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The concept of a normal curve is important in understanding the use of statistics. Most direct measures used in the study, for instance, of varying traits in human beings and most psychological measures, such as IQ scores, have been found to approximate closely a mathematical model called the normal distribution. The graph of this normal distribution is a continuous, symmetrical, bell-shaped curve. Frequencies tend to concentrate around the median and become fewer and fewer at either end, resulting in a frequency curve which is high in the middle and low at the ends. The normal curve is a limited curve which is approached by many distributions when a large number of measurements is made or, as is often said, when there is a large number of cases. It is necessary to assume that these measurements or cases are taken at random, or that there is no bias or systematic error. Of course, frequency distributions are not always of the normal curve type and the term skewness is used in describing abnormal distributions. In a normal curve, the right and left halves of the curve are mirror images of each other. If this is not the case, the curve is said to be skewed, either positively (to the right) or negatively (to the left. Thus, if the scores tend to be concentrated toward the high end of the score scale, the curve is negatively skewed. If they are concentrated toward the low end of the score scale, they are positively skewed. A Practical Illustration of the Normal Probability Curve One way to illustrate the normal curve is by tossing coins or dice. If we represent "heads" by H and "tails" by T, the expression H+T represents the probabilities for any toss of one coin, namely equal probabilities of a head or a tail. If we toss the coin 100 times, the results will approximate 50 heads and 50 tails. If we toss two coins, the possibilities are two heads, head and tail, tail and head, two tails, or H2+2HT+T2. If we toss two coins 100 times we would theoretically get 25H2+50HT+25T2, or both coins heads 25 times, one head and one tail 50 times, and both coins tails 25 times. Similarly, we can predict the theoretical frequency with which each possible combination of any number of coins tossed simultaneously any given number of times will occur. In tossing coins, if there is an equal chance for each coin to fall head or tail each time, every possible combination can occur, but the probabilities of getting ten heads or ten tails when we toss ten coins are less than those for getting other combinations. The most probable combination is five heads and five tails, since each coin has an equal chance of falling heads or tails. By expanding (H+T)10, we get the probabilities of each possible combination occurring if ten coins were tossed an infinite number of times. The expression becomes H10 + 10H9T + 45H8T2 + 120H6T4 + 252H5T5 + 210H4T6 + 120H3T7 + 45H2T8 + 10HT9 + T10. This means that the chances of getting five heads and five tails in tossing ten coins are 252 in 1,024. The probabilities of getting ten heads or ten tails are, respectively, one in 1,024. Where ten coins have thus been tossed, it has been found that the frequencies with which possible combinations do occur approach the theoretical values as limits. The frequency distribution below shos the results of tossing ten pennies 1,000 times, based on an actual experiment. Although the number of tosses is 1,000 instead of 1,024, it is evident that the frequency with which each possible combination actually occurred closely approximated the theoretical values. The resulting curve is very close to the theoretical normal curve of distribution, the perfect "bell" curve. Another situation in which the typical bell-shaped curve occurs is in measurement of natural phenomena. Thousands of measurements have been made of barometric and temperature readings over a long period of time. The results approximate the "bell" curve. The frequency distribution below, a study of height using over 8,000 adult males, also comes very close to a normal "bell" curve. To Page 5 Enrich Your Life With a Philosophy Book... Enrich Your Life With a Philosophy Magazine... Academy Showcase Specials Main Page & Index
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