Mean, Median, Mode: Effect of Outliers

Three measures of center

• Mean (average): Add all values, divide by count. Formula: $\bar{x} = \frac{\sum x_i}{n}$.
• Median: Sort the data, take the middle value (or average of two middle values if $n$ is even).
• Mode: The most frequently occurring value.

Step-by-step with the dataset $\{2, 3, 4, 5, 5, 6, 100\}$

Mean: $\bar{x} = \frac{2+3+4+5+5+6+100}{7} = \frac{125}{7} = 17.86$

Median: Data is already sorted. 7 values → middle is the 4th value: 5.

Mode: 5 appears twice (most frequent): 5.

The outlier effect

The outlier (100) pulls the mean from ~4.2 (without the outlier) to 17.86 — a massive distortion! But the median (5) barely moves. This is why the median is called resistant or robust — it resists the influence of extreme values.

When to use which

• Mean: When data is symmetric with no extreme outliers. Good for: test scores, measurements.
• Median: When data is skewed or has outliers. Good for: income, home prices, reaction times.
• Mode: When you want the most common value. Good for: shoe sizes, survey choices.

Real-world example

If 9 people earn $50K and one person earns $10M: mean income = $1.045M (misleading!), median = $50K (representative). That's why "median household income" is the standard statistic.

Try it in the visualization

Drag the outlier value from 6 to 200. Watch the mean jump wildly while the median barely moves. The number line shows all three measures and how they respond to the outlier.