AP STATISTICS · EXPLORING DATA

Describing Distributions

Shape, center, spread, and outliers — always in context.

13 min read 3 practice questions

When you describe a quantitative distribution on the AP exam, address four things: shape (symmetric, skewed left or right, unimodal or bimodal), center (mean or median), spread (standard deviation, range, or IQR), and unusual features (outliers, gaps).

\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i \qquad\qquad s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}

Sample mean and sample standard deviation

The median and IQR are resistant — outliers barely move them. The mean and standard deviation are not. For skewed data, prefer the median and IQR.

\text{Outlier if } x < Q_1 - 1.5\cdot\text{IQR} \;\text{ or }\; x > Q_3 + 1.5\cdot\text{IQR}

The 1.5 × IQR rule

Worked example · Finding the five-number summary

Data:

2, 4, 5, 7, 8, 9, 12, 15, 30

1
Min $= 2$ , median $= 8$ (the 5th of 9 values), max $= 30$ .
2
$Q_1$ = median of $2, 4, 5, 7$ $= 4.5$ ; $Q_3$ = median of $9, 12, 15, 30$ $= 13.5$ .
3
IQR $= 9$ . Upper fence $= 13.5 + 13.5 = 27$ , so $30$ is an outlier.

CHECK YOUR UNDERSTANDING

Practice

Answer each question, then check.

Find the mean of

3, 7, 8, 10, 12

A distribution has a long tail to the right. Which is most likely true?

Q_1 = 20

and

Q_3 = 32

. What is the upper fence for outliers?

Numeric answers accept decimals or fractions like 3/4.