Hi,

I usually dont do back transformation as the log transfermation at the begaining would shrink the differences among your observations thus shrinking the sd as well, so when you transfer back, you may not get what you thought you could get. I would just interpret  as the log of x is balabala. But i guess different people have different  ways to deal with it

Dear Timothy,

We use normalizing transformations and the back transformations (the transformations are inverse to the normalizing transformations) for solving some problems, for example, anomaly detection in non-Gaussian data. I hope the link below will help to solve your problem.

https://www.researchgate.net/publication/282329599_Statistical_anomaly_detection_techniques_based_on_normalizing_transformations_for_non-Gaussian_data

In additionally, as the normalizing transformation, we recommend to use the Johnson transformation (the Johnson translation system) that has been applied by us for solving many problems.

Conference Paper Statistical anomaly detection techniques based on normalizin...

@Yongdong,

you are right that different people have different ways to deal with transformation.

how about a new approach which provides new descriptives in statistics, instead of transformation

Article TO DETERMINE SKEWNESS, MEAN AND DEVIATION WITH A NEW APPROAC...

I've not had problems doing back-transformations, for the purpose of data presentation. The main issue has been getting the formulas (to do so) right.

-Bob

@Bob, I used a log transformation. What is the right formula to back-transform? My simple attempt seemed to fail. The question could be asked in a different way. I am going out to the field to measure fruit weight. The only help I have is from a single paper that reported back-transformed (log (fruit weight)) values. I gather my fruit weights, find the average and discover that the published value and my value are not very similar. Is the published value wrong?

If you used a log10(x) transformation, then the back-transformation is 10**(x) . But if you used ln(x), then go w/ e**(x).

Re: the published back-transformed values, such mean values won't match the untransformed mean unless the data are perfectly Gausian ('normal'), b/c you transform the data before taking the average. Hence, the back-transformed means should be less biased than if no transformation was used. Did I answer your question?

-Bob Vadas, Jr.

Oops, I need to be more exact in my symbology. Call the transformed value y. So if you used a log10(x) transformation, then the back-transformation is 10**y. But if you used ln(x), then go w/ e**y.

It gets trickier if you used a you used a log10(x+1) transformation, b/c the back-transformation is (10**y) -1. Likewise, if you used ln(x+1), then go w/ (e**y) - 1.

-Bob Vadas, Jr.

Could you elaborate on the bias that the back transformed values have less of?

Could you please elaborate of the nature of the bias that you refer to?

If a parameter isn't normally distributed, then the untransformed ave. will be biased, b/c that violates the Gaussian assumption when you calculate an arithmetic mean. Let's say that there's 1 high outlier, then such a mean will overestimate the central tendency. But if you take a log transformation, then the back-transformed mean will be lower in value & not overly influenced by the high outlier. Does that make sense? Feel free to try this out on data sets at home!

With 1 high outlier I have a few options. It is a mistake and should be deleted. It is a real value and my sample size is insufficient to properly estimate the variability in the data. In this case the data may be Gaussian but I have a poor estimate of the true mean because of small sample size. Possibly I have a large sample size and an arithmetic mean is a poor predictor of the central tendency of the data. I am having trouble with RG, so I will need to return to this in a day or two when I hope the problems will have been solved. I'll have a data set at that point.

Well, you could do a Q-test to objectively determine if the data pt. is a real outlier. But if it isn't, then log transformation reduces statistical bias and makes use of parametric methods reasonable. If there's a general skewing of the data, then more than one data pt. may be outlying & the Q-test won't find a significant outlier. Then the data just aren’t Gaussian in distribution.

-Bob Vadas, Jr.

I came across the same problem, where I transformed data using Ln and log10 for different data sets. When I back transform the means using ex (where e = 2.718 .....) and 10x, mean values tended to be lower than means of the original data. So in this case, I used the back transformed means although they do not normally mirror the original data. The main reason for this is that, in most of the cases you will notice that the standard errors decline after transformation compared to the ones for original data, even after you back transform them they become less than those of the original data, indicating a well dispersed data.