1.7 Summary Statistics for a Quantitative Variable

Measures of Center

The mean is the arithmetic average — sum of values divided by n. It's sensitive to outliers because every value pulls on it. The median is the middle value when data are ordered; it's resistant to outliers and skew. For skewed distributions, the median is usually the better summary of center.

Measures of Spread

The range (max − min) is simple but ignores middle data. The IQR (Q3 − Q1) captures the middle 50% and is resistant to outliers. Standard deviation measures typical distance from the mean, using every value, but is affected by outliers and skew — best used with symmetric data.

Choosing & Describing Shape

For skewed or outlier-heavy data, report median and IQR. For roughly symmetric data with no major outliers, report mean and standard deviation. Always connect your choice to the distribution's shape when describing data (shape, center, spread, unusual features).

Key terms

Mean

Arithmetic average of all data values; sensitive to outliers and skew.

Median

Middle value of ordered data; resistant to outliers and skew.

IQR

Q3 − Q1; measures spread of the middle 50% of data, resistant to outliers.

Standard deviation

Typical distance of data points from the mean; uses all values, sensitive to outliers.

When to use median/IQR vs. mean/SD

Use median/IQR for skewed data or outliers; use mean/SD for symmetric data without major outliers.

Try an AP-style question

What are the mean and median?

1. Mean 5, median 5
2. Mean 4, median 4
3. Mean 5, median 4
4. Mean 4, median 5