Describing the Distribution of a Quantitative Variable
Introduced| English | Meaning |
|---|---|
| Shape/ʃeɪp/ | The overall pattern of a distribution, including its symmetry, tails and peaks. |
| Center/ˈsentə/ | A typical value — roughly where the data pile up (mean or median). |
| Spread/spred/ | Spread |
| Symmetric/sɪˈmetrɪk/ | Having corresponding parts or relationships that remain equivalent under a specified transformation or interchange. |
| Skewed left/skjuːd left/ | A long tail stretches to the *left* (low values). |
| skewed right/skjuːd raɪt/ | skewed right |
| clusters/ˈklʌstəz/ | clusters |
| gaps/ɡæps/ | gaps |
| outliers/ˈaʊtlaɪəz/ | An observation unusually far from the rest of the data or from the pattern expected under a model. |
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.
A complete description of a distribution mentions which features?
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Shape, center, spread, and unusual features — in context.
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
Reading the shape
A long tail to one side means the distribution is skewed that way — skew is named for the tail.
A distribution with a long tail stretching to the left is...
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Skew is named for the tail direction → 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."
The "center" of a distribution refers to...
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Center = typical value (mean or median).
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.
A value far from the rest of the data is a possible ____.
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Flag outliers as unusual features.
A good description states shape/center/spread in context (with the variable and units).
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Always describe in context, not as bare numbers.
Two separate clusters with a gap between them may suggest...
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Clusters/gaps can reveal mixed sub-populations.
Skew is named for the tail, not the peak. A distribution with most data on the right and a long tail trailing to the left is skewed left — even though the "bump" is on the right. And always describe shape/center/spread/unusual features in context (name the variable and units), not as bare numbers.
Describe household incomes in a town.
- Shape: skewed right — most incomes are moderate, with a long tail of a few very high earners.
- Center: a typical income around, say, $45{,}000 (median).
- Spread & unusual: a wide spread, with a few high outliers (the wealthiest households).
Describe a quantitative distribution by its shape (symmetric / skewed left / skewed right — named by the tail), center (typical value), spread (variability), and unusual features (clusters, gaps, outliers) — always in context with the variable and units.
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.
How does the shape affect the mean?
Extreme high values raise the mean.
What happens to the mean in this right-skewed income distribution?
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The long right tail lifts the mean.