When Likert scale data are subjected to statistical analyses, the normal distribution is usually assumed as underlying distribution. Alternatively nonparametric statistical techniques are applied. Other techniques like polychoric correlation make the assumption that the Likert scale divides the sample space of the normal distribution into intervals. In this paper, an alternative distribution is proposed that is based on the normal distribution, but the sample space is discrete, and can take on only the values of the Likert scale. This distribution has two parameters similar to the normal distribution. It however differs from the normal distribution in that the shape of the distribution is dependent on both parameters.
Maximum likelihood estimators for the two parameters are shown, and some desirable properties of the distribution discussed. There are theoretical aspects of the distribution that remains to be researched, and the purpose of this paper is to present the initial concept, and test its acceptability amongst peers.
Results from a study on real world Likert scale data indicated that in 67% of goodness-of-fit tests that were done, the Likert distribution provided an acceptable fit on a 5% significance level.
A test statistic based on the Likert distribution is proposed for comparing means of two groups, and results from a comprehensive simulation study indicated superior power over the standard t-test for small samples.
Keywords: Likert scale; discrete distribution; ordinal
Biography: Martin Kidd is director of the Centre for Statistical Consultation at the University of Stellenbosch, South Africa. He obtained his PhD in statistics at the university of Stellenbosch in 1991, and spent 15 years in the Naval maritime research field before accepting a post as consultant at Stellenbosch university in 2001.