📚 Chi-squared Tests for Edexcel A-Level Statistics | Edexcel A-Level 统计：卡方检验
Chi-squared tests are a major part of the Edexcel A-Level Statistics syllabus. They allow you to test whether categorical data fit a claimed distribution, or whether two categorical variables are associated. This article explains the key ideas, formulas, conditions and worked examples you need for exam success.
卡方检验是 Edexcel A-Level 统计课程的重要组成部分。它可以用来检验分类数据是否符合某个给定的分布，或者两个分类变量是否有关联。本文讲解考试成功所需的关键概念、公式、条件和例题。
1. What Is a Chi-squared Test? | 什么是卡方检验？
A chi-squared test is a hypothesis test for categorical data. It compares observed frequencies with expected frequencies to decide whether any difference is statistically significant. In Edexcel A-Level Statistics, you will meet two main types: the chi-squared goodness of fit test and the chi-squared test for association or independence.
卡方检验是针对分类数据的假设检验。它将观察频数与期望频数进行比较，以判断差异是否具有统计显著性。在 Edexcel A-Level 统计中，你会遇到两种主要类型：卡方拟合优度检验和卡方关联性（独立性）检验。
2. Hypotheses and Significance Levels | 假设与显著性水平
For a goodness of fit test, the null hypothesis states that the observed frequencies follow a claimed distribution. For a test for association, the null hypothesis states that the two categorical variables are independent. The alternative hypothesis says the model does not fit, or that the variables are not independent.
在拟合优度检验中，原假设指出观察频数服从某个给定的分布。在关联性检验中，原假设指出两个分类变量相互独立。备择假设则说明模型拟合不好，或者变量不独立。
The significance level, usually 5%, is the probability of rejecting the null hypothesis when it is actually true. You compare the calculated chi-squared statistic with a critical value from tables.
显著性水平通常为 5%，它是在原假设实际为真时拒绝原假设的概率。你需要将计算出的卡方统计量与表中的临界值进行比较。
3. Observed and Expected Frequencies | 观察频数与期望频数
Observed frequencies are the actual counts recorded in a sample. Expected frequencies are the counts you would expect if the null hypothesis were true. If the observed and expected frequencies are very different, the test statistic becomes large and suggests the model may not be correct.
观察频数是样本中实际记录的计数。期望频数是如果原假设成立时你预计会得到的计数。如果观察频数和期望频数相差很大，检验统计量就会变大，这表明模型可能不正确。
4. Degrees of Freedom | 自由度
The number of degrees of freedom depends on the test. For a goodness of fit test with k categories and no parameters estimated from the data, the degrees of freedom are k − 1.
自由度的数量取决于检验类型。对于拟合优度检验，若有 k 个类别且没有从数据中估计参数，则自由度为 k − 1。
For a test for association in an r × c contingency table, the degrees of freedom are (r − 1)(c − 1), where r is the number of rows and c is the number of columns.
对于列联表中的关联性检验，若表格为 r × c，则自由度为 (r − 1)(c − 1)，其中 r 是行数，c 是列数。
5. The Test Statistic | 检验统计量
The chi-squared test statistic is calculated by summing the squared differences between observed and expected frequencies, divided by the expected frequencies:
卡方检验统计量的计算方法是：将观察频数与期望频数之差的平方除以期望频数，然后求和：
X² = Σ (O − E)² / E
Here O represents an observed frequency and E represents the corresponding expected frequency. The sum is taken over all categories or cells in the table.
其中 O 表示观察频数，E 表示对应的期望频数。求和范围包括表格中的所有类别或单元格。
6. Goodness of Fit Tests | 拟合优度检验
A goodness of fit test is used when one categorical variable is measured and you want to test whether its distribution follows a specified set of proportions. For example, you could test whether a die is fair by rolling it 60 times.
拟合优度检验用于只有一个分类变量，且你想检验其分布是否符合一组给定比例的情形。例如，你可以掷骰子 60 次，检验骰子是否均匀。
The expected frequencies are found by multiplying the total number of observations by the claimed probability for each category. You then compute X², find the degrees of freedom, and compare with the critical value.
期望频数的计算方法是：将总观察次数乘以每个类别声称的概率。然后计算 X²，确定自由度，并与临界值比较。
7. Tests for Association with Contingency Tables | 列联表关联性检验
When two categorical variables are recorded, a contingency table shows the frequencies for each combination. The chi-squared test for independence asks whether the row variable and column variable are associated.
当记录两个分类变量时，列联表显示每个组合的频数。卡方独立性检验研究行变量和列变量是否有关联。
To find each expected frequency, multiply the row total by the column total and divide by the grand total:
计算每个期望频数时，用行合计乘以列合计，再除以总合计：
Expected frequency = (row total × column total) / grand total
This formula follows from the assumption of independence. Once all expected frequencies are found, compute X² and compare it with the critical value at (r − 1)(c − 1) degrees of freedom.
这个公式来源于独立性假设。求出所有期望频数后，计算 X²，并在自由度为 (r − 1)(c − 1) 下与临界值比较。
8. Conditions for Validity | 有效性条件
The chi-squared test requires a random sample and frequency data. The observations must be independent, and the data should be counts rather than percentages or measurements.
卡方检验要求随机样本和频数数据。观测值必须相互独立，数据应为计数，而不是百分比或测量值。
In Edexcel questions, you are usually required to check that all expected frequencies are at least 5. If any expected frequency is less than 5, you may need to combine adjacent categories in a goodness of fit test, or use a different approach for contingency tables.
在 Edexcel 的考题中，通常要求检查所有期望频数是否至少为 5。如果某个期望频数小于 5，可能需要在拟合优度检验中合并相邻类别，或对列联表采用其他处理方法。
9. Worked Example: Goodness of Fit | 例题：拟合优度检验
A die is rolled 60 times. The observed frequencies for faces 1 to 6 are 7, 11, 9, 8, 10 and 15. Test at the 5% significance level whether the die is fair.
一枚骰子被掷 60 次。1 到 6 点的观察频数分别为 7、11、9、8、10 和 15。在 5% 显著性水平下检验该骰子是否均匀。
If the die is fair, the expected frequency for each face is 60 ÷ 6 = 10. Then:
如果骰子均匀，每个面的期望频数为 60 ÷ 6 = 10。因此：
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