The mean height for American adult men is5feet10inches(5'10").The standard deviation is roughly3inches.Is a5'7"man considered abnormally short?Discuss your reasoning.

tn Naked Statistics by Wheelan. Copyright Elli 3 5y i farlrs heelan. Permission must be obtained before further reproduction. ' Descriptive Smtmm' ' - 23 third grader in an Illinois elementary school scored 43 out of 60 on the mathematics portion of the Illinois State Achievement Test, that absolute score doesn't have much meaning. But when I convert it to a percen- tilemeaning that I put that raw score into a distribution with the math scores for all other Illinois third gradersthen it acquires a great deal of meaning. If 43 correct answers falls into the 83rd percentile, then this .student is doing better than most of his peers statewide. If he's in the 8th percentile, then he's really struggling. In this case, the percentile (the relative score) is more meaningful than the number of correct answers (the absolute score). Another statistic that can help us describe what might otherwise be a jumble of numbers is the standard deviation, which is a measure of how dispersed the data are from their mean. In other words, how spread out are the observations? Suppose I collected data on the weights of 250 people on an airplane headed for Boston, andI also collected the weights of a sample of 250 qualiers for the Boston Marathon. Now assume that the mean weight for both groups is roughly the same, say 155 pounds. Anyone who has been squeezed into a row on a crowded ight, ght ing for the armrest, knows that many people on a typical commercial ight weigh more than 155 pounds. But you may recall from those same unpleasant, overcrowded ights that there were lots of crying babies and poorly behaved children, all of whom have enormous lung capacity but not much mass. When it comes to calculating the average weight on the ight, the heft of the 320pound football players on either side of your middle seat is likely offset by the tiny screaming infant across the row and the six-yearold kicking the back of your seat from the row behind. On the basis of the descriptive tools introduced so far, the weights of the airline passengers and the marathoners are nearly identical. But they're not. Yes, the weights of the two groups have roughly the same \"middle,\" but the airline passengers have far more dispersion around that midpoint, meaning that their weights are spread farther from themidpoint. My eight- year-old son might point out that the marathon runners look like they all weigh the same amount, while the airline passengers have some tiny people and some bizarrely large people. The weights of the airline passengers are \"more spread out," which is an important attribute when it comes to 24 . naked statistics describing the weights of these two groups. The standard deviation is the descriptive statistic that allows us to assign a single number to this disper- sion around the mean. The formulas for calculating the standard deviation and the variance (another common measure of dispersion from which the standard deviation is derived) are included in an appendix at the end of the chapter. For now, let's think about why the measuring of dispersion matters. Suppose you walk into the doctor's ofce. You've been feeling fatigued ever since your promotion to head of North American printer quality. Your doctor draws blood, and a few days later her assistant leaves a message on your answering machine to inform you that your HCbZ count (a ctitious blood chemical) isl34. You rush to the Internet and discover that the mean HCbZ count for a person your age is 122 (and the ' median is about the same). Holy crap! If you're like me, you would nally draft a will. You'd write tearful letters to your parents, spouse, children, and close friends. You might take up skydiving or try to write a novel very fast. You would send your boss a hastily composed email comparing him to a certain part of the human anatomy[N ALL CAPS. None of these things may be necessary (and the e-mail to your boss could turn out very badly). When you call the doctor's ofce back to arrange for your hospice care, the physician's assistant informs you that your count is within the normal range. But how could that be? \"My count is 12 points higher than average!\" you yell repeatedly into the receiver. \"The standard deviation for the HCbZ count is 18,\" the technician informs you curtly. What the heck does that mean? There is natural variation in the HCbZ count, as there is with most: biological phenomena (e.g., height). While the mean count for the fake chemical might be 122, plenty of healthy people have counts that are higher or lower. The danger arises onlywhen the HCbZ count gets exces sively high or low. So how do we gure out what \"excessively\" means in this context? AS, we've already noted, the standard deviation is a measure of dispersion, meaning that it reects how tightly the observations cluster around the mean. For many typical distributions of data, a high propor- Desmptive Statistic: - 25 tion of the observations lie within one standard deviation of the mean (meaning that they are in the range from one standard deviation below the mean to one standard deviation above the mean). To illustrate with a simple example, the mean height for American adult men is 5 feet 10 inches. The standard deviation is roughly 3 inches. A high proportion of adult men are between 5 feet 7 inches and 6 feet 1 inch. Or, to put it slightly differently, any man in this height range would not be considered abnormally short or tall. Which brings us back to your troubling HCbZ results. Yes, your count is 12 above the mean, but that's less than one standard deviation, which is the blood chemical equivalent of being about 6 feet tallnot particularly unusual. Of course, far fewer observations lie two standard deviations from the mean, and fewer still lie three or four standard deviations away. (In the case of height, an American man who is three standard deviations above average in height would be 6 feet 7 inches or taller.) Some distributions are more dispersed than others. Hence, the stan~ dard deviation of the weights of the 250 airline passengers will be higher than the standard deviation of the weights of the 250 marathon runners. A frequency distribution with the weights of the airline passengers would literally be fatter (more spread out) than a frequency distribution of the weights of the marathon runners. Once we know the mean and stande deviation for any collection of data, we have some serious intellectual traction. For example, suppose I tell you that the mean score on the SAT math'test is 500 with a standard deviation of 100. As with height, the bulk of students taking the test will be Within one standard deviation of the mean, or between 400 and 600. How many students do you think score 720 or higher? Probably not very many, since that is more than two standard deviations above the mean. In fact, we can do even better than \"not very many.\" This is a good time to introduce one of the most important, helpful, and common dis tributions in statistics: the normal distribution. Data that are distributed normally are symmetrical around their mean in a bell shape that will look familiar to you. The normal distribution describes many common phenomena. 26 - naked stall-atlas Imagine a frequency distribution describing popcorn popping on a stove top. Some kernels Start to pop early, maybe one or two pops per second; after ten or fteen seconds, the kernels are exploding 'eneiically. Then gradually the number of kernels popping per Second fades away at roughly the same rate at which the popping began. The heights of American men are distributed more or less normally, meaning that they are roughly sym metrical around the mean of 5 feet 10 inches. Each SAT test is specically designed to produce a normal distribution of scores with mean 500 and standard deviation of 100. According to the Wall Smetjoumal, Americans even tend to park in a normal distribution at shopping malls; most cars park directly opposite the mall entrancethe \"peak\" of the normal curvewith \"tails\" of cars going off to the right and left of the entrance. The beauty of the normal distributionAte Michael Jordan power, nesse, and elegancecomes horn the fact that we know by denition exactly what proportion of the observations in a normal distribution lie within one standard deviation of the mean (68.2 percent), within two standard deviations of the mean (95.4 percent), Within three standard deviations (99.7 percent), and so on. This may sound like trivia. In fact, it is the foundation on which much of statistics is built. We Will come back to this point in much great depth later in the book. The Normal Distribution pZo 1110 F u+10 p+20