For more information
Call 1-800-351-0222

NORMAL CURVE.
  Term Paper ID:21099 		Get This Paper Free! or
Essay Subject:
Defines & examines this distribution curve & its role in statistical analysis. Tables, graph.... More...
6 Pages / 1350 Words
6 sources, 10 Citations, TURABIAN Format
$24.00

More Papers on This Topic


Paper Abstract:
Defines & examines this distribution curve & its role in statistical analysis. Tables, graph.

Paper Introduction:
Statisticians work with large masses of data. Before any conclusions can be drawn from such data, it must be condensed and arranged in a usable form. One of the most common ways to summarize and describe a mass of data is to arrange a frequency distribution table. These tables can then be graphed with the frequency scale on the y-axis and the interval being graphed on the x-axis. Above each interval a horizontal line is drawn which corresponds to the frequency of the interval, resulting in a stair-step histogram pattern. Connecting the midpoints of these class intervals produces a frequency polygon and an interval curve. Distribution curves which can be "folded" vertically so that the two halves of the curve are essentially the same are said to be bilaterally symmetrical. Perfectly symmetrical curves which have a bell shape are said to be normal curves, or Gaussian curve

Text of the Paper:
The entire text of the paper is shown below. However, the text is somewhat scrambled. We want to give you as much information as we possibly can about our papers and essays, but we cannot give them away for free. In the text below you will find that while disordered, many of the phrases are essentially intact. From this text you will be able to get a solid sense of the writing style, the concepts addressed, and the sources used in the research paper.


186 | 695 | |. Englewood Cliffs: Prentice-Hall, 1979.Downing, Douglas. The HarperCollins Dictionary of Statistics. Basic Business Statistics: Concepts and Applications. Perfectlysymmetrical curves which have a bell shape are said to be normal curves, orGaussian curves (after the German mathematician Frederic Gauss). 196 < . 184 < . New York: HarperCollins, 1991.Steel, Robert G. In thenatural world, human height and weight, weights of animals, velocity ofmolecules in gases and myriad other phenomena can be explained with normaldistribution curves. [7]Porkess, 2 5. All normal distributions have the same overall shape: they are bell-shaped with a single peak and symmetric. 19 < . Normalcurves occur frequently both in nature and in human events, and, as such,form the basis for much statistical analysis. 182 < . New York: McGraw-Hill Book Company, 1976. By calculating the area under the curve from the meanto one standard deviation for the standardized curve, the statisticianeffectively calculates the standard deviation for the random variable.Tables have been developed which allow for the easy calculation of thisinformation. 198 < . The normal distribution is also important because it occurs sofrequently in the natural world as well as in human endeavors. 188 |1198 | |. This research examines thenature of the normal curve, when it occurs, how it is developed, and itscharacteristics and significant limitations and abnormalities. Distribution curveswhich can be "folded" vertically so that the two halves of the curve areessentially the same are said to be bilaterally symmetrical. 2 < . The mean is calculated by summing all theobservations and dividing by the number of observations. Using a normaldistribution and calculating the standard deviation, the probability thatthe manufacturer's claims are accurate (or not) can be established.[3] Combining the normal distribution with principles of probability isone of the most common uses of the normal distribution. [3]Ibid., 147. 192 |1896 | |. Its measures of central tendency (the mean, median andmode) are the same, and it is bell-shaped and symmetrical. Thebulk of the frequencies occur in the middle of the range, with thefrequencies decreasing as the measures approach either extreme.[1] Thenormal distribution is also found in nonhuman events, such as the length ofcockroaches in a house, the weights of carrots in a field or the velocitiesof molecules in a gas.[2] Normal distributions have many applications, and are used in a widevariety of fields. [5]Ibid. The heightof the density curve at the point x is described by the followingequation:[6] [pic] At the heart of the normal distribution is the calculation for themean and the standard deviation. [2]Roger Porkess, The HarperCollins Dictionary of Statistics, NewYork: HarperCollins, 1991, 144. 2 |325 | |. 191 and . Statisticians must be careful to test for thevalidity of the normal distribution before making probability statementsrelated to it since there are bell-shaped symmetrical curves which are notnormal. To illustrate, itis helpful to use the following data, which can be grouped: 12, 9, 9, 9, 8,6, 6, 5, 4 and 2. The exact density curve for agiven distribution is described by its mean () and its standard deviation(). As a result, any normal distributionhas the following properties: 68 percent of the observations fall withinone standard deviation of the mean; 95 percent of the observations fallwithin two standard deviations of the mean; 99.7 percent of theobservations fall within three standard deviations of the mean.[5] Thisrule illustrates that there can be symmetric bell-shaped density curveswhich are not normal. In business, a manufacturer may use normaldistributions to establish an acceptable failure rate for his product. 192 < . Freeman and Company, 1989, 56. [6]Ibid. New York: Barron's Educational Series, Inc., 1989.Friday, Frank A. To illustrate the construction of a normal distribution curve,calculate its mean and standard deviation and derive additional probabilitydata, the following frequency data (thicknesses of brass washers) can beused:[8] Thickness |Frequency (n = 1 , ) | |< . These curves are indicated by a bell that either tootall or too short to have the characteristics outlined above. A curve can then be drawn through the midpoints of thehistogram in order to construct a normal distribution curve based on thisdata. Above each interval a horizontal line is drawnwhich corresponds to the frequency of the interval, resulting in a stair-step histogram pattern. Introduction to Statistics. and James H. [9]Ibid., 172. 18 | 48 | |. A customermay find that 5 bulbs had a mean life of 79 hours and claims that themanufacturer is misstating the reliability of the bulbs. 18 < . One of the most common ways to summarize and describe a mass of datais to arrange a frequency distribution table. [pic] Calculating the mean and the standard deviation for this distribution,we arrive at . BibliographyBerenson, Mark and David Levine. H. The standard deviation controls the spread of a normal curve.Changing the mean without changing the standard deviation moves the normalcurve to a new location without altering its spread. In human events, normal distributions can be found in coin tosses,heights of sample populations, and scores on human intelligence tests. Freeman and Company, 1989.Porkess, Roger. 194 |1664 | |. When statisticians use this information, theyspeak in degrees of confidence. Within the realm of human endeavor, measures such asIQ scores, tosses of a coin and how individuals vote can be described innormal distribution curves. This practice hasgained particular popularity with political pollsters who use the normaldistribution and principles of sampling to determine how voters will voteand who will win an election based on early returns.[4] The reason that statisticians are able to apply probability theory tonormal distributions is because of the peculiar nature of the normal curve. The mean for this data isthe sum of the observations divided by the number of observations, or 7 /1 , which equals 7. New York: W. There are ten observations. Torrie. Normal curves can bestandardized so that a standardized random variable Z can be calculated asequal to a normal random variable X less the mean divided by the standarddeviation:[9] [pic]The mean of the standardized variable Z is always zero and the standarddeviation of the standardized variable is always one.[1 ] With thisinformation, it is possible to graph the random and standardized variableson the same graph. 188 < . Connecting the midpoints of these class intervalsproduces a frequency polygon and an interval curve. H. Forexample, a light bulb manufacturer may claim that his light bulbs have amean life of 8 hours with a standard deviation of 4 hours. 182 | 122 | |. 196 |1198 | |. The normal distribution is important to statisticians because of itsunique properties. [8]Mark Berenson and David Levine, Basic Business Statistics:Concepts and Applications, Englewood Cliffs: Prentice-Hall, 1979, 146. New York: Barnes & Noble, 1967.Moore, David and George McCabe. Statistics the Easy Way. [1 ]Ibid., 173. These tables can then begraphed with the frequency scale on the y-axis and the interval beinggraphed on the x-axis. 198 |695 | |. All normaldistributions are the same if measurements are made in units of thestandard deviation about the mean. Statisticians work with large masses of data. 2 2 |122 | |> . 2 2 |48 | |We can construct a histogram based on this data using the frequencydivided by n to represent the probability of X, and the midpoint of thethicknesses. 186 < . It shouldbe noted that this data is used to provide an illustration of how thestandard deviation is calculated, not an illustration of a normaldistribution curve. ----------------------- [1]Douglas Downing, Statistics the Easy Way, New York: Barron'sEducational Series, Inc., 1989, 115. Constructing a frequency tablebased on this data results in the following: x |f |fx |[pic] |[pic] |[pic] | |12 | 1 |12 | 5 |25 |25 | | 9 | 3 |27 | 2 | 4 |12 | | 8 | 1 | 8 | 1 | 1 | 1 | | 6 | 2 |12 |-1 | 1 | 2 | | 5 | 1 | 5 |-2 | 4 | 4 | | 4 | 1 | 4 |-3 | 9 | 9 | | 2 | 1 | 2 |-5 |25 |25 | | |1 |7 | | |78 | |Finishing the remainder of the calculation, 78 / 1 = 7.8, and thestandard deviation is equal to the square root of 7.8, or 2.79. Before any conclusions canbe drawn from such data, it must be condensed and arranged in a usableform. 194 < . 184 | 325 | |. The mean is located at the center of the curve and is the same as themedian. Introduction to the Practice of Statistics. Itsrelationship with the standard deviation is equally important: themajority of observations occur within one standard deviation of the mean,and nearly all observations are accounted for within three standarddeviations. D . The Elements of Probability and Sampling. Added to the 68-97-99.7 rule (68 percent of observations willbe within one standard deviation of the mean, 95 percent within twostandard deviations and 99.7 percent within three standard deviations), andthe statistician can calculate the probability that a given X will fallwithin a certain range. [4]David Moore and George McCabe, Introduction to the Practice ofStatistics, New York: W. 19 |1664 | |. 179, respectively. The standard deviation is defined by the equation:[7] [pic][pic]where i is the frequency observation, f the number of frequencies, x theinterval, and n the number of observations.

If this paper is not what you are looking for, you can search again:

Search for:

or

We can write a Custom Essay just for you.


Browse Essays by Subject