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Describing Data & Frequency Distribution |
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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 curves (after the German mathematician Frederic Gauss). Normal curves occur frequently both in nature and in human events, and, as such, form the basis for much statistical analysis. This research examines the nature of the normal curve, when it occurs, how it is developed, and its characteristics and significant limitations and abnormalities. In human events, normal distributions can be found in coin tosses, heights of sample populations, and scores on human intelligence tests. The bulk of the frequencies occur in the middle of the range, with the frequencies decreasing as the measures approach either e
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of the observations fall within one standard deviation of the mean; 95 percent of the observations fall within two standard deviations of the mean; 99.7 percent of the observations fall within three standard deviations of the mean. This rule illustrates that there can be symmetric bell-shaped density curves which are not normal. These curves are indicated by a bell that either too tall or too short to have the characteristics outlined above. The height of the density curve at the point x is described by the following equation:
At the heart of the normal distribution is the calculation for the mean and the standard deviation. The mean is calculated by summing all the observations and dividing by the number of observations. To illustrate, it is helpful to use the following data, which can be grouped: 12, 9, 9, 9, 8, 6, 6, 5, 4 and 2. There are ten observations. The mean for this data is the sum of the observations divided by the number of observations, or 70 / 10, which equals 7.
The standard deviation is defined by the equation:
where i is the frequency observation, f the number of frequencies, x the interval, and n the number of observations. Constructing a frequency table based on this data results in the fo
Category: Misc - D
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Gauss Normal, , standard deviation, normal distribution, 1, 1 , , 1 , normal distributions, standard deviation mean, 2, standard deviations, 4, deviation mean, mean standard deviation, Barnes Noble, Cliffs Prentice-Hall, Book Company, York HarperCollins, Series Inc, 2 , percent observations fall, Freeman Company, Statistics York,
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