# How to check Likert scale summaries for plausibility

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Suppose you are reading a paper that uses Likert scale responses. The paper reports the mean, standard deviation, and number of responses. If we are — for some reason — suspicious of a paper, we might ask, “Are these summary statistics possible for this number of responses, for this Likert scale?” Someone asked me this recently, so I wrote some simple code to help check. In this blog post, I outline how the code works.

Suppose we are reading a paper that uses a 5-category Likert scale response (0-7) and they report that for 100 responses, the mean response was .5, and the standard deviation of the responses was 5. I have intentionally chosen these numbers to be impossible: for the mean to be .5, the responses have to be on average near the bottom of the scale. But if the responses are near the bottom of the scale, the standard deviation also has to be very low, because there is a bound at 0. The standard deviation of 5 is much to large for the mean.

Another possible inconsistency arises due to the discreteness of Likert scales. For 10 Likert responses on a 3 point scale (0-2), the mean must be a multiple of .1. This, in turn, imposes a complicated constraint on the standard deviation.

Checking whether a response pattern is possible may be simple for low N, but it gets complex as N increases. Suppose we are wondering, for a 6-item Likert scale (0-5), whether N=94, M=.83, and SD=1.21 are possible summary statistics. There are several ways we could go about this using R.

The code for both is available here: https://gist.github.com/richarddmorey/787d7b7547a736a49c3f

### Option 1: Brute force

> res[1:10,] |

### Option 2: Linear Inverse Models

- Proportions must sum to 1 (equality constraint).
- Our summary statistics should be as close as possible to our target summaries (approximate equality constraint)
- Proportions must be between 0 and 1 (inequality constraint).

> xs <- limSolve::xsample(A = A, B = B, E = E, F = F, G = G, H = H, sdB = 1) |

> xs$X[1:10,] |

> xs$X[1:10,]*94 |

### Conclusion

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