# Transforming different Likert scales to a common scale

## Problem

I have data from a questionnaire which used a 5-point Likert scale last year. But, although the questions are the same, this year the answers were recorded using a 7-point Likert scale. I want to analyze the combined data from both years. How can I transform the data to a common scale?

## Resolving The Problem

Here is how to easily find the right linear transformation to convert one Likert scale to another. This is best done in two stages. Notice that a Likert scale is determined by its minimum, which is usually 1, and its maximum, for example 5. First, find the linear transformation so that in the new scale, the minimum is 0 and the maximum is 1. Second, find the transformation which undoes this. That is, starting with a scale with a minimum of 0 and a maximum of 1, transform it so it has whatever minimum and maximum is required.

It is easy to check that for a scale with minimum a and maximum b, the transformation

X = (x - a) / (b - a)

is the one we want. Just substitute a for x to see that the result is 0, and then substitute b for x to see that the result is 1.

To go in the other direction, let's say we want the new minimum to be A and the new maximum to be B. The transformation we want is

Y = (B - A) * X + A

Substitute 0 for X to see that the result is A, and 1 for X to see that the result is B.

Now to put this all together, substitute the whole first transformation in place of X in the second:

Y = (B - A) * (x - a) / (b - a) + A.

That looks just a little messy, but let's apply it to the example of a 5-point scale to be converted to a 7-point scale. Since the minimum of the 5-point scale is 1, we have a=1, b=5 in the first transformation. Similarly for the second transformation, we have A=1, B=7. Putting them together we get:

(7 - 1) * (x - 1) / (5 - 1) + 1

Of course this looks a lot less scary if we do the subtractions:

6 * (x - 1) / 4 + 1

A little rearrangement gives:

(6/4) * x - (6/4) + 1

A little more rearrangement and we get:

1.5 * x - 0.5

So in SPSS we just need

COMPUTE x2 = 1.5 * x1 - 0.5 .
EXECUTE.

You can check the results on a small dataset:

x1 x2
1 1.0
2 2.5
3 4.0
4 5.5
5 7.0

In particular, notice that 1 is sent to 1, and 5 to 7.

You should be able to quickly work through the formula to convert a 4-point into a 5-point scale:

(5 - 1) * (x - 1) / (4 - 1) + 1 = 4 * (x - 1) / 3 + 1 = (4/3) * x - (4/3) + 1 = (4/3) * x - (1/3)

Again, notice that if we substitue x=1 the result is 1, and x=4 results in 5.

To convert a 5-point to a 4-point scale:

(4 - 1) * (x - 1) / (5 - 1) + 1 = 3 * (x - 1) / 4 + 1 = (3/4) * x - (3/4) + 1 = (3/4) * x + (1/4)

As intended, x=1 results in 1, and x=5 results in 4.

## Related Information

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79980

Modified date:
16 April 2020

swg21482329