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The chi-square (χ2) statistic is a test that measures the comparison between a model and actual observed data.
The chi-square (χ2) statistic is a test that measures the comparison between a model and actual observed data. The data used in calculating the chi-square statistic must be random, original, mutually exclusive, and derived from independent variables, and the sample size must be large enough. For example, the outcome of tossing a fair coin meets these criteria.
The chi-square test is often used to test hypotheses. The chi-square statistic compares the size of any difference between expected results and actual results, taking into account sample size and number of variables.
In these tests, degrees of freedom are used to determine whether a null hypothesis can be rejected based on the variables and total number of samples in the experiment. As with any statistic, the larger the sample size, the more reliable the results.
There are two main types of chi-square tests, and they provide different information:
Note: Chi-square analysis is particularly useful when applied to categorical variables, when these variables are nominal (i.e., ordinal, such as marital status or gender).
The χ2 independence test can be used when considering student gender and course selection. To conduct this test, the researcher will collect data on two variables (gender and course chosen) and then compare the frequency with which boys and girls choose the courses offered using the above formula and the χ2 statistics table.
If there is no relationship between gender and course choice (i.e., they are independent), then the actual frequency of boys and girls taking each course should be approximately equal, or the proportion of boys and girls in each chosen course should be approximately equal to the proportion of boys and girls in the sample.
The χ2 independence test tells us how likely it is that random chance explains any observed differences between the actual frequencies in the data and these theoretical expectations.
In an independence test, a company might want to evaluate whether its new productan herbal supplement that claims to boost energyreaches the most likely to be interested.
It advertises on sports and fitness-related websites, assuming that active health-conscious people are most likely to buy. It conducted an extensive survey to gauge interest in the product among various demographic groups. The findings showed no correlation between interest in the product and the most health-conscious demographics.
χ2 provides a way to test how well the sample data match (known or assumed) characteristics of the larger population, which is called goodness of fit.
If the sample data do not match the expected characteristics of the population of interest, then the sample should not be used to draw conclusions about the larger population.
As an example of a goodness-of-fit test, a marketing professional is considering launching a new product that the company believes will appeal to women over the age of 45. The company has already tested the product with 500 potential buyers.
Marketing professionals had information on the test group's age and gender. This makes it possible to construct a chi-square test showing the age and gender distribution of people who said they would buy the product.
The results will show whether the most likely buyer is a woman over the age of 45. If testing shows that men over 45 or women ages 18 to 44 are equally likely to buy the product, marketing professionals will adjust the product's advertising, promotion and placement strategies to appeal to this broader group of customers.
For example, consider a fictional coin that has a 50/50 chance of coming up heads and tails, and you toss a real coin 100 times. If the coin is fair, then it will be equally likely to land on either side, and the expected outcome of tossing the coin 100 times is 50 heads and 50 tails.
In this case, χ2 tells us how the actual outcome of 100 coin tosses compares to the 50/50 outcome expected by the fair coin theory model. The actual result may be 50/50, it may be 60/40, or even 90/10.
The further the actual outcome deviates from 50/50, the worse the goodness of fit for this set of coin tosses, making it more likely to be concluded that the coin is not actually a fair coin.
The chi-square test is used to help determine whether observations are consistent with expected results and to rule out the possibility that observations are due to chance.
The chi-square test is appropriate when the data being analyzed comes from a random sample and the question variable is categorical. Categorical variables include choices such as car type, race, education, gender, or how much someone likes a political candidate (from very much liked to very much disliked).
This type of data is typically collected through survey responses or questionnaires, so chi-square analysis is particularly useful when analyzing this type of data.
Here are the basic steps for conducting a goodness-of-fit test or independence test:
The chi-square test is very sensitive to sample size. When using very large samples, a relationship may appear to be significant when in fact it is not.
Furthermore, the chi-square test cannot determine whether one variable has a causal effect on another variable. It can only determine whether two variables are related.
Chi-square is a statistical test that examines differences between categorical variables from random samples to determine how well the expected results fit the observed results.
Because chi-square works with categorical variables, it is most commonly used by researchers studying survey response data. This type of research can involve areas such as demography, consumer and market research, political science, and economics.
A nominal variable is a qualitatively classified variable whose numerical order may be insignificant. For example, asking someone about their favorite color yields a nominal variable. On the other hand, asking someone for their age produces an ordered data set. Chi-square works best with nominal data.
The chi-square statistic measures the difference between the observed and expected frequencies of a set of variables. It is particularly useful when analyzing differences in categorical variables, especially those of a nominal nature. Two different types of chi-square testsindependence tests and goodness-of-fit testswill answer different relationship questions.
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主要有两种卡方检验，它们提供不同的信息：
注意： 卡方分析应用于分类变量，特别是在这些变量为名义变量（即顺序无关，如婚姻状况或性别）时特别有用。
在考虑学生性别与课程选择时，可以使用 χ2 独立性检验。进行此测试时，研究者将收集关于两个变量（性别和选择课程）的数据，然后使用上述公式和 χ2 统计表比较男生和女生选择所提供课程的频率。
如果性别与课程选择之间没有关系（即它们是独立的），那么男生和女生选择每个课程的实际频率应大致相等，或者说在每个选定课程中的男生和女生比例应大约等于样本中男生和女生的比例。
χ2 独立性检验可以告诉我们随机机会解释数据实际频率与这些理论期望之间任何观察差异的可能性。
在独立性检验中，一家公司可能希望评估其新产品一种声称能提升能量的草本补充剂是否能接触到最可能感兴趣的人群。
它在与运动和健身相关的网站上做广告，假设积极的健康意识人群最有可能购买。它进行了一项广泛的调查，以评估各人口群体对该产品的兴趣。调查结果表明该产品的兴趣与最注重健康的人群之间没有相关性。
χ2 提供了一种测试样本数据如何匹配（已知或假定的）较大人群特征的方法，这被称为拟合优度。
如果样本数据与所关注的人群的预期特性不符，那么就不应使用该样本来对较大人群得出结论。
作为拟合优度检验的一个示例，营销专业人士正在考虑推出一款新产品，该公司认为将对45岁以上女性具有吸引力。公司已经对500名潜在买家进行产品测试。
营销专业人士掌握了测试小组的年龄和性别信息。这使得可以构建一个卡方检验，展示表示他们愿意购买该产品的人的年龄和性别分布。
结果将显示最有可能的买家是否为45岁以上的女性。如果测试显示45岁以上的男性或18至44岁的女性同样可能购买该产品，那么营销专业人士将会调整产品的广告、推广和摆放策略，以吸引这一更广泛的客户群体。
例如，考虑一个虚构的硬币，其正反面出现的机会各占50/50，而你用一枚真实硬币投掷100次。如果这枚硬币是公平的，那么它也将同样有可能落在任一面上，抛掷100次硬币的期望结果是正面出现50次，反面出现50次。
在这种情况下，χ2 可以告诉我们100次抛硬币的实际结果与公正硬币理论模型所预期的50/50结果之间的比较。实际结果可能为50/50，也可能为60/40，甚至90/10。
实际结果与50/50的偏差越大，这组抛硬币的拟合优度就越差，从而越可能得出该硬币实际上不是公正硬币的结论。
卡方检验用于帮助确定观察结果是否与预期结果一致，并排除观察结果是偶然造成的可能性。
当分析的数据来自随机样本，且问题变量是分类变量时，卡方检验是合适的。分类变量包括诸如汽车类型、种族、教育程度、性别或某人对政治候选人的喜欢程度（从非常喜欢到非常不喜欢）等选择。
这类数据通常通过调查响应或问卷收集，因此卡方分析在分析这类数据时尤其有用。
以下是进行拟合优度检验或独立性检验的基本步骤：
卡方检验对样本大小非常敏感。当使用非常大的样本时，关系可能看似显著，实际上却并非如此。
此外，卡方检验无法确定一个变量是否对另一个变量有因果关系。它只能确定两个变量是否相关。
卡方是一种统计检验，用于检查来自随机样本的分类变量之间的差异，以判断预期结果与观察结果的拟合优度。
由于卡方适用于分类变量，因此最常被研究调查响应数据的研究人员使用。这类研究可以涉及人口统计学、消费者和市场研究、政治学和经济学等领域。
名义变量是一种按质量分类的变量，其数字顺序可能是无关紧要的。例如，询问某人最喜欢的颜色会产生一个名义变量。另一方面，询问某人的年龄则会产生一个有序数据集。卡方可最佳应用于名义数据。
卡方统计量用于衡量一组变量的观察与期望频率之间的差异。它在分析分类变量的差异时尤其有用，尤其是那些名义性质的变量。两种不同类型的卡方检验独立性检验和拟合优度检验将回答不同的关系性问题。
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