# Thread: Benford's law and transformation.

1. ## Benford's law and transformation.

So, usually when Benford's law is presented, as is in the wikipedia article on it, it is said that it does not hold if the data is not over several orders of magnitude. And, this may be a dumb question, but the next natural question that occurs to me is, that one can always transform any data that is not to so that it is. Will the law apply then? If so, why?

Thanks.

2. Im no expert on benfords law (in fact i never heard of it!).

Having said that, it is exactly true if your data is log-uniformly distributed.

If benford's law does not hold for your data then (ignoring sampling error) it is probably because it is not log-uniformly distributed. Now, you can apply an arbitrary transformation to your data so that it appears to be log-uniformly distributed and benford's law will apply...but then it isn't really the same data anymore.

The point is that by applying your transformation, you are changing the statistical properties of the data; you wont be able to learn much about the underlying data by examining the transformed version.

3. but, isn't it so, that if that were true, one could never use transformation in any case in statistics? If I transform the data so that all the data points hold a similar porportional distance to each other, only now the numbers run over several magnitudes of order, wouldn't I still be able to use the law to examine whether or not the data is likely to be random or, say faked?

4. You can use a transformation where:

1) You understand the effects that the transformation will have on the propoerties of the data
2) Those effects are desirable.

If you want to test your transformed data using benfords law, you will have to seperately verify that this gives you some meaningful information about the untransformed data.

5. ## rewording

I guess my original question sort of was, is there any reason to believe that such a thing could not be done?

Example: say I was handed data of the populations of 500 towns all ranging from 30000 to 40000. Could I transform the data in some manner and then apply Benford's law meaningfully or not?

How exactly would one justify this act?

Is there any reason to believe that the data would not be log-uniformly distributed because of the transform or on the basis of the original data? Is it safe to assume that if the answer to the two above questions is negative, that one can conlude that what I suggest can be done?

I've tried to search for information on this in literature, but failed to come up with any ideas. AFAIK there is no reason why this could not be done. But I am not very good at math...

6. probably should've posted in the other forum...

7. I would have thought you could apply it to the second digit of the data in your example.

8. Originally Posted by SpringFan25
I would have thought you could apply it to the second digit of the data in your example.
why?

9. if you intentionally sampled in the range 30,000 - 39,999 then the first digit is not a random variable. I would look at the part of your data that is random.