25.5 Comparing distributions and outliers — Learn

Prerequisites: find medians, ranges and IQRs, and read box plots.

Objectives: compare distributions using suitable measures and discuss possible outliers.

Compare a suitable measure of central tendency, such as the median, to describe a typical value. The median resists the effect of an unusually high or low value. Compare the spread using range or IQR. The range uses the extremes; the IQR describes the middle half.

Example 1. Group A has median 18 and IQR 6. Group B has median 21 and IQR 10.

Group B has the higher typical value because $21>18$.

Group B has greater spread in its middle half because $10>6$.

Use the context in a comparison: “Group B's typical time is higher, and its middle half varies more.”

An outlier is a value far from most others. A common rule flags values below $Q_1-1.5\times\mathrm{IQR}$ or above $Q_3+1.5\times\mathrm{IQR}$. This is a rule for spotting possible outliers, not proof that a measurement is wrong.

Example 2. For the travel times, $Q_1=17.5$, $Q_3=26.5$, so IQR $=9$.

Lower fence $=17.5-1.5\times9=4$.

Upper fence $=26.5+1.5\times9=40$.

The value 42 is above 40, so it is a possible outlier. Check the original record before deciding what it means.

Watch out. A larger range can come from one unusual value. Compare the IQR as well, and explain what each measure says about the data.

Progress check

The possible outlier 42 minutes may affect the range more than the IQR. Explain why the IQR uses only the middle half.

You should now be able to

• compare typical values and spread using suitable measures;
• interpret differences in the context of the data;
• identify a possible outlier using the IQR rule;
• explain why an unusual value needs checking.