Summary Statistics for a Quantitative Variable

Numbers that summarize the data

• A graph shows the shape; summary statistics boil a distribution down to a few key numbers.
• Two jobs: measure the center (a typical value) and the spread (how variable).
• Which numbers you use depends on the shape — especially whether there are outliers.
• These summaries are the backbone of every later inference.

Measures of center

• The mean 均值 $\bar{x}$ is the arithmetic average: add all values, divide by how many.
• The median 中位数 is the middle value when the data are sorted (average the two middle if even).
• For a symmetric distribution they're close; for a skewed one they differ.
• The mean gets pulled toward a long tail; the median stays put.

The five-number summary and the IQR on a box plot

Measures of spread

• Range 极差 = maximum − minimum (simple, but sensitive to extremes).
• IQR (interquartile range) $=Q_3-Q_1$ — the spread of the middle $50\%$.
• Standard deviation 标准差 $s_x$ — the typical distance of values from the mean.
• Bigger spread numbers mean more variability.

Resistant vs. sensitive

• Median and IQR are resistant 稳健 — outliers barely move them.
• Mean and standard deviation are sensitive — one extreme value can shift them a lot.
• So for skewed data or data with outliers, report the median and IQR.
• Outlier rule: a value is an outlier if it's below $Q_1-1.5\times\text{IQR}$ or above $Q_3+1.5\times\text{IQR}$.

Write from a mean of 5

• The values are 4, 4, 4 and 8.
• The lesson summaries include a mean of 6 and a spread of 2.