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Chi squared distribution

What Is Chi-Squared Distribution?

The chi-squared distribution (often denoted as (\chi^2)) is a continuous probability distribution that plays a fundamental role in inferential statistics and hypothesis testing. It is a specific type of gamma distribution that arises when independent standard normal distribution variables are squared and summed. This distribution is commonly used to analyze categorical data and test hypotheses about population variances, among other applications, falling under the broader financial category of Quantitative Analysis. The shape of the chi-squared distribution is determined by a single parameter known as its degrees of freedom.

History and Origin

The concept of the chi-squared distribution and its application to statistical testing was pioneered by Karl Pearson. In 1900, Pearson published a seminal paper introducing the chi-squared test, which provided a method to assess the "goodness of fit" between observed and expected frequencies in data. This groundbreaking work laid much of the foundation for modern hypothesis testing and the development of statistics as a distinct discipline. Pearson's chi-squared test, which relies on the chi-squared distribution, became one of his most significant contributions to statistical theory.10,9

Key Takeaways

  • The chi-squared distribution is a continuous probability distribution derived from the sum of squared standard normal variables.
  • Its shape is defined by its degrees of freedom, which increase with the number of independent variables contributing to the sum.
  • It is primarily used in the chi-squared test to evaluate discrepancies between observed frequencies and expected frequencies in categorical data.
  • The chi-squared distribution is a key component of several statistical tests, including tests of goodness of fit, independence, and homogeneity.
  • As the degrees of freedom increase, the shape of the chi-squared distribution approaches that of a normal distribution.

Formula and Calculation

The chi-squared distribution itself is characterized by its probability density function (PDF). However, its practical application often involves the calculation of a chi-squared test statistic, which then follows this distribution under certain conditions. For a chi-squared test of goodness of fit or independence, the test statistic ((\chi^2)) is calculated as:

χ2=∑i=1k(Oi−Ei)2Ei\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}

Where:

The sum is taken over all categories or cells. The resulting (\chi^2) value is then compared against a critical value from the chi-squared distribution with the appropriate degrees of freedom to determine statistical significance.

Interpreting the Chi-Squared Distribution

Interpreting the chi-squared distribution involves understanding its role in statistical inference. When performing a chi-squared test, the calculated (\chi^2) statistic is compared to a critical value from the chi-squared distribution corresponding to a chosen significance level and the relevant degrees of freedom. A higher calculated (\chi^2) value indicates a greater discrepancy between observed frequencies and expected frequencies. If the calculated (\chi^2) statistic exceeds the critical value, it suggests that the observed differences are unlikely to have occurred by chance, leading to the rejection of the null hypothesis. The associated p-value from the chi-squared distribution further quantifies the strength of evidence against the null hypothesis. The shape of the chi-squared distribution is asymmetric, starting at zero and extending indefinitely, becoming more symmetric as the degrees of freedom increase.8,

Hypothetical Example

Imagine a financial analyst wants to determine if the daily closing prices of a particular stock follow a uniform distribution across four price ranges (e.g., $0-$25, $25.01-$50, $50.01-$75, $75.01-$100) over a period of 100 trading days.

  1. Define Hypothesis:

    • Null Hypothesis ((H_0)): The stock's closing prices are uniformly distributed across the four price ranges.
    • Alternative Hypothesis ((H_A)): The stock's closing prices are not uniformly distributed.

  2. Collect Observed Frequencies ((O_i)):

    • Range 1 ($0-$25): 18 days
    • Range 2 ($25.01-$50): 32 days
    • Range 3 ($50.01-$75): 25 days
    • Range 4 ($75.01-$100): 25 days
    • Total observed days = 100

  3. Calculate Expected Frequencies ((E_i)):
    If uniformly distributed over 100 days across 4 ranges, each range would be expected to have (100 / 4 = 25) days.

    • Range 1 Expected: 25
    • Range 2 Expected: 25
    • Range 3 Expected: 25
    • Range 4 Expected: 25

  4. Calculate Chi-Squared Statistic:

    χ2=(18−25)225+(32−25)225+(25−25)225+(25−25)225χ2=(−7)225+(7)225+(0)225+(0)225χ2=4925+4925+0+0χ2=1.96+1.96=3.92\chi^2 = \frac{(18-25)^2}{25} + \frac{(32-25)^2}{25} + \frac{(25-25)^2}{25} + \frac{(25-25)^2}{25} \\ \chi^2 = \frac{(-7)^2}{25} + \frac{(7)^2}{25} + \frac{(0)^2}{25} + \frac{(0)^2}{25} \\ \chi^2 = \frac{49}{25} + \frac{49}{25} + 0 + 0 \\ \chi^2 = 1.96 + 1.96 = 3.92

  5. Determine Degrees of Freedom:
    For a goodness of fit test, degrees of freedom = (number of categories - 1) = (4 - 1 = 3).

  6. Compare to Critical Value:
    Assuming a significance level ((\alpha)) of 0.05, the critical value for a chi-squared distribution with 3 degrees of freedom is approximately 7.815. Since the calculated (\chi^2) of 3.92 is less than 7.815, the analyst would fail to reject the null hypothesis. This suggests that there isn't enough evidence to conclude the stock prices are not uniformly distributed across these ranges.

Practical Applications

The chi-squared distribution is integral to numerous applications across various fields, including finance, economics, and healthcare, particularly when dealing with categorical data.

  • Market Research: Analysts use chi-squared tests to determine if there's a relationship between consumer demographics (e.g., age groups, income levels) and preferences for different financial products or investment strategies. This can inform product development and targeted marketing.
  • Risk Management: In assessing credit risk, banks might use a chi-squared test to see if there's an association between a borrower's credit score category and their loan default status. This helps in refining credit scoring models.
  • Auditing and Compliance: Auditors can apply the chi-squared goodness of fit test to determine if observed patterns in transaction data (e.g., frequency of certain types of errors) conform to expected distributions, helping to identify anomalies or potential fraud. This is part of financial forensics.
  • Public Health and Epidemiology: The chi-squared test is widely used to analyze the relationship between categorical variables, such as exposure to a risk factor and the incidence of a disease. For instance, it can assess if there's an association between smoking status and the prevalence of lung cancer.7 This highlights its importance in evidence-based decision-making in sectors like healthcare.6
  • Portfolio Analysis: While primarily dealing with quantitative data, a chi-squared test could, in specific scenarios, assess if the observed distribution of portfolio returns across certain predefined categories (e.g., positive, negative, neutral) deviates significantly from a theoretical or historical distribution. This can be a rudimentary check in portfolio theory.
  • Regulatory Compliance: Regulators might use the chi-squared distribution to test if the distribution of reported data from financial institutions adheres to certain regulatory expectations or benchmarks, ensuring data integrity.

Limitations and Criticisms

Despite its widespread utility, the chi-squared distribution and its associated tests have several limitations and areas for criticism. A primary concern is the requirement for a sufficiently large sample size and adequate expected frequencies in each cell of the contingency table. If these conditions are not met (e.g., many cells have expected counts less than five), the chi-squared approximation to the true underlying distribution may be unreliable, leading to inaccurate p-values and an increased risk of Type I errors (false positives).5,4

Another limitation is its sensitivity to violations of the assumption of independence among observations. If data points are not independent, traditional chi-squared tests can produce artificially low p-values, leading to erroneous conclusions.3 Furthermore, the chi-squared test only indicates whether a statistically significant relationship exists; it does not measure the strength or direction of that relationship. For very large sample sizes, even very small, practically insignificant differences can yield a statistically significant chi-squared result, underscoring the need for accompanying effect size measures. The chi-squared test is also designed for nominal or ordinal categorical data and is not appropriate for continuous variables without prior categorization, which can lead to a loss of information.2

Chi-Squared Distribution vs. Chi-Squared Test

It is crucial to distinguish between the chi-squared distribution and the chi-squared test. The chi-squared distribution is a theoretical probability distribution that describes the probabilities of different chi-squared values occurring given a certain number of degrees of freedom. It's the mathematical framework against which results are compared. The chi-squared test, on the other hand, is a non-parametric statistics hypothesis test that calculates a statistic whose distribution approximates the chi-squared distribution under the null hypothesis. This test is used to evaluate relationships between categorical data or to determine how well observed data fit an expected distribution. In essence, the chi-squared distribution is the theoretical tool, while the chi-squared test is the practical application that leverages this tool for statistical inference.

FAQs

What is the primary purpose of the chi-squared distribution in statistics?

The primary purpose of the chi-squared distribution is to serve as a reference distribution for various statistical tests, particularly those involving categorical data. It allows statisticians to determine the probability of observing a particular test statistic value under the null hypothesis.

How does the number of degrees of freedom affect the chi-squared distribution?

The number of degrees of freedom is the sole parameter that defines the shape of the chi-squared distribution. As the degrees of freedom increase, the distribution shifts to the right and becomes more symmetrical, approaching the shape of a normal distribution.

Can the chi-squared distribution have negative values?

No, the chi-squared distribution cannot have negative values. It is a sum of squared terms, and squared real numbers are always non-negative. Therefore, the chi-squared distribution always starts at zero and extends into positive values.1

What is a "goodness of fit" test using the chi-squared distribution?

A goodness of fit test using the chi-squared distribution evaluates how well an observed sample distribution of categorical data matches a theoretical or expected frequencies distribution. For example, it can test if customer preferences for different investment platforms are uniformly distributed or follow a specific pattern.