Measures of Variability

Range

The range of a set of data is the difference between the largest and smallest value in the data set; it is calculated very simply by using this formula:

$$\text{Range}=\text{maximum}\text{value}-\text{minimum}\text{value},$$

and it is very useful not only for showing dispersion in a data set but also for comparing variability between and among similar data sets. In the previous example, the professor of the college algebra course may want to compare dispersion of Exam 1 scores for three different classes or to compare dispersion of scores for Exams 1, 2, and 3 for the same class.

Although the range is the easiest measure of variability to find, using range to describe dispersion is not always the most appropriate choice. Calculation of range relies on only two values, so it is very sensitive to extreme values. If the highest and/or lowest value is/are an outlier(s), the range will provide an inaccurate picture of how much spread actually exists in the data set. Using the previous college algebra example, the professor wants to find the amount of dispersion in the set of scores from the first exam; there are 25 students in the class. If the highest score in the class is 90 of 100 and the lowest score is 20 of 100, the range is 90 – 20 = 70. This indicates a great deal of dispersion on a 100-point scale. However, if the majority of the scores cluster around the class average of 75 on the exam, and the scores of 20 and 90 are both outliers, then the range of 70 is misleading.

Interquartile Range

For data sets that contain outliers, the interquartile range can be calculated as a measure of variability; the interquartile range is not sensitive to outliers because it considers variability only within the middle 50% of the data set. The interquartile range is related to the median, which is a measure of central tendency that divides a distribution in half with 50% of the scores below and 50% of the scores above it. The distribution or data set can be further divided into fourths or quartiles. If there are 16 values in the data set, each quartile will contain 4 values; the data are not divided into 25% portions of total value but rather 25% of the number of observations. If the values in the data set are in order of ascending value, the first quartile contains the lowest 25% of the values; the second quartile contains the next 25% of the values; the third quartile contains the second highest 25% of the values; and the fourth quartile contains the highest 25% of the values. The interquartile range can be calculated by subtracting the first quartile from the third

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