Mathematics/Statistics/Chi-Square Test

The Chi-Square test, alternatively called $\chi ^{2}$ test, is used to measure two possible things:

• Chi-Square Goodness of Fit test - Determines if a set of sample data matches a larger population.
• Chi-Square Test for Independence - Determines if two variables are at all correlated.

General Notation

In general, Chi-Square is represented by a formula. The notations are as follows:

• O stands for "Observed/Actual".
• E stands for "Expected".
• Spans across some data set of size n.

With this in mind we have the following formulas:

Scientific Notation

Formal:

$\chi ^{2}=\sum _{i=1}^{n}{\frac {(O_{i}-E_{i})^{2}}{E_{i}}}$

Less Formal:

$\chi ^{2}=\sum _{i=1}^{n}{\frac {(Actual_{i}-Expected_{i})^{2}}{Expected_{i}}}$

Direct Notation

Formal:

$\chi ^{2}={\frac {(O_{1}-E_{1})^{2}+(O_{2}-E_{2})^{2}+...+(O_{n-1}-E_{n-1})^{2}+(O_{n}-E_{n})^{2}}{E_{1}+E_{2}+...+E_{n-1}+E_{n}}}$

Less Formal:

$\chi ^{2}={\frac {(Actual_{1}-Expected_{1})^{2}+(Actual_{2}-Expected_{2})^{2}+...+(Actual_{n-1}-Expected_{n-1})^{2}+(Actual_{n}-Expected_{n})^{2}}{Expected_{1}+Expected_{2}+...+Expected_{n-1}+Expected_{n}}}$