Skip to content

residuals, standardized

(NL: residuen, gestandaardiseerde)

Standardized residuals are a measure to determine whether a category in a frequency or contingency table is over- or underrepresented in the sample.

A residual is the difference between the observed and the expected value. Standardizing means that the resulting value does not depend on the magnitude of the frequencies, the sample size, or the size of the frequency or contingency table.

\[ r_i = \frac{o_i-e_i}{\sqrt{e_i (1-\pi_i)}} \]

Where:

  • \(i\) = index of the category
  • \(r_i\) = standardized residual of category \(i\)
  • \(o_i\) = observed frequency
  • \(e_i\) = expected frequency
  • \(\pi_i\) = proportion of the \(i\)-th category in the sample (relative frequency)

You can interpret the value \(r_i\) as follows:

  • If \(r_i < -2\), then category \(i\) is underrepresented in the sample.
  • If \(-2 \leq r_i \leq 2\), then category \(i\) is proportionally represented in the sample.
  • If \(r_i > 2\), then category \(i\) is overrepresented in the sample.