Quartiles

After sorting observations, locate the positions associated with the 25th, 50th, and 75th percentiles. If a position falls between two observed values, an interpolation rule supplies the cutpoint rather than selecting an existing value automatically. Applying one consistent calculation convention is important when quartiles are compared across datasets.

The IQR focuses on the distance between Q1 and Q3, so it describes the spread of the central half without being determined by the most extreme observations. This makes it useful for comparing variability across groups when unusually large or small values could distort a summary based on the mean.

Compare the distance from Q1 to Q2 with the distance from Q2 to Q3. Unequal spacing signals that observations are not distributed symmetrically around the median, helping indicate skewness. This interpretation adds shape information beyond a single center value and can show whether one side of the distribution is more extended than the other.

First compute Q1, Q3, and the IQR, then evaluate observations using an IQR-based rule. Values judged unusually distant from the central interval can be flagged as potential outliers rather than automatically discarded. This approach provides a distribution-based screening method for identifying observations that warrant closer statistical examination.

Quartile summaries remain informative when extreme observations make the mean less representative of the dataset. Q2 supplies a middle-position summary, while Q1 and Q3 show how the central observations are distributed around it. Together, these values describe location and spread with less emphasis on extremes than a mean-based summary.

Apply the same quartile calculation rule to each ordered dataset, then compare Q1, Q2, Q3, and the resulting IQRs. Differences in Q2 indicate different central positions, whereas differences in IQR indicate different central-half spreads. This supports distributional comparisons rather than relying on one summary statistic alone.