描述统计：中心、离散、形状与离群值

One sentence that hides a distribution

• "Average delivery: 27.4 minutes" is true — and almost empty. Forty orders sit behind it, from 16.8 to 38.5 minutes, and description is where you meet them.
• Four questions, every dataset: centre, spread, shape, outliers.

Centre: two answers, two jobs

• Mean 均值 = sum ÷ n = 1,097.1 ÷ 40 = 27.43. The median 中位数 (middle order) = 27.6. So close because the data's stretch is mild.
• The mean follows the tail; the median resists it. For skewed money or delay data, report the median — and say why you did.

Spread: sd and IQR

• Standard deviation 标准差 $= \sqrt{940.9/39} = \mathbf{4.91}$ — the typical distance from the mean. Divide by $n-1$: one degree of freedom spent estimating the mean.
• The quartiles 24.5 / 27.6 / 30.9 give the IQR 6.3 — the middle half's width, immune to tails.

Shape and the honest axis

• Single peak, mild right stretch — and the mean sits slightly BELOW the median (27.43 < 27.6): the left side has its own cluster (three runs under 18 minutes). Check the slogan "right skew ⇒ mean above median" against the data, never recite it.
• Chart honesty: an axis starting at 24 minutes triples visual differences. Draw from zero or declare the break.

Outliers: catch, investigate, decide

• The fence: Q1 − 1.5×IQR = 15.0, Q3 + 1.5×IQR = 40.3. No delivery crosses it — 38.5 is high but inside.
• Change one digit to 168 minutes: the mean jumps to 30.43, the sd quadruples to 22.0, the median barely moves — and the fence catches it. Never delete silently: investigate, document, decide.

Three descriptive slips

Recap