---
title: "chi-square"
type: concept
related: [Z Test, Chi Square, Hypothesis Testing, Homogeneity, Stastistic]
source: https://www.jemoka.com/posts/kbhchi_square/
confidence: high
status: active
---

\(\chi^2\) is a test statistic for hypothesis testing.
motivation for chi-square The motivation for chi-square is because t-test (means, &ldquo;is the value significantly different&rdquo;) and z-test (proportion, &ldquo;is the incidence percentage significantly different&rdquo;) all don&rsquo;t really cover categorical data samples: &ldquo;the categories are distributed in this way.&rdquo;
Take, for instance, if we want to test the following null hypothesis:
Category Expected Actual A 25 20 B 25 20 C 25 25 D 25 25 \(\alpha = 0.05\). What do we use to test this??
(hint: we can&rsquo;t, unless&hellip;)
Enter chi-square.
chi-square test chi-square test is a hypothesis test for categorical data. It is responsible to translate differences in distributions into p-values for significance.
Begin by calculating chi-square after you confirmed that your experiment meets conditions for inference (chi-square test).
Once you have that, look it up at a chi-square table to figure the appropriate p-value. Then, proceed with normal hypothesis testing.
Because of this categorical nature, chi-square test can also be used as a homogeneity test.
conditions for inference (chi-square test) random sampling expected value for data must be \(\geq 5\) sampling should be \(&lt;10\%\) or independent chi-square test for homogeneity The chi-square test for homogeneity is a test for homogeneity via the chi-square statistic.
To do this, we take the probability of a certain outcome happening&mdash;if distributed equally&mdash;and apply it to the samples to compare.
Take, for instance:
Subject Right Hand Left Hand Total STEM 30 10 40 Humanities 15 25 40 Equal 15 5 20 Total 60 40 100 We will then figure the expected outcomes:
Right Left 24 16 24 16 12 8 Awesome! Now, calculate chi-square with each cell of measured outcomes. Calculate degrees of freedom by (num_row-1)*(num_col-1).
chi-square test for independence The chi-square test for independence is a test designed to accept-reject the null hypothesis of &ldquo;no association between two variables.&rdquo;
Essentially, you leverage the fact that &ldquo;AND&rdquo; relationships are multiplicative probabilities. Therefore, the expected outcomes are simply the multiplied/fraction of sums:
calculating chi-square \begin{equation} \chi^2 = \frac{(\hat{x}_0-x_0)^2}{x_0} +\frac{(\hat{x}_1-x_1)^2}{x_1} + \cdots + \frac{(\hat{x}_n-x_n)^2}{x_n} \end{equation}
Where, \(\hat{x}_i\) is the measured value and \(x_i\) is the expected value.
