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8: Statistics, Graphing, and Java
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Whenever the sample is less than the population, there is the possibility for error, and often the error factor can be known. For this chapter, we will be assuming that the sample equals the population, and we will not be concerned with issues surrounding sample errors. Statistical information is affected by the way events are distributed in the population. Several distributions are possible, but the most common, and the only one we will be using, is the normal distribution, or bell curve, with which you are undoubtedly familiar. In a bell curve, elements are symmetrically distributed around the middle (or peak) of the curve. In any statistical process, there is a dependent variable, which is the number under study, and an independent variable, which is the quantity that determines the dependent variable. This chapter uses time for the independent variable that is, the stepwise incremental passage of events as measured in whole number units. The use of time for the independent variable is quite common. For example, in watching a stock portfolio, you might monitor its value on a daily basis.
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At the core of most statistical analysis are three quantities: the mean, the median, and the mode. Each is useful on its own, but combined they paint a fairly clear picture of the characteristics of a sample. The statistical methods in this chapter assume that the elements that comprise the sample are stored in an array of double. All of the statistical methods are static methods stored in a class called Stats, which is shown in its entirety later in the chapter. Because they are static, they can be called without having to create a Stats object.
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The Mean
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The mean is the most commonly used statistic because it is the arithmetic average of a set of values. Thus, the mean is the center of gravity of the data. To compute the mean, divide the sum of the elements in the sample by the number of the elements. For example, the sum of these values 1 2 3 4 5 6 7 8 9 10 is 55. Dividing that value by the number of elements in the sample, which is 10, yields the mean, which is 5.5 in this example. Thus, the formula for finding the mean is
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Here, Di represents an element of data, and N is the number of elements in the sample. The following method called mean( ) computes the mean of the elements contained in the array referred to by its parameter. The mean is returned.
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// Return the average of a set of values. public static double mean(double[] vals) {
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double avg = 0.0; for(int i=0; i < vals.length; i++) avg += vals[i]; avg /= vals.length; return avg; }
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To use mean( ), simply call it with a reference to an array containing a set of values, and the mean is returned.
The Median
The median of a sample is the middle value based on order of magnitude. For example, in the sample set 123456789 the median is 5. For samples with an even number of elements, the median is the average of the two middle values. For example: 1 2 3 4 5 6 7 8 9 10 Here, the median is 5.5. For a sample that has a normal distribution, the median and the mean will be similar. However, as the distribution of elements within a sample moves further away from a normal distribution curve, the difference between the median and the mean increases. An easy way to obtain the median of a sample is first to sort the data and then take the middle value. This is the way the median( ) method, shown here, works:
// Return the median of a set of values. public static double median(double[] vals) { double temp[] = new double[vals.length]; System.arraycopy(vals, 0, temp, 0, vals.length); Arrays.sort(temp); // sort the data // Return the middle value. if((vals.length)%2==0) { // If even number of values, find average return (temp[temp.length/2] + temp[(temp.length/2)-1]) /2; } else return