Is the mean of 1-5 Likert satisfaction responses a legitimate summary?

## The setup A Likert item offers a small set of ordered response options — for example `strongly dissatisfied`, `dissatisfied`, `neutral`, `satisfied`, `strongly satisfied` — which are stored as 1 through 5 for convenience. The question is whether that storage convenience licenses arithmetic. ## Why it is ordinal The respondent chose a *category*. The categories are ordered, so we know 4 sits above 3, which sits above 2. What we do not know is how far above. The psychological distance between `neutral` and `satisfied` — the step where a person crosses from indifference into approval — may be much larger, or much smaller, than the distance between `satisfied` and `strongly satisfied`. Respondents also differ from each other in how they use the endpoints; some never pick 1 or 5 at all. None of that is captured by the integers. An interval scale requires equal, meaningful gaps. A Likert item gives you order and nothing more, so by the rules of measurement scales the mean is not defined on it. ## The decisive demonstration Because an ordinal scale is defined only up to an order-preserving relabelling, any legitimate statistic must be unchanged when you relabel while keeping the order. Try it. Take ten responses: four 5s, three 4s, two 3s and one 1. - Coded 1, 2, 3, 4, 5, the mean is `(4*5 + 3*4 + 2*3 + 1*1) / 10 = 39 / 10 = 3.9`. - Recode the options as 1, 2, 3, 4, 10 — still strictly increasing, so the ordinal content is identical. The mean becomes `(4*10 + 3*4 + 2*3 + 1*1) / 10 = 59 / 10 = 5.9`. The responses did not change. The mean did. Now check a statistic the scale actually supports: the median response is the fourth option in both codings, and the proportion answering in the top two options is 0.7 in both codings. Those are properties of the data; the mean is a property of your spreadsheet. ## Why people compute it anyway The practice is widespread, and the arguments for it are not empty: - It compresses a distribution into one trackable number, which is what a weekly dashboard wants. - Movement in the mean does usually reflect movement in the distribution, so as a *trend* signal it is often serviceable. - Multi-item scales are a stronger case: when several related items are averaged into one composite intended to measure a single underlying construct, the composite takes many more distinct values and behaves much more like an interval quantity than any single item does. Analysing composites as interval data is standard practice in survey research and is far more defensible than averaging one item. The honest position is therefore not "never compute it" but "know what you are assuming, and never let it travel alone". ## What the mean hides One number cannot distinguish a population that is uniformly lukewarm from one that is sharply split. Five respondents all answering 3, and a group where half answer 1 and half answer 5, can both produce a mean of 3 while describing completely different situations — and the second one is the situation you needed to act on. Reporting the mean without the response counts throws away exactly the structure that makes survey data worth collecting. ## What to report instead - **A frequency and proportion table** over all five options: each option with its count and its share. This is fully licensed by an ordinal scale and preserves the whole distribution. - **Top-two-box and bottom-two-box proportions**: the share choosing the two most positive options, and the share choosing the two most negative. These are proportions of clearly-defined groups, they survive any recoding, and they are directly interpretable by non-technical readers. - **The median response**, which needs only order. - **The sample size**, always, since survey subgroups get small fast. ## A ratio trap to avoid Even if you accept the mean, do not divide two of them. "Our score rose from 3.5 to 4.2, a 20 percent improvement" imports a true zero that the scale does not have — the coding could have started at 0 or at 10 with the same responses, and the percent change would differ. Report the movement in the units of the scale, or better, report the movement in the top-two-box proportion, which is a genuine proportion and supports ratio statements. ## How to answer this in an interview State the scale, state the assumption that averaging makes, give the recoding demonstration in one sentence, then say what you would actually put in the report. Interviewers are looking for a candidate who neither parrots "you can never average ordinal data" nor shrugs and averages it without noticing — the credible answer is that you know the assumption, you disclose it, and you always ship the distribution beside it.