Spread · 离散程度
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| quartiles/ˈkwɔːtaɪlz/ | 四分位数 | sì fēn wèi shù |
| range/reɪndʒ/ | 全距 | quán jù |
| interquartile range/ˌɪntəˈkwɔːtaɪl reɪndʒ/ | 四分位距 | sì fēn wèi jù |
| box plot/bɒks plɒt/ | 箱线图 | xiāng xiàn tú |
| standard deviation/ˈstændəd ˌdiːvɪˈeɪʃn/ | 标准差 | biāo zhǔn chà |
One average hides two score patterns
- The lists 58, 60, 62 and 20, 60, 100 both have mean 60. Their different spread affects what that shared mean tells you.
- A spread summary adds information without explaining the cause of a difference. Keep measurement units and population scope the same when comparing.
一个平均值掩盖了两种分数模式
- 列表58, 60, 62和20, 60, 100的均值均为60。它们不同的离散程度会影响该共同均值所传达的信息。
- 离散程度汇总提供了信息但未解释差异的原因。比较时请保持测量单位和总体范围一致。
Two classes both average 60. Why report the spread as well? · 两个班级平均分均为 60。为何还要报告离散程度？
58, 60, 62 and 20, 60, 100 share a mean and share nothing else. · 58, 60, 62 与 20, 60, 100 具有相同均值，但其余毫无相似之处。
State how the quartiles are calculated
- The range 全距 is maximum minus minimum. The quartiles 四分位数 mark positions in ordered data; interquartile range 四分位距 is $Q_3-Q_1$.
- Sheet 2.7 uses medians of ordered halves, excluding the overall median for an odd count. Software can use other conventions; specify yours before comparing results.
说明四分位数的计算方法
- 全距是最大值减最小值。四分位数标记有序数据中的位置；四分位距是$Q_3-Q_1$。
- 练习纸2.7使用有序半列的中位数，且当计数为奇数时排除总中位数。软件可能采用其他惯例；比较结果前请明确指定您的方法。
Why is the interquartile range often preferred to the range? · 为何四分位距常优于全距？
The range is defined entirely by the two most extreme values, which are the least typical. · 全距完全由两个极端值定义，而这些值最不具代表性。
A box plot uses a defined whisker rule
- A box plot 箱线图 shows quartiles and median. Under this sheet's rule, whiskers extend to minimum and maximum.
- Other plots use outlier fences and separately plotted values. A five-number summary does not reveal every observation, count or explanation for an extreme.
箱线图使用定义的须线规则
- 箱线图显示四分位数和中位数。根据本练习纸的规则，须线延伸至最小值和最大值。
- 其他图表使用离群值边界并单独绘制数值。五数概括无法揭示每个观测值、数量或极端值的成因。
Two data sets, same median · 两组数据，中位数相同
The box shows the middle half; the whiskers show how far the extremes reach. · 箱体显示中间一半数据，须须显示极值延伸范围。
Distinguish population and sample spread
- Population standard deviation 标准差 is $\sigma=\sqrt{\sum(x-\mu)^2/n}$. For the stated sample-estimate convention use $s=\sqrt{\sum(x-\bar x)^2/(n-1)}$.
- Squared deviations make spread nonnegative; taking the square root restores the original units. Variance has squared units and is not the same quantity.
区分总体与样本的离散程度
- 总体标准差是$\sigma=\sqrt{\sum(x-\mu)^2/n}$。对于指定的样本估计惯例，请使用$s=\sqrt{\sum(x-\bar x)^2/(n-1)}$。
- 平方偏差使离散程度非负；开方还原原始单位。方差具有平方单位，并非同一量度。
Treat 58, 60, 62 as the complete population. Find its population standard deviation to 2 decimal places. · 将58、60、62视为完整总体。求其总体标准差，保留2位小数。
Use the population divisor 3: √((4 + 0 + 4)/3) = √(8/3) ≈ 1.63, in the original units. · 使用总体除数3：√((4 + 0 + 4)/3) = √(8/3) ≈ 1.63，单位为原单位。
Treat the supplied three-value lists as complete populations. For 58, 60, 62, $\sigma_A=\sqrt{(4+0+4)/3}\approx1.63$. For 20, 60, 100, $\sigma_B=\sqrt{(1600+0+1600)/3}\approx32.7$. The second population has twenty times the standard deviation under the same convention.
将提供的三值列表视为完整总体。 对于58, 60, 62，$\sigma_A=\sqrt{(4+0+4)/3}\approx1.63$。对于20, 60, 100，$\sigma_B=\sqrt{(1600+0+1600)/3}\approx32.7$。在相同惯例下，第二个总体的标准差是第一个的二十倍。
Two classes both average 60, with standard deviations 1.6 and 33. Write one sentence reporting this properly. · 两个班级平均分均为 60，标准差分别为 1.6 和 33。写一句话恰当报告此结果。
Example: Both groups have mean score 60, but the group with standard deviation 33 is more spread around the mean than the group with standard deviation 1.6. · 示例：两组平均分均为60，但标准差为33的一组比标准差为1.6的一组围绕平均值的离散程度更大。
Choose an informative pair of summaries. Report a centre and a spread where they answer the comparison, along with units, n and the calculation convention. Sheet 2.7 examines how an extreme changes range without changing its supplied middle-half span.
选择信息丰富的摘要对。 报告中心位置和离散程度以回答比较问题，并注明单位、n及计算惯例。练习纸2.7考察了极端值如何改变全距而不改变其提供的中间一半跨度。
The standard deviation has the same units as the original data. · 标准差与原数据具有相同单位。
The square root undoes the squaring. The variance, its square, does not share the units. · 平方根抵消了平方运算。方差是其平方，不具有相同单位。
Do not discard an extreme value merely because it changes a summary. Check its origin and report any justified correction. A sample standard deviation and population standard deviation can differ even for the same printed list.
不要仅因极端值改变了摘要就将其剔除。检查其来源并报告任何合理的修正。即使是同一印刷列表，样本标准差和总体标准差也可能不同。