# Box-Cox Transformation on Single Variable and Interpretation of the Transformed Variable's Mean and Standard Deviation

I'm revisiting the Box-Cox transformation in one of my stats books, and I started playing around with the SAS macro %BCTRANS2 (Source: http://support.sas.com/resources/papers/proceedings12/430-2012.pdf). I applied that macro on a data set and it gave me a lambda value of 0.8. Using that value, I transformed my original variables and calculated the mean and standard deviation of those transformed variables and came up with the table below.

ID Orig Trans MuTrans SigmaTrans

01 0.19 0.26 0.39 0.21

02 0.69 0.74 0.39 0.21

03 0.74 0.79 0.39 0.21

04 0.68 0.73 0.39 0.21

05 0.62 0.68 0.39 0.21

06 0.31 0.39 0.39 0.21

07 0.47 0.55 0.39 0.21

08 0.56 0.63 0.39 0.21

09 0.09 0.15 0.39 0.21

10 0.28 0.36 0.39 0.21

11 0.32 0.40 0.39 0.21

12 0.18 0.25 0.39 0.21

13 0.38 0.46 0.39 0.21

14 0.22 0.30 0.39 0.21

15 0.05 0.09 0.39 0.21

16 0.09 0.15 0.39 0.21

17 0.22 0.30 0.39 0.21

18 0.13 0.20 0.39 0.21

19 0.13 0.20 0.39 0.21

20 0.10 0.16 0.39 0.21

21 0.33 0.41 0.39 0.21

22 0.42 0.50 0.39 0.21

23 0.04 0.08 0.39 0.21

24 0.25 0.33 0.39 0.21

25 0.59 0.66 0.39 0.21

26 0.18 0.25 0.39 0.21

27 0.49 0.57 0.39 0.21

28 0.45 0.53 0.39 0.21

29 0.20 0.28 0.39 0.21

Then, I got to thinking, "How could I use the mean and standard deviation to say, for example, observation 04 is N standard deviations away from the mean?" Do I have to back-transform MuTrans and SigmaTrans to obtain the correct interpretation? I've only dealt with Box-Cox transformations in a linear regression context; I've never transformed a single variable and asked to interpret it that way.

Any help is appreciated! Thank you.

• Assuming $\lambda=0.8$ corresponds to a Box-Cox parameter of $0.8,$ since that is so close to $1$ your transformation isn't accomplishing much of note. Why apply it at all? – whuber Jun 6 '18 at 18:27
• I agree. This is a play data set whose sole purpose was to test this macro, and I wasn't intending on thinking about the question I posed above. However, let's pretend that I did apply it, and I wanted to interpret the results correctly. How would I go about doing that? Would I back-transform or do something else to ensure the correct interpretation is obtained? – B.J. Guerrero Jun 6 '18 at 18:43
• One idea behind this transformation is that the numbers used to convey the data to you may be arbitrary. Possibly, the transformed numbers have more meaning than the originals. From this point of view there's little point to back-transforming: you start interpreting the data in terms of the transformed values. The use of logarithms in chemistry is a good example: often the log concentration is more meaningful than the concentration itself. – whuber Jun 6 '18 at 19:20
• Ah, I see. I'll have to do some more research about this, but the possibility of the Box-Cox-transformed numbers having more meaning than the originals is very interesting and something I never thought about. Thanks, whuber! How do I give you kudos? – B.J. Guerrero Jun 15 '18 at 0:20
• Thank you--it is enough to explain your insights and understanding to us. If, after doing your research, you have a similar or follow-up question, please don't hesitate to edit this post: that will bring it forward for renewed attention from the community. – whuber Jun 15 '18 at 13:22