Chapter 1 The Chi-squared distribution

Before we go too much further, we will look at the sampling distribution used for chi-squared tests: the chi-squared distribution. "Chi" is a Greek letter, \(\chi\), and is pronounced, "ky".

The chi-squared distribution is defined by the degrees of freedom (df). So, supposing a random variable \(X^2\) follows a chi-squared distribution, we would write this as \(X^2 \sim \chi^2_{\text{df}}\). If, for example, we had df = 5, we would write \(X^2 \sim \chi^2_5\). The below figure shows some example density curves of the \(\chi^2\) distribution for varying degrees of freedom:

As we can see, the chi-squared distribution is positively skewed, however as the degrees of freedom increases, the density curve begins to look flatter and more like a density that resembles the normal distribution. The chi-squared distribution only takes on positive values.

When we carry out a chi-squared test, the observed test statistic, \(\chi^2\) is placed within the context of the corresponding sampling distribution and we calculate the \(p\)-value as \(p = P(X^2 \geq \chi^2)\). This means that a large test statistic will result in a small \(p\)-value (and subsequently a significant result), whereas a small test statistic will result in a large \(p\)-value (and subsequently a non-significant result).

---

These notes have been prepared by Amanda J. Shaker. The copyright for the material in these notes resides with the author named above, with the Department of Mathematics and Statistics and with La Trobe University. Copyright in this work is vested in La Trobe University including all La Trobe University branding and naming. Unless otherwise stated, material within this work is licensed under a Creative Commons Attribution-Non Commercial-Non Derivatives License CC BY-NC-ND