The Effect of Outliers on Summary Statistics and Data Displays

What is an Outlier?

An outlier is a data value that is unusually far from the rest of the data.

Formal definition (using fences):
$$\text{Value is an outlier if } x < Q_1 - 1.5 \times \text{IQR} \quad \text{or} \quad x > Q_3 + 1.5 \times \text{IQR}$$

Effect on Measures of Location

Worked Example

Original data: 10, 12, 13, 14, 15, 16, 17

• Mean = $\frac{97}{7} = 13.86$, Median = 14

After adding outlier 52: 10, 12, 13, 14, 15, 16, 17, 52

• New mean = $\frac{149}{8} = 18.63$ ← inflated significantly
• New median = $\frac{14 + 15}{2} = 14.5$ ← barely changed

KEY TAKEAWAY: An outlier drags the mean toward it but has minimal effect on the median. This is why median is preferred for skewed data.

Effect on Measures of Spread

Effect on Data Displays

Histogram

• A tall, isolated bar far from the main cluster appears
• May compress the scale, making the main distribution harder to read

Boxplot

• Outlier appears as a separate dot (×) beyond the whisker
• The box and whiskers themselves are unaffected

Dot Plot / Stem-and-Leaf

• An isolated point/leaf far from the cluster is visually obvious
• May require extra rows (stems) that are otherwise empty

What To Do With Outliers

1. Check for errors: Was the value recorded correctly? Is it a data entry mistake?
2. Investigate: Is there a legitimate reason for this value?
3. Report: Always acknowledge outliers in your analysis
4. Choose robust statistics: If keeping the outlier, use median + IQR

EXAM TIP: VCAA commonly asks: “What effect does the outlier have on the mean and median?” Always state the direction (increases/decreases) and explain why. Then identify which statistic is more appropriate.

VCAA FOCUS: Never just say “remove the outlier” without justification. In a VCAA context, you must investigate and explain — outliers often represent real and important data points (e.g. one student with a very high income in a household survey).