Averages, and which to use
| English | Chinese | Pinyin |
|---|---|---|
| mean | 平均数 | píng jūn shù |
| median | 中位数 | zhōng wèi shù |
| mode | 众数 | zhòng shù |
| symmetric | 对称 | duì chèn |
| skewed | 偏态 | piān tài |
Three averages, three different claims
- "The average income here is 42,000" can be true and misleading at once.
- The mean 平均数, the median 中位数 and the mode 众数 all answer "what is typical", differently.
- Choosing between them is the marked skill; computing them is arithmetic.
What each one does
- The mean is the total divided by the count. It uses every value — including every outlier.
- The median is the middle value in order. It ignores how extreme the extremes are.
- The mode is the most frequent value, and it is the only average that works for categories.
Which average can be used for favourite subject?
Categories cannot be added or ordered meaningfully, so only the most frequent value applies.
When they disagree
- In a symmetric 对称 distribution the three sit together.
- In a skewed 偏态 one they separate, and the mean is pulled furthest towards the long tail.
- That separation is information: mean far above median means a few very large values.
A mean much larger than the median suggests a few unusually large values.
The mean is pulled towards the long tail; the median is not. The gap is the information.
Nine people earn 30,000. One earns 150,000.
The mean describes nobody in the room. Nine of the ten earn 30,000, and the tenth earns five times that.
For income, the median describes a typical person and the mean does not — which is why national statistics report both.
Nine people earn 30000 and one earns 150000. What is the median?
Ordered, the middle values are both 30000. The single large value does not move it.
Which average best describes a typical income in that group?
The mean of 42000 describes nobody in the group. Saying why is what completes the answer.
Choose the average by the shape of the data, and say why. "The median is 30,000 because one large value pulls the mean above what most people earn" is a complete answer. A number alone is half of one.
You cannot recover the mean from a grouped frequency table exactly, only estimate it — because the individual values are gone. Questions with grouped data expect the word "estimate", and answering without it loses the mark even when the arithmetic is right.
What can you say about the mean of grouped frequency data?
The individual values are gone, so midpoints are used. The word "estimate" is part of the mark.