Unfortunately, I collected my data using different Likert scales instead of using a uniform standard scale. For example for Q1 of the questionnaire I have used a scale from 1 (Low) - 4 (High), Q2: 0 (Low) - 3 (High), Q3: 1 (Low) - 5 (High). Can I linearize the scales to one scale (e.g. 1 (low) - 4 (high)) without recollecting the data.? To be more specific, after linearizing the scales I will be able to compare mean (M) of Q1, Q2, and Q3 (my objective).
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$\begingroup$ I think this is backwards. If you decide that the mean is interesting and useful and defensible then you need to put variables on the same scale to compare their means. I'd still want to keep track of the frequency distributions. For example, you could show histograms or bar charts of all such variables but with shorter scales and longer scales scaled geometrically. $\endgroup$– Nick CoxCommented Apr 25, 2018 at 9:08
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$\begingroup$ Other relevant threads include stats.stackexchange.com/questions/117771/… $\endgroup$– Nick CoxCommented Apr 25, 2018 at 9:29
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$\begingroup$ Note any comparing any other measures apart from the minimum is fraught if scales are different in this way, so e.g. maximum will typically be different and medians may be hard to compare too. $\endgroup$– Nick CoxCommented Apr 25, 2018 at 9:57
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In principle you can do that. You will lose some information if your scales are of different length as you seem to imply (one of them has 5 possible answers instead of 4).
Be aware though that comparing means of Likert scales is often frowned upon for at least two reasons:
Does the mean have a real meaning here? E.g. if you have a sample of two people where one agrees and one disagrees are they on average neutral?
Likert scales are ordinal data, meaning that they can be ranked but not really manipulated in the way you would if you calculate a mean.
Mind you, the fact that it is frowned upon has not prevented many people from doing it.
Alternative ways of checking for differences after you brought the answers back to a similar scale are a chi-square test for association or if you have a small sample a Fisher exact test. These compare the distributions of the answers for two questions based on relative frequencies, and thereby avoid the problems mentioned above. On the other hand these tests ignore the order of the answers so you lose more information when you apply them.
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1$\begingroup$ Many universities run systems of averaging grades, in essence often ordered values on a coarse scale, even while several academics in some disciplines within are explaining to their students that this is a really bad idea. $\endgroup$– Nick CoxCommented Apr 25, 2018 at 9:10
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$\begingroup$ On your #1 the same remark could be made about heights, weights, blood pressure, inflation rates, etc. The mean of two different values will always be a number that doesn't correspond to a data point. I guess you mean to say that the average of grades 1 and 2 is 1.5 which is not even a possible data point, but so is the average of 1 children and 2 children for #children in a family. $\endgroup$– Nick CoxCommented Apr 25, 2018 at 9:15
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$\begingroup$ For more on means of ordinal variables see e.g. stats.stackexchange.com/questions/67551/… $\endgroup$– Nick CoxCommented Apr 25, 2018 at 9:18
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$\begingroup$ @NickCox I edited my answer. The grades issue you raise really depends on the country. We grade 1 to 10, typically uncurved in my country so the mean really is the mean. And yes for #1 the same applies to height but there I find it less of a problem. The point is that you work with opinions from people and it is often argued e.g. in hapiness research that averaging opinions is not a good measure and it is better to look at proportions. That being said I do mention in my answer that a lot of people still compare means. $\endgroup$ Commented Apr 25, 2018 at 9:25
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$\begingroup$ Also as I understand it, the finer the resolution the better this actually comes close to an interval scale. But a scale of 1 to 5 is rather coarse. $\endgroup$ Commented Apr 25, 2018 at 9:27