Population Distributions and Parameters
For a well-defined population of units and a variable, say X, the collection of all possible values of the variable X formed by measuring all of the units in the target population forms the population associated with the variable X. When multiple variables are recorded, each of the variables will generate its own population. Furthermore, since a variable may take on many different values, an important question concerning the population of values of the variable is “How can the population of values of a variable be described or summarized?” The two different approaches that can be used to describe the population of values of the variable are (1) to describe explicitly how the variable is distributed over its values and (2) to describe a set of characteristics that summarize the distribution of the values in the population.
2.2.1 Distributions
A statistical analysis of a population is centered on how the values of a variable are distributed, and the distribution of a variable or population is an explicit description of how the values of the variable are distributed often described in terms of percentages. The distribution of a variable is also called a probability distribution because it describes the probabilities that each of the possible values of the variable will occur. Moreover, the distribution of a variable is often presented in table or chart or modeled with a mathematical equation that explicitly determines the percentage of the population taking on each possible value of the variable. The total percentage in a probability distribution is 100%. The distribution of a qualitative or a discrete variable is generally displayed in a bar chart or in a table, and the distribution of a continuous variable is generally displayed in a graph or is represented by a mathematical function.
Example 2.9
The four basic classifications of blood type are O, A, B, and AB. The distribution of blood type, according to the American Red Cross, is given in Table 2.1, and a bar chart representing this distribution is shown in Figure 2.3. Based on the information in Table 2.1, 45% of Americans have type O blood, 40% have type A, 11% have type B, and 4% have type AB blood.
Figure 2.3 A bar chart of the distribution of blood types in the United States.
Table 2.1 The Distribution of Blood Type According to the American Red Cross
| Blood Type | Percentage |
|---|---|
| O | 45% |
| A | 40% |
| B | 11% |
| AB | 4% |
Another method of classifying blood types is to represent blood type by type and Rh factor. A bivariate distribution of blood type for the variables type and Rh factor is given in Table 2.2 and the bar chart in Figure 2.4.
Figure 2.4 A bar chart of the distribution of blood types and Rh factor in the United States.
Table 2.2 The Distribution of Blood Types with Rh Factor
| Rh Factor | ||
|---|---|---|
| Type | + | − |
| O | 38% | 7% |
| A | 34% | 6% |
| B | 9% | 2% |
| AB | 3% | 1% |
Example 2.10
One of the goals of the 1989 Wisconsin Behavioral Risk Factor Surveillance System (BRFS) was to estimate the distribution of adults who count calories. The distribution of male and female adults in Wisconsin who count calories is given in Table 2.3. Based on the information in Table 2.3, the percentage of females who do not count calories is 69.6% and the percentage of males who do not count calories is 84.8%. Note that there are actually two distributions given in Table 2.3.
Table 2.3 The Distribution of Adults who Count Calories Based on the 1989 Wisconsin BRFS by Age and Gender
| Sex | Calories Eaten Per Day | ||
|---|---|---|---|
| % 1200 or Less | % > 1200 | % Do Not Count | |
| Male | 4.6 | 10.6 | 84.8 |
| Female | 19.0 | 11.5 | 69.6 |
The distribution of a continuous quantitative variable is often modeled with a mathematical function called the probability density function. The probability density function explicitly describes the distribution of the values of the variable. A plot of the probability density function provides a graphical representation of the distribution of a variable, and the area under the curve defined by the probability density function corresponds to the percentage of the population falling between these two values. The height of the curve at a particular value of the variable measures the percentage per unit in the distribution at this point and is called the density of the population at this point. Regions where the values of the variable are more densely grouped are areas in the graph of a probability density function where it is tallest. Examples of the most common shapes of the distribution of a continuous variable are given in Figures 2.5–2.8.
Figure 2.5 An example of a mound-shaped distribution.
Figure
