Describing Distributions

The SOCS Framework

When describing any distribution, address all four components:

Shape

• Symmetric: Mean ≈ Median; bell-shaped or uniform
• Skewed Right: Tail extends right; Mean > Median
• Skewed Left: Tail extends left; Mean < Median
• Bimodal: Two distinct peaks
• Uniform: Roughly equal frequencies

Outliers

Values that fall far from the bulk of the data. Use the 1.5 × IQR rule: $\text{Outlier if } x < Q_1 - 1.5 \cdot IQR \text{ or } x > Q_3 + 1.5 \cdot IQR$

Center

• Mean $\bar{x}$: Arithmetic average; sensitive to outliers
• Median: Middle value; resistant to outliers

Spread

• Range: Max − Min (sensitive to outliers)
• IQR: $Q_3 - Q_1$ (resistant)
• Standard Deviation $s$: Average distance from the mean

Comparing Distributions

When comparing two distributions, always use comparative language:

• "Distribution A has a higher center than Distribution B"
• "Distribution A is more spread out than Distribution B"
• "Both distributions are approximately symmetric"

Five-Number Summary

$\text{Min}, \; Q_1, \; \text{Median}, \; Q_3, \; \text{Max}$

This summary is used to create boxplots. Side-by-side boxplots are excellent for comparing distributions.

AP Tip: On free-response questions, you must use context (mention the actual variable). Don't just say "the distribution is skewed right" — say "the distribution of test scores is skewed right."