Normal Distribution and Natural Phenomena, Research Paper Example

Introduction

When going into the math world, there are quite a few concepts and terms that to the average person would be confusing. One of the biggest math theories that puzzles even the most mathematically advanced student from time to time is normal distribution curve. The normal distribution curve is one of the most important theories which is studied in statistics.The imperativeness of the normal distribution stems from the way that the conveyances of numerous natural phenomena are in any event more or less typically circulated. One of the first applications of the typical dissemination was to the dissection of slips of estimation made in galactic perceptions, errors that occurred in view of defective instruments and flawed eyewitnesses.

Galileo in the seventeenth century noted that these lapses were symmetric, and that little errors happened more every now and again than expansive mistakes. This prompted a few speculated circulations of lapses; however, it was not until the early nineteenth century that it was found that these blunders emulated an average dispersion. Freely, the mathematicians Adrian in 1808 and Gauss in 1809 created the recipe for the typical dissemination and demonstrated that blunders were fit well by this circulation. The importance of normal distribution is far reaching than the math world, but to understand its importance along with other math phenomena, the paper will provide a well-researched background on the concepts, along with one of the biggest contributors to the math world, Carl Gauss.

Carl Gauss

Much of what has been learned about statistics and other significant math equations can be owed to the mathematician Carl Gauss. Gaussis known for making an incredible arrangement advancements forward in the various works in both physics and mathematics. He is in charge of tremendous commitments to the fields of analysis, differential geometry, magnetism, optics, number theory, astronomy, and additionally a lot more. This list includes, “probability theory, geodesy, planetary astronomy, the theory of functions, and potential theory (including electromagnetism)” (Encyclopedia Britannica).The ideas that he himself made have had a huge impact in numerous regions of the mathematics and exploratory world of science. Born as Johann Carl Friedrich Gauss in now Germany, he was raised in an impoverished family, and despite his humble beginnings he shone brilliance at an early age. Upon noticing how exceptional Gauss was, his eleventh grade teacher had the insight to provide Gauss with books, and education to continue his development. He studied with Martin Bartles, who later became the teacher to Lobachevsky. He entered the Gymnasium in 1788 and progressed rapidly in his subject, which much attention paid to mathematics and classics. (Encyclopedia) After receiving a scholarship from the Duke of Brunswick, Gauss in 1792, entered into Brunswick Collegium Carolinumm, he had a logical and established instruction a long ways past what was typical for his age at the time. He was acquainted with basic geometry, polynomial math, and analysis (frequently having found imperative hypotheses before arriving at them in his studies), however what’s more he had an abundance of arithmetical data and numerous number-theoretic bits of knowledge. Far reaching estimations and perception of the results, regularly recorded in tables, had headed him to a close acquaintance with individual numbers and to generalizations that he used to amplify his computing capability. As of now his deep rooted heuristic example had been set: far reaching experimental examination prompting guesses and new bits of knowledge that guided further trial and perception. By such means he had as of now freely found Bode’s law of planetary separations, the arithmetic-geometric mean, and the binomial theorem for rational exponents.

Gauss’ first noteworthy disclosure, in 1792, was that a standard polygon of 17 sides could be developed by ruler and compass alone. Its noteworthiness lies not in the result however in the confirmation, which rested on a significant dissection of the factorization of the polynomial, mathematical statements and opened the way to later thoughts of Galois hypothesis. His doctoral postulation of 1797 provided a verification of the principal theory of polynomial math: each polynomial comparison with genuine or complex coefficients has as numerous roots (results) as its degree (the most astounding force of the variable). Gauss’ distinguished as a positively surprising ability, however, came about because of two important distributions in 1801. First was his production of the first precise course reading on logarithmic number theory, DisquisitionesArithmeticae. This book starts with the first record of measured modular arithmetic, gives a careful record of the results of quadratic polynomials in two variables in whole numbers, and finishes with the theory of factorization specified previously. This decision of themes and its natural generalizations set the plan in number hypothesis for a significant part of the nineteenth century, and Gauss’ proceeding with enthusiasm toward the subject impelled much research, particularly in German colleges.

He eventually left Gottingen without getting his diploma, however he continued his career in mathematics, but advancing the field. He helped Zach an astronomer publish his book on orbital positions using his least squares approximation method, and also helped him to become involved in the field of astronomy. He published his second book on the motion of celestial bodies in 1809, Theoria motus corporumcoelestium in sectionibusconiscisSolemambientium.(Encyclopedia Britannica) He then went to develop the heliotrope which aided him in developing a geodesic, in which he predicted the exact areas sizes of the earth’s surface. He published over seventy papers during the decade of 1820 to 1830, and gave proof to the existence of non-Euclidean Geometry (although he was afraid to admit its existence). He later worked with Wilhelm Weber on his theory on terrestrial magnetism, and together discovered the Kirchhoff’s law, the MagnetischerVerien, and produced an atlas of geomagnetism. (Encyclopedia) His health slowly deteriorated from 1850, in which his actively greatly decreased. Although he died in 1855, his influence in both the mathematics and scientific worlds is immeasurable. His intellect and work has transformed the way in which we study the world, although a lot of his work was left unpublished including his work towards non-Euclidean geometry, complex variable, and mathematical foundations of physics, was not credited, his achievements are unforgettable. More importantly his discovery and works towards normal distribution, which is also referred to as Gaussian distribution.

Normal Distribution

The normal distribution is a general state of likelihood appropriation with a specific symmetrical level of bunching around the mean, which resembles a bell curve.Normal distribution was first used as part of astronomical perceptions, where they discovered errors of estimation. (Stahl 97)  In the seventeenth century, Galileo finished up the results, with connection to the estimation of separations from the star. He recommended that little blunders are more prone to happen than expansive mistakes, irregular lapses are symmetric to the last slips, and his perceptions typically accumulate around the genuine qualities. (Stahl 97) Galileo’s hypothesis of the lapses was found to be the attributes of the normal distribution and the equation for a normal distribution, which was found by Adrian and Gauss, overall connected with the errors.As noted by the Wolfram MathWorld, “A normal distribution in a variate X with mean mu and variance sigma^2 is a statistic distribution with probability density function:

The conveyance is likewise regularly known as the Gaussian distribution. The typical dispersion is an essential appropriation because a lot of people arbitrarily happening occasions tail its dissemination of results. What’s more, focal breaking point hypothesis expresses that the dissemination of the specimen method for a sufficiently vast number of examples will constantly surmised to the normal distribution. The characteristics of normal distribution bell curve includes, the mean, mode, and median which are qual. The total area under the normal curve should be equal to 1, while it is also symmetric to the mean. The normal curve will approach but never touch the x axis the further it travels from the mean. Case in point, if a coin is thrown 50 times and the quantity of heads recorded then, if this test is rehashed an extensive number of times, the ensuing conveyance of the quantity of heads recorded for every 50 tosses would be pretty nearly ordinarily dispersed with a mean of 25 heads for every 50 tosses. This equation would like: P(x=50) = (100/50) (.5) ^50 (.5) ^50.

The normal distribution itself is constantly symmetric around its mean and reaches out to infinity in every bearing. The aggregate likelihood under the bend is constantly equivalent to 1. Likewise, the way that the normal distribution has a particular shape implies that any normal distribution is completely depicted by its standard deviation and mean. The standard deviation and the mean of a given conveyance permit that appropriation to be changed over into a standard typical circulation, with mean 0 and standard deviation 1, where each information point is now portrayed and known. This likewise implies that all normal distribution have around 68% of their information inside 1 standard deviation each faction of the mean. With 95% of the information inside around 2 standard deviations each one side the mean and 99.7% of the information inside 3 standard deviations either side of the mean. The aspects of normal distribution is significant to the statistical data distribution pattern that often occurs in natural phenomena.

Standard Deviation

In providing a better definition of standard deviation to get a better understanding of normal distribution, standard deviation is defined “as the square root of the variance” (Wolfram MathWorld).  This can be seen in the equation.

The Population Standard Deviation

Standard deviation is a broadly utilized estimation of variability or assorted qualities utilized as part of facts and likelihood hypothesis. It indicates the amount variety or “scattering” there is from the “normal” (mean, or expected worth). A low standard deviation shows that the information indicates tend be near the mean, though elevated requirement deviation demonstrates that the information is spread out over a vast scope of qualities.

Examples of Normal Distribution and Natural Phenomena

In looking for the far-reaching capability of a normal distribution, it is used heavily in other fields outside of mathematics, this includes biomedical engineering. In looking at the probability models that help in the manufacturing and biological process, normal distribution is the most frequently used in the probability distribution to describe populations, but also in statistical analysis. This includes heart rate variability and in random processes such as given in the central limit theorem. In looking at the usage of Gaussian distribution or normal distribution, in regards to cholesterol Ecc (enzymatic cholesterol cycling) can be calculated to find the propensity to hypertension can cause reduced ventricular function. To statistically find the normal distribution in ECC is by performing the Kolmogorov-Smirnov test to find the total cholesterol, triglycerides, and plasma glucose levels. From there, one can study the correlations between the risk factors and Ecc in patients. (Rosen, Saad, Shea, Nasir, Edvardsen, Burke, Jerosch-Herold, Arnett, Lai, Bluemke, and Lima 1152) Biomedical phenomena are used in the probability of biomedical variables, which includes the number of patients experiencing secondary effects using pharmacological treatments, the probability distribution of the measurement errors, and also other independent factors related to biomedical variable resulting.

Physical Phenomena

Normal distribution is also used in calculating the sample size or height of humans in a given population, also known as physical phenomena. The normal distribution of the observed frequencies has a prominent role in statistics as it provide the outcomes of the results. In looking at a familiar example, measuring the heights and weight of the large sample of adult men and women. The µ stands for the mean, while the standard deviation is represented by ?, and e approximately is the 2.17, while n equals 3.1415. This was determined by using the information in which, average height of the males is 5’10 or 70 inches with a 4 inch standard deviation. The adult women were on average shorter at 5’5 or 65 inches, with 3.5 of standard deviation.  When plugged into the equation this would equal, .

Social Phenomena

This type of the normal distribution is frequently used in other examples which include social phenomena in which, is used dealing in social sciences. Many can be derived using a statistical test, but also using the normal distribution of the standard deviation of IQ scores in adolescents. Or it can be used to calculate scores from tests, and others that can be viewed for observational errors and the propagation of uncertainty. Social scientists usually use the normal distribution because it can calculate the most basic continuous probability. Specifically using the central limit theorem in which gives the sum of a number of random variable with variances and finite means that approach the normal distribution as the amount of variables increases. This can be seen in the example in which 184 students with a mean of 72.3 and 8.9 standard deviation, will try to calculate an expectant score between 80 and 90. The mean plus 2 standard deviation 90.1 and with 1 standard deviation 81.2. In trying to calculate 13.6 percent or 11.5 percent with graphic calculator, of the students will have the score given (approximately 25 students). Using a graphic calculator this is what the bell curve:

Income and Normal Distribution

Along with other uses for the normal distribution, income is also calculated using the equation. In calculating the income distribution, normal distribution is sometimes used in more varied distribution, which will largely depict income skewed. This is something of great difficulty just because normal distribution is usually symmetrical to the mean. The problem with calculating an accurate result for income distribution using this method is because realistically there is no perfect distributions. In using a normal distribution it provides a statistical misrepresentation in which is the exploitation of “normal” distribution. In the calculation the income, using the example of salary, the mean, mode, and median will be all differently, or greatly skewed from the normal distribution. This will on average be an abnormal distribution in which the number of values will be at different distance on both sides of the mean, which won’t be equal. Income usually follows the power law distribution, while normal distribution does not, i.e. (if percentage of the population makes x dollars per year, then 1/n of the percentage of the population will make n^x).

Pareto Principle

While normal distribution is used variedly in several sciences, it fails to provide an accurate statistical representation of income distribution. One concept which provides a better result and used more frequently is, Pareto’s Principle. Pareto Principle also known as the 80-20 rule, in which provides an understanding for most events only 80 percent of the effects comes from 20 percent of the causes (Rodd 78). The Pareto Principle is used mostly for the law of income distribution because it relies on the principle in which a small amount of people are accountable for a large amount of income, or “80 percent of sales comes form 20 percent of the customers.” Seen in this equation: F(x)=1?1xa,1?x<?, it was found in 1906 by Vilfredo Pareto, he developed a mathematical formula in which to explain the unequal distribution of wealth in Italy, when observing how just 20 percent of the people (mainly wealthy) owned 80 percent of the land. Many continued to adopt his principle when they realized similar phenomena in their fields. Used substantially in business, software, occupational health and safety, economics, and mathematical equations, the Pareto principle is essential in income distribution. The reason that many prefer Pareto’s Principle over normal distribution when configuring income distribution is due to its ability to show that variables or quantities will grow exponentially, and are more often skewed to the right. Pareto is largely favored over normal distribution because it focuses on the complex, recursive pattern in which most wealth is disproportionally shared in society. Not everything is the same or equal such as in normal distribution, but rather differences in frequency and in the mean, mode, and median.

Conclusion

In exploring the complexities and the concepts in math, there are several theorems and principles that are largely confusing to the average person. However, unknowingly many use or reference to these concepts each day as they read the Business section, or look at the results of studies conducted by social scientists. The discoveries of such great innovators such as Karl Gauss have helped to usher in a new way of thinking that was new to the 19th century. Normal distribution is a significant portion of the scientific and mathematics world, not only does it involve statistical analysis, but also measuring observational patterns, and provides examinations and calculation of the distributed pattern in random variables. Normal distribution is used in observation in astronomy and other scientific fields where errors were plentiful, however working along with a standard deviation it provides a more accurate picture of the average or normal value of the deviation and measurement. While normal distribution is used widely for many means including social, physical, and biological phenomena, where it usually fails is in income distribution. Normal distribution deals largely with numbers that close to the mean, however, when dealing with income the numbers tend to grow exponentially in which will provide a skewed curve. This is where Pareto Principle is applied and does a better job. Although it might not provide an accurate statistical representation for income distribution, it is greatly significant in other fields, in which will continue to be used throughout the scientific world.

Works Cited

K O, May. “Biography of Karl Gauss.” Dictionary of Scientific Biography (New York 1970-1990).  N.d. Web. 29 Aug 2014.
http://www.encyclopedia.com/doc/1G2-2830901590.html

Gauss, Karl. “Biography of Karl Gauss.” Encyclopedia Britannica.N.d. Web. 28 Aug, 2014.
http://www.britannica.com/eb/article-9109423/Carl-Friedrich-Gauss

Rodd, J. “Pareto’s law of income distribution, or the 80/20 rule.”Int. J. Nonprofit Volunt. Sect. Mark. 1: 77–89. Print. 1996.

Rosen, Boaz D., Mohammed F. Saad., Steven, Shea, Khuarram, Nasir, Edvardsen, Thor, Gregory, Burke, Michael Jerosch-Herold, Donna K. Arnett, Shenghan La, David A. Bluemke, Joao, A.C. Lima. “Hypertension and Smoking Are Associated With Reduced Regional Left Ventricular Function in Asymptomatic Individuals. The Multi-Ethnic Study of Atherosclerosis.” Journal of the American College of Cardiology. Vol. 47, No.6. 2006. Print.

Stahl, Saul. “The Evolution of the Normal Distribution.” Mathematics Magazine. Vol. 79, No.2. 2006. Web. 28 Aug 2014. https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/stahl96.pdf

Weisstein, Eric W. “Normal Distribution.” MathWorld–A Wolfram Web Resource.N.d. Web. 29 Aug 2014. http://mathworld.wolfram.com/NormalDistribution.html

Weisstein, Eric W. “Standard Deviation.” MathWorld–A Wolfram Web Resource.N.d. Web. 28 Aug 2014. http://mathworld.wolfram.com/StandardDeviation.html