I don't think means would mean much here, so what basic summary statistics are considered usefull?
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A frequency table is a good place to start. You can do the count, and relative frequency for each level. Also, the total count, and number of missing values may be of use.
You can also use a contingency table to compare two variables at once. Can display using a mosaic plot too.
I'm going to argue from an applied perspective that the mean is often the best choice for summarising the central tendency of a Likert item. Specifically, I'm thinking of contexts such as student satisfaction surveys, market research scales, employee opinion surveys, personality test items, and many social science survey items.
In such contexts, consumers of research often want answers to questions like:
For these purposes, the mean has several benefits:
1. Mean is easy to calculate:
2. Mean is relatively well understood and intuitive:
3. The mean is a single number:
4. It doesn't make much difference
For basic summaries, I agree that reporting frequency tables and some indication about central tendency is fine. For inference, a recent article published in PARE discussed t- vs. MWW-test, Five-Point Likert Items: t test versus Mann-Whitney-Wilcoxon.
For more elaborated treatment, I would recommend reading Agresti's review on ordered categorical variables:
It largely extends beyond usual statistics, like threshold-based model (e.g. proportional odds-ratio), and is worth reading in place of Agresti's CDA book.
Below I show a picture of three different ways of treating a Likert item; from top to bottom, the "frequency" (nominal) view, the "numerical" view, and the "probabilistic" view (a Partial Credit Model):
The data comes from the
Conventional practice is to use the non-parametric statistics rank sum and mean rank to describe ordinal data.
Here's how they work:
M/R is a more sophisticated statistic than R/S because it compensates for unequal sizes in the groups you are comparing. Hence, in addition to the steps above, you divide each sum by the number of members in the group.
Once you have these two statistics, you can, for instance, z-test the rank sum to see if the difference between the two groups is statistically significant (I believe that's known as the Wilcoxon rank sum test, which is interchangeable, i.e., functionally equivalent to the Mann-Whitney U test).
R Functions for these statistics (the ones I know about, anyway):
wilcox.test in the standard R installation
meanranks in the cranks Package
Based on the abstract This article may be helpful for comparing several variables that are Likert scale. It compares two types of non-parametric multiple comparison tests: One based on ranks and one based on a test by Chacko. It includes simulations.
I agree with Jeromy Anglim's evaluation. Remember that Likert responses are estimates — you are not using a perfectly reliable ruler to measure a physical object with stable dimensions. The mean is a powerful measure when using reasonable sample sizes.
In business and product R&D, the mean is by far the most common statistic used with Likert scales. When using Likert scales I have usually chosen a measure that ideally fits the research question. For instance, if you are talking about "preference" or "attitudes" you can use multiple Likert-based indicators, with each indicator providing slightly different insight.
To evaluate the question "how will people in segment $i$ react to service offering $X$," I may look at (1) arithmetic mean, (2) exact median, (3) percentage most favorable response (top box), (4) % top two boxes, (5) ratio of top two boxes to bottom two boxes, (6) percentage within mid-range boxes... etc. Each measure tells a different piece of the story. In a very critical project, I use multiple Likert-based indicators. I will also use multiple indicators with small samples and when a specific cross tab has an "interesting" structure or looks information-rich. Ahhh... the art of statistics.
I usually like to use Mosaic plot. You can create them by incoorporating other covariates of interest (such as: sex, stratified factors etc.)
"Box scores" are often used to summarize ordinal data, particularly when it comes with meaningful verbal anchors. In other words, you might report "top 2 box", the percentage that chose either "agree" or "strongly agree".