Applied Biostatistics for the Health Sciences. Richard J. Rossi

deviation empirical rule; roughly 99% of a mound-shaped distribution lies between the values μ−3σ and μ+3σ.

       THE EMPIRICAL RULES

      For populations having mound-shaped distributions,

      1 Roughly 68% of all of the population values fall within 1 standard deviation of the mean. That is, roughly 68% of the population values fall between the values μ−σ and μ+σ.

      2 Roughly 95% of all the population values fall within 2 standard deviations of the mean. That is, roughly 95% of the population values fall between the values μ−2σ and μ+2σ.

      3 Roughly 99% of all the population values fall within 3 standard deviations of the mean. That is, roughly 99% of the population values fall between the values μ−3σ and μ+3σ.

2.2.6 The Coefficient of Variation

      The standard deviations of two populations resulting from measuring the same variable can be compared to determine which of the two populations is more variable. That is, when one standard deviation is substantially larger than the other (i.e., more than two times as large), then clearly the population with the larger standard deviation is much more variable than the other. It is also important to be able to determine whether a single population is highly variable or not. A parameter that measures the relative variability in a population is the coefficient of variation. The coefficient of variation will be denoted by CV and is defined to be

      The coefficient of variation is also sometimes represented as a percentage in which case

The coefficient of variation compares the size of the standard deviation with the size of the mean. When the coefficient of variation is small, this means that the variability in the population is relatively small compared to the size of the mean of the population. On the other hand, when the coefficient of variation is large, this indicates that the population varies greatly relative to the size of the mean. The standard for what is a large coefficient of variation differs from one discipline to another, and in some disciplines a coefficient of variation of less than 15% is considered reasonable, and in other disciplines larger or smaller cutoffs are used.

      Because the standard deviation and the mean have the same units of measurement, the coefficient of variation is a unitless parameter. That is, the coefficient is unaffected by changes in the units of measurement. For example, if a variable X is measured in inches and the coefficient of variation is CV = 2, then coefficient of variation will also be 2 when the units of measurement are converted to centimeters. The coefficient of variation can also be used to compare the relative variability in two different and unrelated populations; the standard deviation can only be used to compare the variability in two different populations based on similar variables.

       Example 2.18

Use the means and standard deviations given in Table 2.6 for the three variables that were measured on a population to answer the following questions:

Table 2.6 The Means and Standard Deviations for Three Different Variables

Variable	µ	σ
I	100	25
II	10	5
III	0.10	0.05

      1 Determine the value of the coefficient of variation for population I.

      2 Determine the value of the coefficient of variation for population II.

      3 Determine the value of the coefficient of variation for population III.

      4 Compare the relative variability of each variable.

       Solutions

      1 The value of the coefficient of variation for population I is CVI=25100=0.25.

      2 The value of the coefficient of variation for population II is CVII=510=0.5.

      3 The value of the coefficient of variation for population III is CVIII=0.050.10=0.5.

      4 Populations II and III are relatively more variable than population I even though the standard deviations for populations II and III are smaller than the standard deviation of population I. Populations II and III have the same amount of relative variability even though the standard deviation of population III is one-hundredth that of population II.

The previous example illustrates how comparing the absolute size of the standard deviation is relevant only when comparing similar variables. Also, interpreting the size of a standard deviation should take into account the size of a typical value in a population. For example, a standard deviation of σ = 0.01 might appear to be a small standard deviation; however, if the mean was µ = 0.006, then this would be a very large standard deviation (CV =167%); on the other hand, if the mean was µ = 5.2, then σ = 0.01 would be a small standard deviation (CV =0.2%).

      2.2.7 Parameters for Bivariate Populations

      In most biomedical research studies, there are many variables that will be recorded on each individual in the study. A multivariate distribution can be formed by jointly tabulating, charting, or graphing the values of the variables over the N units in the population. For example, the bivariate distribution of two variables, say X and Y, is the collection of the ordered pairs

      These N ordered pairs form the units of the bivariate distribution of X and Y and their joint distribution can be displayed in a two-way chart, table, or graph.

      When the two variables are qualitative, the joint proportions in the bivariate distribution are often denoted by p_ab, where

The joint proportions in the bivariate distribution are then displayed in a two-way table or two-way bar chart. For example, according to the American Red Cross, the joint distribution of blood type and Rh factor is given in Table 2.7 and presented as a bar chart in Figure 2.21.

      Figure 2.21 The joint distribution