What Is the Chi Square Formula?

The chi-square statistic is a fundamental quantity in inferential statistics, particularly in hypothesis tests involving categorical data. It quantifies the discrepancy between observed frequencies and the frequencies expected under a specific hypothesis.

Mathematical Expression for the Chi-Square Statistic

Let $O_i$ denote the observed frequency for the $i$-th category, and $E_i$ denote the expected frequency for the $i$-th category under the null hypothesis. The chi-square statistic, denoted as $\chi^2$, is defined by the formula:

\[\chi^2 = \sum_{i=1}^k \frac{(O_i - E_i)^2}{E_i}\]

Here, $k$ is the total number of mutually exclusive categories. Each term in the summation quantifies the squared standardized difference between observed and expected frequencies for one category.

Determination of Expected Frequencies in Contingency Tables

In a contingency table of $r$ rows and $c$ columns, the expected frequency $E_{ij}$ for cell $(i, j)$ is calculated under the assumption of independence as follows:

\[E_{ij} = \frac{(\text{Row total for }i) \times (\text{Column total for }j)}{N}\]

Here, $N$ denotes the total number of observations, while the row and column totals refer to the respective sums over the observed frequencies. For more principles on this calculation, refer to Statistics and Probability Overview.

Interpretation of the Chi-Square Value in Hypothesis Testing

The computed value of $\chi^2$ can be used to evaluate the null hypothesis by comparing it with the tabulated critical value from the chi-square distribution with an appropriate degrees of freedom, given by:

\[\text{Degrees of freedom (df)} = (r-1) \times (c-1)\]

For a goodness-of-fit test with $k$ categories, the degrees of freedom is $k-1$.

Calculation of the p-Value in the Chi-Square Test

Once the statistic $\chi^2$ is computed, the p-value is determined as the probability that a value at least as extreme as the observed statistic occurs under the null hypothesis, i.e., $P(\chi^2_{\text{df}} \geq \chi^2_{\text{obs}})$. A low p-value (commonly below 0.05) provides significant evidence against the null hypothesis. A high p-value indicates consistency with the null hypothesis.

Detailed Example: Computing the Chi-Square Statistic for Observed vs. Expected Car Ownership

Given: The distribution of $n=50$ families according to the number of cars they own:

One car: $O_1 = 30$, $E_1 = 25.6$
Two cars: $O_2 = 14$, $E_2 = 15.1$
Three cars: $O_3 = 6$, $E_3 = 5.2$

For each category, calculate the contribution to $\chi^2$:

\[ \frac{(O_1 - E_1)^2}{E_1} = \frac{(30 - 25.6)^2}{25.6} = \frac{19.36}{25.6} = 0.75625 \] \[ \frac{(O_2 - E_2)^2}{E_2} = \frac{(14 - 15.1)^2}{15.1} = \frac{1.21}{15.1} = 0.08013 \] \[ \frac{(O_3 - E_3)^2}{E_3} = \frac{(6 - 5.2)^2}{5.2} = \frac{0.64}{5.2} = 0.12308 \]

Sum these contributions:

\[ \chi^2 = 0.75625 + 0.08013 + 0.12308 = 0.95946 \]

Result: $\chi^2 = 0.96$ (to two decimal places).

For the tabulated critical value at appropriate degrees of freedom and the desired significance level, the conclusion on the null hypothesis can be made by comparison. For further computational tools, see Chi Square Formula.

Stepwise Expansion: Derivation of the Chi-Square Statistic

Consider a discrete random experiment categorized into $k$ mutually exclusive classes. Under the null hypothesis $H_0$, let the expected frequency for the $i$-th class be $E_i = N p_i$ where $p_i$ is the expected probability of the $i$-th class, and $N$ is the sample size. For $i = 1, 2, \ldots, k$:

Step 1: Compute the difference between observed and expected values:

\[ d_i = O_i - E_i \]

Step 2: Square this difference:

\[ d_i^2 = (O_i - E_i)^2 \]

Step 3: Normalize the squared difference by dividing by $E_i$:

\[ \frac{d_i^2}{E_i} = \frac{(O_i - E_i)^2}{E_i} \]

Step 4: Sum over all categories to obtain the chi-square statistic:

\[ \chi^2 = \sum_{i=1}^k \frac{(O_i - E_i)^2}{E_i} \]

Critical Use Cases and Statistical Considerations of the Chi-Square Statistic

The chi-square test is employed for evaluating categorical data in cases such as the goodness-of-fit test, test of independence in a contingency table, and test of homogeneity. Valid statistical inference requires the data to be in frequency form (actual counts, not percentages), and for each expected frequency $E_i$ to be sufficiently large (generally $E_i \geq 5$) for the approximation to the chi-square distribution to remain accurate. For deeper foundational understanding, see Understanding Probability.

Frequently Examined Points on the Chi-Square Formula

The chi-square formula presupposes mutually exclusive categories. Its statistic is always non-negative since each squared difference is non-negative. A small value of $\chi^2$ affirms close alignment between observed and expected frequencies. A large value suggests a statistically significant discrepancy, frequently leading to the rejection of the null hypothesis at the chosen significance level. The exact threshold for rejection is determined via the chi-square distribution table based on the test’s degrees of freedom.

The chi-square procedure is widely used in biology, genetics, and medical sciences for inference regarding categorical data, as well as in numerous statistical methods for hypothesis testing involving qualitative data. For application to distributions and combinatorial data, see Permutations and Combinations.