When comparing different papers it might be that the papers have numbers about the same thing, but that the numbers are on different scales. Forr example, many different questionnaires exists measuring the same constructs such as the NEO-PI and the BFI both measure the Big Five personality traits. Say, we want to compare reported means and standard deviations (SDs) for these questionnaires, which both use a Likert scale.
In this post, the equations to rescale reported means and standard deviations (SDs) to another scale are derived. Before that, an example is worked trough to get an intuition of the problem.
In this post, I stick to the set theory convention of denoting sets by uppercase letters. So, denotes the number of items in the set and denotes the absolute value of the number . To say that predicate holds for all elements in , I use the notation , for example: if contains all integers above 3, then we can write .
For some study, let the set of participants and questions be respectively denoted by and with and . Let the set of responses be denoted by with and let denote the set of the summed scores per participant, that is, , see the table below.
Let and denote respectively the reported mean and sample SD. We assume that the papers calculated the mean and SD with
Note here that Bessel's correction is applied, because instead of . This seems to be the default way to calculate the standard deviation.
Lets consider one study consisting of only one question and three participants. Each response is an integer () in the range [1, 3], that is, . So, the lower and upper bound of are respectively and .
We can rescale these numbers to a normalized response in the range [0, 1] by applying min-max normalization,
The rescaled responses become
Now, suppose that the study would have used a scale in the range [0, 5]. Let these responses be denoted by . We can rescale the normalized responses in the range [0, 1] up to in the range [0, 5] with
This results in
Since we know all the responses, we can calculate the means and standard deviations:
Now, suppose that was part of a study reported in a paper and the scale of was the scale we have for our own study. Of course, a typical study doesn't give us all responses . Instead, we only have and and want to know and . This can be done by using the equations derived below. We could first normalize the result, by Eq. (14),
and, by Eq. (15),
Next, we can rescale this to the range of . By Eq. (16),
and, by Eq. (17),
We could also go from to in one step. By Eq. (18),
and, by Eq. (19),
Consider a random variable with a finite mean and variance, and some constants and . Before we can derive the transformations, we need some equations to be able to move and out of and .
For the mean, the transformation is quite straightforward,
Note that the position of constant makes intuitive sense: for example, if you add a constant to all the elements of a sample, then the mean will move by . To scale the standard deviation, we can use the equation for a linear transformation of the variance (Hogg et al. (2018)),
We can use this to derive that
Next, we derive the equations for the transformations. Let and be respectively the lower and upper bound for the Likert scale over all the answers; specifically, . Let and be respectively the lower and upper bound for the Likert scale per answer; specifically, . Now, for the normalized mean ,
and for the normalized SD ,
where since we know that both are positive and .
To change these normalized scores back to another scale in the range , we can use
We can also transform the mean and SD into one step from the range to with
Hogg, R. V., McKean, J., & Craig, A. T. (2018). Introduction to mathematical statistics. Pearson Education.