quartile is located standard deviation below the mean. This means that in this example the first quartile should be close to 515 –  114 ≈ 438 points. But again, the first quartile has to be an actual test score (the only time this is not the case is when N is divisible by 4), so a reasonable guess is that the first quartile of the test scores is either 430 or 440 points. Example SAT Math Scores: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Third Quartile. We know that the third quartile is located standard deviation above the mean. In this case this gives  114 ≈ 592 points. The most reasonable guess is that the third quartile of the test scores is either 590 or 600 points. Example SAT Math Scores: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. In all three cases our guesses were very good–as reported by the College Board, the 2007 SAT math scores had median M = 510, first quartile Q 1 = 430, and third quartile Q 3 = 590. We are now going to go analyze the 2007 SAT scores in a little more depth–using the rule. Example SAT Math Scores: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The Middle 68. Approximately 68% of the scores should have fallen within plus or minus one standard deviation from the mean. In this case, this range of scores goes from 515 – 114 = 401 to = 629 points. Since SAT scores can only come in multiples of 10, we can estimate that a little over two-thirds of students had scores between 400 and 630 points. Example SAT Math Scores: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The Middle 68. The remaining third were equally divided between those scoring 630 points or more (about 16% of test takers) and those scoring 400 points or less (the other 16%). Example SAT Math Scores: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The Middle 95. Approximately 95% of the scores should have fallen within plus or minus two standard deviations from the mean, that is, between 515 – 228 = 287 and 515 – 228 = 743 points. In practice this means SAT scores between 290 and 740 points. The remaining 5% of the scores were 740 points or above (about 2.5%) and 290 points or below (the other 2.5%). Example SAT Math Scores: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Everyone. The 99.7 part of the rule is not much help in this example. Essentially, it says that all test scores fell between 515 – 342 = 173 points and = 857 points. Duh! SAT mathematics scores are always between 200 and 800 points, so what else is new? Example SAT Math Scores: Part 2