Are My Results Unexpected? · 我的结果出乎意料吗？
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| counts/kaʊnts/ | 计数 | jì shù |
| observed counts/ɒbˈzɜːvd kaʊnts/ | 观测频数 | guān cè pín shuò |
| expected counts/ekˈspektɪd kaʊnts/ | 期望频数 | qī wàng pín shuò |
| chi-square/kaɪ skweə/ | 卡方 | kǎ fāng |
| goodness-of-fit/ˈɡʊdnəs ɒv fɪt/ | 拟合优度 | nǐ hé yōu dù |
When counts surprise us
- Some questions ask whether observed counts 计数 differ from what we'd expect.
- "Do the colors in a candy bag match the advertised mix?"
- We compare the observed counts 观测频数 to the expected counts 期望频数 under a claim.
- Big gaps between observed and expected are the signal of something unusual.
当计数让我们意外
- 有些问题问的是观察到的计数是否与我们所期望的不同。
- “一袋糖果里的颜色是否符合广告宣称的配比？”
- 我们把观察计数与某个主张下的期望计数作比较。
- 观察与期望之间的大差距，是某种异常的信号。
The chi-square family
- The tool is the chi-square 卡方 family of tests, written $\chi^2$.
- It's built for categorical data — counts in categories, not measurements.
- Every chi-square test compares observed counts to expected counts.
- A larger total gap → a larger $\chi^2$ → stronger evidence against the claim.
卡方族
- 工具是卡方族检验，记作 $\chi^2$。
- 它是为分类数据设计的——各类别里的计数，而非测量值。
- 每个卡方检验都把观察计数与期望计数作比较。
- 总差距越大 → $\chi^2$ 越大 → 反对主张的证据越强。
Which chi-square question?
- Goodness-of-fit 拟合优度: does one categorical variable match a claimed distribution?
- Two-way table questions: are two categorical variables related (across groups or within one)?
- One variable, one sample → goodness-of-fit; a table of two variables → homogeneity/independence.
- Spotting the type is the first decision.
哪一种卡方问题？
- 拟合优度：****一个分类变量是否符合某个宣称的分布？
- 双向表问题：两个分类变量是否相关（跨各组或在一组内）？
- 一个变量、一个样本 → 拟合优度；两个变量的表 → 齐性/独立性。
- 认出类型是第一个决定。
Observed vs. expected
- Every chi-square test rests on one comparison: observed vs. expected counts.
- Expected = what the null hypothesis predicts each count should be.
- Observed = what the data actually show.
- The test measures how far, overall, observed strays from expected.
观察对期望
- 每个卡方检验都建立在一个比较上：观察对期望计数。
- 期望 = 零假设预测每个计数应该是多少。
- 观察 = 数据实际显示的。
- 检验度量观察总体上偏离期望多远。
Chi-square tests use COUNTS, not proportions or percentages. If a problem gives percentages, convert them back to actual counts before computing anything. And the whole logic is observed vs. expected: expected counts come from the null model, and a big total discrepancy is what makes $\chi^2$ large.
卡方检验用计数，而非比例或百分比。如果题目给的是百分比，先把它们换回实际计数再算任何东西。而整个逻辑是观察对期望：期望计数来自零假设模型，一个大的总体差距正是使 $\chi^2$ 变大的原因。
A bag claims equal numbers of $4$ candy colors; you count $100$ candies.
- Expected (equal mix): $25$ of each color.
- Observed: $30, 20, 28, 22$ — close, but not exact.
- A chi-square test judges whether these gaps are surprising or just chance.
一袋糖宣称 $4$ 种颜色数量相等；你数了 $100$ 颗。
- 期望（等量混合）：每种 $25$ 颗。
- 观察：$30, 20, 28, 22$——接近，但不精确。
- 卡方检验判断这些差距是意外的还是仅仅出于偶然。
Chi-square tests ask whether observed counts differ from expected counts for categorical data. A goodness-of-fit test checks one variable against a claimed distribution; two-way table tests check whether two variables are related. Every $\chi^2$ test compares observed vs. expected counts.
卡方检验问的是观察到的计数是否与期望计数不同，用于分类数据。拟合优度检验用一个变量对照宣称的分布；双向表检验用两个变量是否相关。每个 $\chi^2$ 检验都比较观察对期望计数。
A categorical count breakdown · 分类计数的构成
Chi-square tests compare observed slices to expected slices. · 卡方检验把观察到的分块与期望的分块作比较。
Chi-square tests are designed for which kind of data? · 卡方检验是为哪种数据设计的？
Chi-square compares counts in categories. · 卡方比较各类别里的计数。
A goodness-of-fit test involves how many categorical variables? · 拟合优度检验涉及几个分类变量？
GOF checks one variable against a claimed distribution. · 拟合优度用一个变量对照宣称的分布。
Chi-square tests should be run on percentages, not counts. · 卡方检验应对百分比、而非计数运行。
Convert percentages back to counts first — chi-square uses counts. · 先把百分比换回计数——卡方用计数。
A bag of 100 candies claims 4 equally likely colors. What is the expected count per color? · 一袋 100 颗糖宣称 4 种等可能的颜色。每种颜色的期望计数是多少？
100 ÷ 4 = 25 expected per color. · 100 ÷ 4 = 每种期望 25。
Every chi-square test compares observed counts to ___ counts (one word). · 每个卡方检验都把观察计数与 ___ 计数作比较（填英文一词 expected）。
Observed vs. expected is the heart of chi-square. · 观察对期望是卡方的核心。