Describing the Distribution of a Quantitative Variable

Telling the story of a distribution

• Once you've graphed a quantitative variable, you describe what you see — in words, in context.
• The four things to mention: shape, center, spread, and unusual features.
• Cover all four and you've summarized the distribution completely.
• "In context" means naming the variable and its units, not just abstract numbers.

Shape

• Shape 形状 describes the overall form of the distribution.
• Symmetric 对称: the left and right halves roughly mirror each other.
• Skewed left 左偏: a long tail stretches to the left (low values); skewed right 右偏: a long tail to the right (high values).
• The tail's direction names the skew — skew points where the tail goes.

Describe shape: symmetric or skewed right (tail to the right) or skewed left

Center and spread

• Center 中心: a typical value — roughly where the data pile up (mean or median).
• Spread 分布范围 (variability): how stretched out the values are (range, IQR, standard deviation).
• A distribution with small spread is tightly clustered; large spread is widely scattered.
• Together, center and spread answer "typical value, and how much it varies."

Unusual features

• Note distinctive features: clusters 群 (separate groups), gaps 间隙 (empty stretches), and possible outliers 离群值 (values far from the rest).
• Outliers may be errors, or genuinely unusual cases — flag them.
• A gap or two clusters may hint at two different sub-populations mixed together.
• Always describe these in context of the real variable.

Write from a mean pulled right

• A distribution of incomes has a long tail to the right.
• A few very high incomes sit in that tail.