Extreme values can stretch the full range without substantially changing the bounds of the central half. Because the Middle 50 Percent is bounded by Q1 and Q3, its spread is summarized by the IQR rather than by the smallest and largest observations. This makes it a more stable basis for describing typical dispersion when unusually low or high values are present.
The distance from Q1 to Q3 measures how widely the central observations are distributed. A larger IQR indicates that this portion of the data occupies a wider interval, whereas a smaller IQR indicates greater concentration within the central half. Reporting both the interval and its width shows where the observations lie and how much central spread they have.
To compare data sets, examine their Q1-to-Q3 intervals and compare their IQRs. The interval identifies the location of each data set’s central half, while the IQR compares the amount of spread within that portion. Thus, one data set may have a wider or narrower central distribution, even when the comparison does not focus on extreme observations.
The full range is determined by the lowest and highest values, so it reflects the entire span of a data set. The Middle 50 Percent instead focuses on the interval from Q1 to Q3 and summarizes central spread through the IQR. Using both measures can distinguish overall span from the behavior of the central observations.
When presenting this summary, give Q1 and Q3, state the interval between them, and report the IQR as Q3 − Q1. These pieces answer complementary questions: the quartiles identify the boundaries, the interval displays the central portion, and the IQR gives a single numerical measure of its width. This format supports clear comparisons between data sets.
Ordering is essential because quartiles divide the data according to position, not merely according to the numerical values viewed in arbitrary sequence. Once the observations are arranged from low to high, Q1 and Q3 can be identified as the boundaries associated with the 25th and 75th percentiles. Without that order, the central interval cannot be located consistently.