Averages, and which to use · 平均数与如何选择
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| mean/miːn/ | 平均数 | píng jūn shù |
| median/ˈmiːdiːən/ | 中位数 | zhōng wèi shù |
| mode/məʊd/ | 众数 | zhòng shù |
| symmetric/sɪˈmetrɪk/ | 对称 | duì chèn |
| skewed/skjuːd/ | 偏态 | piān tài |
A centre answers a specified question
- The mean 平均数 measures total per observation; the median 中位数 locates the ordered middle; the mode 众数 identifies a most frequent value.
- No single summary answers every question. A mode can describe a category, while mean and median need a suitable numerical or ordered scale.
Combine totals before combining groups
- Compute the mean as $\bar x=\sum x/n$, or $\sum fx/\sum f$ for exact-value frequency data. For an even count, the numerical median averages the middle two ordered values.
- To combine group means, weight them by group size. Ten scores with mean 60 and twenty with mean 75 have combined mean 70, not 67.5.
Which average can be used for favourite subject? · "最喜欢的科目"可以用哪个平均数?
Categories cannot be added or ordered meaningfully, so only the most frequent value applies. · 类别既不能相加,也无法有意义地排序,所以只有"出现最多"适用。
Interpret shape without inventing observations
- A symmetric 对称 distribution need not have its mode at the centre if it has two peaks. Do not assume mean, median and mode always coincide.
- A skewed 偏态 pattern or extreme values can separate the mean from the median. Inspect the supplied data rather than treating one gap between summaries as a complete shape description.
A mean much larger than the median suggests a few unusually large values. · 平均数远大于中位数,说明存在少数异常大的值。
A mean well above the median can suggest high extreme values or a right-skewed pattern. It does not uniquely determine the full data shape; inspect the observations. · 平均值显著高于中位数可能暗示存在极端高值或右偏分布。它并不能唯一确定整体数据形状；需观察具体数据点。
Nine incomes of 30000 and one of 150000. The mean is $\bar x=[9(30000)+150000]/10=42000$, while median is 30000. The mean correctly describes total income per person; it does not claim anybody earns 42000. The median identifies the ordered middle.
Nine people earn 30000 and one earns 150000. What is the median? · 九人年收入 30000,一人 150000,中位数是多少?
Ordered, the middle values are both 30000. The single large value does not move it. · 排序后中间的两个值都是 30000。那一个极大值移动不了它。
Which average best describes a typical income in that group? · 哪个平均数最能描述这群人中典型的收入?
In this supplied group, nine of ten incomes equal the median 30000. The mean 42000 is still a valid total-per-person summary, but is not what most individuals earn. · 在该给定组别中，十份收入中有九份等于中位数30000。平均值42000仍是一个有效的每人总收入摘要，但这并非大多数人的实际收入水平。
Explain what your summary represents. In the supplied income example the median also equals what nine people earn. That particular fact comes from the actual list, not from a general definition of median. Sheet 2.6 compares weighted means and the effect of a changed extreme.
A grouped frequency table with numeric intervals usually loses individual values. Using class midpoints gives an estimated mean unless further information makes it exact. Do not confuse interval groups with a frequency table of exact distinct values, whose mean can be calculated exactly.
What can generally be calculated from a grouped frequency table of numeric intervals when the individual values are not supplied? · 当不提供个体数值时，通常可以从分组频数表中计算出什么？
Class midpoints give an estimated mean unless additional information makes it exact. A table of exact distinct values and their frequencies is different and permits an exact mean. · 类中值可提供估计平均值，除非有额外信息使其精确。提供确切不同数值及其频数的表格则不同，允许计算精确平均值。