Distributions Help

By — McGraw-Hill Professional
Updated on Aug 26, 2011


A distribution is a description of the set of possible values that a random variable can take. This can be done by noting the absolute or relative frequency. A distribution can be illustrated in terms of a table, or in terms of a graph.

Discrete Versus Continuous

Table 2-1 shows the results of a single, hypothetical experiment in which a die is tossed 6000 times. Figure 2-3 is a vertical bar graph showing the same data as Table 2-1. Both the table and the graph are distributions that describe the behavior of the die. If the experiment is repeated, the results will differ. If a huge number of experiments is carried out, assuming the die is not ''weighted,'' the relative frequency of each face (number) turning up will approach 1 in 6, or approximately 16.67%.



Table 2-2 shows the number of days during the course of a 365-day year in which measurable precipitation occurs within the city limits of five different hypothetical towns. Figure 2-4 is a horizontal bar graph showing the same data as Table 2-2. Again, both the table and the graph are distributions. If the same experiment were carried out for several years in a row, the results would differ from year to year. Over a period of many years, the relative frequencies would converge towards certain values, although long-term climate change might have effects not predictable or knowable in our lifetimes.


Both of the preceding examples involve discrete variables. When a distribution is shown for a continuous variable, a graph must be used. Figure 2-5 is a distribution that denotes the relative amount of energy available from sunlight, per day during the course of a calendar year, at a hypothetical city in the northern hemisphere.


Frequency Distribution

In both of the above examples (the first showing the results of 6000 die tosses and the second showing the days with precipitation in five hypothetical towns), the scenarios are portrayed with frequency as the dependent variable. This is true of the tables as well as the graphs. Whenever frequency is portrayed as the dependent variable in a distribution, that distribution is called a frequency distribution.

Suppose we complicate the situation involving dice. Instead of one person tossing one die 6000 times, we have five people tossing five different dice, and each person tosses the same die 6000 times. The dice are colored red, orange, yellow, green, and blue, and are manufactured by five different companies, called Corp. A, Corp. B, Corp. C, Corp. D, and Corp. E, respectively. Four of the die are ''weighted'' and one is not. There are thus 30,000 die tosses to tabulate or graph in total. When we conduct this experiment, we can tabulate the data in at least two ways.

Ungrouped frequency distribution

The simplest way to tabulate the die toss results as a frequency distribution is to combine all the tosses and show the total frequency for each die face 1 through 6. A hypothetical example of this result, called an ungrouped frequency distribution, is shown in Table 2-3. We don't care about the weighting characteristics of each individual die, but only about potential biasing of the entire set. It appears that, for this particular set of die, there is some bias in favor of faces 4 and 6, some bias against faces 1 and 3, and little or no bias either for or against faces 2 and 5.


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