What Distribution Actually Describes a 1–5 Rating Scale?
We asked survey respondents to rate, on a scale of 1 to 5, how likely they are to be upset by criticism. The average response came out to 3.5 — right in the middle-upper range. But an average alone doesn't tell you much about the shape of people's opinions: are responses clustered tightly around the middle, spread out toward the extremes, or skewed toward one end? To answer that, we fit eight candidate probability distributions to the raw response data using maximum likelihood estimation (MLE), then compared them using log-likelihood, AIC, and BIC. Now, being a 1-to-5 scale with human responses, the data won't truly fit any theoretical distribution perfectly, but it's interesting to see which distribution would fit it best, to serve as an approximate (but simple) model for the data.
To analyze the data, we performed the MLE on Hypothesize.io, our sister project web app that assists users who would like to perform sophisticated statistical analyses online, even if they don’t have in-depth knowledge of the analyses themselves. To perform your own Maximum Likelihood Estimation analysis with Hypothesize, you can click here.
The Results
The Binomial distribution (shown in red in the chart) was the clear winner, with parameters n = 5 and p = 0.7, an AIC of 588.6, and a BIC of 591.9 — meaningfully better than every other candidate. The next-best fits were the Normal distribution (AIC 608.0) and the Gamma distribution (AIC 653.1), both roughly 20–65 points worse. Poisson and Negative Binomial, despite sharing the same mean of 3.5, trailed well behind (AIC ≈ 695–699), and Exponential was a poor fit across the board. The Beta distribution couldn't even be estimated, since it requires data on a continuous (0, 1) interval rather than discrete integer ratings.
Of course, we don't expect this result to hold for all rating scale responses, as it will depend on the rating scale itself and the underlying distribution. This is just intended to be a specific example of how this MLE approach can be used.
Why the Binomial Distribution Wins In This Case
This result actually makes a lot of intuitive sense once you think about what a 5-point Likert scale really is. A binomial distribution models the number of "successes" out of a fixed number of independent trials. That's not really what is going on with survey responses, but with n = 5 "trials" and a success probability of p = 0.7, the binomial naturally produces a distribution that's bounded between 0 and 5, skewed toward the higher end (like our data is), and centered near 3.5 (since the binomial mean is n × p = 5 × 0.7 = 3.5).
That boundedness is a key advantage over the runner-up models. The Normal distribution is unbounded and continuous, so it technically allows for impossible values like a rating of 7 or -1, which costs it some likelihood. The Poisson and Negative Binomial distributions are discrete and match the mean well, but they assume an unbounded count (there's no natural ceiling), so they spread probability mass out toward ratings that can never actually occur on this scale. The binomial, by contrast, respects the actual structure of the data: a fixed, small number of discrete, bounded outcomes.
The Takeaway
When you're analyzing ordinal survey data like a 1–5 rating scale, it's tempting to default to a Normal distribution because it's the most familiar. But this analysis is a good reminder that treating a bounded, discrete scale as if it were continuous and unbounded can quietly bias your model. Here, the binomial's AIC advantage of roughly 20 points over the Normal distribution isn't negligible — it reflects a real, better-supported structural interpretation of the data: respondents behaving as if they're accumulating "votes" of agreement across 5 discrete opportunities, with a lean toward being upset by criticism (p = 0.7, well above the neutral midpoint of 0.5). That isn't, of course, what's really underlying the way people answer Likert scale questions, but it serves as a pretty good model for the distribution.
The broader lesson: before fitting a model to survey data, ask what constraints it has. Matching the distribution's mechanics to the real constraints of your measurement — bounded, discrete, ordinal — often beats reaching for the most familiar bell curve.
