6.1 Describing Variability
6.1: Describing Variability
6.1.1: Range
The range is a measure of the total spread of values in a quantitative dataset.
Learning Objective
Interpret the range as the overall dispersion of values in a dataset
Key Takeaways
Key Points
- Unlike other more popular measures of dispersion, the range actually measures total dispersion (between the smallest and largest values) rather than relative dispersion around a measure of central tendency.
- The range is measured in the same units as the variable of reference and, thus, has a direct interpretation as such.
- Because the information the range provides is rather limited, it is seldom used in statistical analyses.
- The mid-range of a set of statistical data values is the arithmetic mean of the maximum and minimum values in a data set.
Key Terms
- dispersion
- the degree of scatter of data
- range
- the length of the smallest interval which contains all the data in a sample; the difference between the largest and smallest observations in the sample
In statistics, the range is a measure of the total spread of values in a quantitative dataset. Unlike other more popular measures of dispersion, the range actually measures total dispersion (between the smallest and largest values) rather than relative dispersion around a measure of central tendency.
Interpreting the Range
The range is interpreted as the overall dispersion of values in a dataset or, more literally, as the difference between the largest and the smallest value in a dataset. The range is measured in the same units as the variable of reference and, thus, has a direct interpretation as such. This can be useful when comparing similar variables but of little use when comparing variables measured in different units. However, because the information the range provides is rather limited, it is seldom used in statistical analyses.
For example, if you read that the age range of two groups of students is 3 in one group and 7 in another, then you know that the second group is more spread out (there is a difference of seven years between the youngest and the oldest student) than the first (which only sports a difference of three years between the youngest and the oldest student).
Mid-Range
The mid-range of a set of statistical data values is the arithmetic mean of the maximum and minimum values in a data set, defined as:
The mid-range is the midpoint of the range; as such, it is a measure of central tendency. The mid-range is rarely used in practical statistical analysis, as it lacks efficiency as an estimator for most distributions of interest because it ignores all intermediate points. The mid-range also lacks robustness, as outliers change it significantly. Indeed, it is one of the least efficient and least robust statistics.
However, it finds some use in special cases:
- It is the maximally efficient estimator for the center of a uniform distribution
- Trimmed mid-ranges address robustness
- As an L-estimator, it is simple