measured in ordinal scales are often found in clinical research. Figure 1.2 shows three examples of ordinal scales: the item list, where the subjects select the item that more closely corresponds to their opinion, the Likert scale, where the subjects read a statement and indicate their degree of agreement, and the visual analog scale, where the subjects mark on a 100 mm line the point that they feel corresponds to their assessment of their current state. Psychometric, behavioral, quality of life, and, in general, many questionnaires commonly used in clinical research have an ordinal scale of measurement.
When the values are numeric, ordered, and the difference between consecutive values is the same all along the scale, that is an interval scale. Interval scales are very common in research and in everyday life. Examples of attributes measured in interval scales are age, height, blood pressure, and most clinical laboratory results. Some interval‐measured attributes are continuous, for example, height, while others are not continuous and they are called discrete. Examples of discrete attributes are counts, like the leukocyte count. Because all values in the scale are at the same distance from each other, we can perform arithmetic operations on them. We can say that the difference between, say, 17 and 25 is of the same size as that between, say, 136 and 144. In practical data analysis, however, we often make no distinction between continuous and discrete variables.
Figure 1.2 Examples of commonly used ordinal scales.
Figure 1.3 Difference between an ordinal and an interval scale.
Figure 1.3 illustrates the difference between ordinal and interval scales: in ordinal scales, consecutive values are not necessarily equidistant all over the range of values of the scale, while that is always true in interval scales. One easy way to differentiate ordinal from interval scales is that interval scales almost always have units, like cm, kg, l, mg/dl, mmol/l, etc. while ordinal scales do not.
If an interval scale has a meaningful zero, it is called a ratio scale. Examples of ratio scales are height and weight. An example of an interval scale that is not a ratio scale is the Celsius scale, where zero does not represent the absence of temperature, but rather the value that was by convention given to the temperature of thawing ice. In ratio scales, not only are sums and subtractions possible, but multiplications and divisions as well. The latter two operations are meaningless in non‐ratio scales. For example, we can say that a weight of 21 g is half of 42 g, and a height of 81 cm is three times 27 cm, but we cannot say that a temperature of 40°C is twice as warm as 20°C. With very rare exceptions, all interval‐scaled attributes that are found in research are measured in ratio scales.
1.3 Central Tendency Measures
We said above that one important purpose of biostatistics is to determine the characteristics of a population in order to be able to make predictions on any subject belonging to that population. In other words, what we want to know about the population is the expected value of the various attributes present in the elements of the population, because this is the value we will use to predict the value of each of those attributes for any member of that population. Alternatively, depending on the primary aim of the research, we may consider that biological attributes have a certain, unknown value that we attempt to measure in order to discover what it is. However, an attribute may, and usually does, express variability because of the influence of a number of factors, including measurement error. Therefore, the mean value of the attribute may be seen as the true value of an attribute, and its variability as a sign of the presence of factors of variation influencing that attribute.
There are several possibilities for expressing the expected value of an attribute, which are collectively called central tendency measures, and the ones most used are the mean, the median, and the mode.
The mean is a very common measure of central tendency. We use the notion of mean extensively in everyday life, so it is not surprising that the mean plays an extremely important role in statistics. Furthermore, being a sum of values, the mean is a mathematical quantity and therefore amenable to mathematical processing. This is the other reason why it is such a popular measure in statistics.
As a measure of central tendency, however, the mean is adequate only when the values are symmetrically distributed about its value. This is not the case with a number of attributes we study in biology and medicine – they often have a large number of small values and few very large values. In this case, the mean may not be a good measure of central tendency, because it will indicate the expected value of an attribute. A better measure of central tendency is, in these cases, the median.
The median is the quantity that divides the sample into two groups with an equal number of observations: one group has all the values smaller than that quantity, and the other group has all the values greater than that quantity. The median, therefore, is a quantity that has a straightforward interpretation: half the observations are smaller than that quantity. Actually, the interpretation of the median is exactly the same as the mean when the values are symmetrically distributed about the mean and, in this case, the mean and median will have the same value. With asymmetric distributions, however, the median will be smaller than the mean.
Figure 1.4 presents a plot of the length of stay in days in an intensive care unit. The distribution of values is very asymmetrical, or skewed to the left. In a distribution like that, the median normally reflects better than the mean the expected value of the attribute. Suppose one wanted to predict the length of stay of someone just admitted to intensive care: a guess of 13 days (the median) would be more likely to be close to the actual length of stay than 23 days (the mean).
Figure 1.4 Comparison of the mean and the median in an asymmetrical distribution.
One problem with the median is that it is not a mathematical result. To obtain the median, first we must count the number of observations, as we do to compute the mean. Then we must sort all the values in ascending order, divide the number of observations by 2, and round the result to the nearest integer. Then we take this result, go to the observation that occupies that position in the sorted order, and obtain the value of that observation. The value is the median value. Further, if the number of observations is even, then we must take the value of the observation that has a rank in the sorted order equal to the division of the number of observations by 2, then add that value to the value of the next observation in the sorted order, and divide the result by 2 to finally obtain the median value.
The median, therefore, requires an algorithm for its computation. This makes it much less amenable to mathematical treatment than the mean and, consequently, less useful. In many situations, however, the median is a much better measure of central tendency than the mean. For example, attributes that are measured on ordinal scales – recall that with ordinal scales sums and differences are meaningless – should almost always be summarized by the median, not the mean. One possible exception to this rule is when an ordinal scale has so many distinct values, say,