CONSIDERATIONS IN SELECTING APPROPRIATE STATISTICS 59Table 5.2 Recommended Graphs and Statistics According to the Stevens System
Scale Graph Location Variability
Nominal Pie and bar graphs Mode p( 1 - p)/n
Ordinal Box plots Median Quartiledrange
Interval Histograms Mean Standard deviation
Ratio Histograms Mean Coefficient of variationscale has a fixed unit of measurement (degrees). The difference between 14" and 15"
is the same as that between 16" and 17", or 1". Note that 0" Fahrenheit or Celsius
does not represent no heat. Interval variables do not have natural zero points.
If the variable not only has equal intervals but also has a natural zero point, it
is called a ratio variable. Variables such as height, weight, and density have true
zero points. Height has a naturally defined zero point on a ruler. We can multiply
height in inches by a constant, say 2.54, and get height in centimeters and still have a
ratio variable. If we reported temperature using the Kelvin scale, it would be a ratio
variable. In addition to the mean defined in this chapter, two other types of means
can be computed for ratio variables (see Afifi et al. [2004]).
Recommended statistics and graphs for nominal, ordinal, interval, and ratio data
are given in Table 5.2. It is important to note that for the statistics, the descriptive
method is appropriate to that type of variable listed in the first column and to all
below it. For example, the median is suitable for ordinal data and also for interval
and ratio data. If ordinal data take on only a few values, box plots and quartiles are
not very useful. We discuss graphics and statistics for nominal data in Chapters 10
and 11. Box plots are discussed in Section 5.5.
In the Stevens system, the mean and standard deviation are not recommended for
nominal or ordinal data. This clearly makes sense for nominal data, where even if
the data is coded with successive numbers, the ordering is artificial. For example, if
we coded hospital A as 1, hospital B as 2, and hospital C as 3, then if we compute
and report a mean and a standard deviation for a variable called hospital, it obviously
lacks any rational meaning. Stevens has shown that problems exist in using means
and standard deviations for ordinal data. Stevens recommends medians, quartiles,
fourths, or ranges for ordinal data.
The coescient of variation is a measure that is often used with ratio data when
authors want to describe how variable their data are. It is computed by dividing the
standard deviation by the mean. The coefficient of variation provides a measure of
variation that is corrected for the size of the mean. If the coefficient of variation was
a large value such as 1.0, most investigators would decide that the observations are
highly variable. But if the coefficient of variation was < .lo, then for most work
this would be considered a small variation (although probably not small enough for
quality control results). The coefficient of variation should not be used unless there
is a natural zero point.