Considering the number of words in Mandarin, you have quite the undertaking. I assume you are basing it on standard Mandarin (i.e. the dialect spoken in Harbin). Or are you basing it on tones indicated in dictionaries. I have noted that there are mistakes on occasion so what you are attempting is no small task. I have noted some highly educated friends from China themselves do not always know how some words are properly pronounced.

Tones can vary somewhat depending on where in China it is spoken. For example I learned Mandarin in Hangzhou and tend to have inflections from that city (not to mention pronounciation) - my "southern drawl".

I am not sure waht a statistical analysis wil indicate. I am certain some tones were attributed to words as per the "music" of the language. This would have been a natural development as opposed to a deliberate act. The advent of Hanyü Pinyin an attempt to standardise attributed tones to words was done. I am not sure that it necessarily represents historical tones over time. Interested to see how your study works out.

Thanks Jeffery. No there is no simple way to accomplish this kind of thing using all words of a language. Some day using computer storage we might be able to approximate it.

For now it would have to be with samples [100, 1000, 10,000... or other amounts]. Probability and statistics have developed to the point of letting people make pretty good predictions about what may well be happening in a group of data as a whole based solely on the use of smaller, well defined and well-chosen samples. 

Yes, there is regional variation in how the tones are produced in Mandarin [and in other tonal languages]. I am thinking that after developing a good starting methodology, I'd address that issue too. I've been enchanted many a time over the past 30 years by visitors' locally accented Mandarin, It is a wonderful thing. When I do interpreting, it is always an interesting thing to me ['what is this person's Mandarin on assignment today going to be like]?'.

My Mandarin for example would be closer to the Beijing standard than the Harbin standard that you mention [it is more 'standard' sounding than it is local-to-Beijing sounding though; it is not overtly of Beijing]. The varieties of Taiwan Mandarin have a definite S. Mandarin influence. Your Hangzhou Mandarin might be similar in the apparent influence of S Mandarin.

I'd be interested in focusing on one standard at a time and informing users of the data the details as to the what and how of everything that was done [data collection, analysis, how the sample groups were chosen...].

I'd want to separate out in groups the various kinds of speakers [L1 from China, North; L1 from China, South; L1 from the US, Heritage Speakers; L1 from the US, Immigrants who were educated in China...].

L1 means native speakers. I'm an L2.

L2 speakers would have their own language variation things to throw into the mix. They also could be classified in various groupings, and the most obvious would be the question of how much formal language training did they receive and how much contact did they have with L1 speakers...

Language variation issue aside -- I'm interested in applying math, probability, and statistics to the linguistics of tonal languages, beginning with the standard Putonghua Mandarin [MSC] as recognized by experts.

I studied briefly at Qinghua and did learn the Peking "twang". I found they sound like the had pebbles under their tongues. Your observation about Taiwan is quite correct. They do have less issues with my pronuncition. I presently speak Mandarin as my main language outside of work and do not even think about the language when I am speaking it or hearing it. I can attest though that the clearest Mandarin spoken is in Harbin. It is similar to my experience with Italian where I found that the Milanese speak the clearest Italian. I would just call it Mandarin though as Putonghua, as you know, only means "common speach". My Pekinese friends tend to believe they speak the best Mandarin but the Parisians also believe they speak the best French, which is in fact etrocious. The Pekinese just love to put "R" on everything, which the Harbin Mandarin speakers do not. That said the Pekinese spoken by my grandfather is almost unknown today and would be incomprehensible to a Mandarin speaker. Mandarin was brought to Peking. It did not originate there.

Statistically a sample size of 27 or greater is considered significant. However, for what you are doing you would need a much larger sample size. Since to be able to have basic conversational abilty requires a knowledge of around 200 words, I would suggest that would be a number to start with. God luck with this.

Thanks. 

Your earlier mention of the mistakes in tone of L1 speakers: I have also thought that this fact is interesting and could eventually be analyzed statistically. But not at the beginning.

R-less and R-full Mandarin: The Beijing rhotacization issue has an interesting sociolinguistic angle.

I'm not exactly answering the immediate question.  But in Hefei (Anhui province) many people swap or used to swap (1980's) "L" and "N". 

Dear Martha, My father in law swapped his L and Ns and spoke only a mixture of Szechwan and Anhui dialect despite being a teacher (Colonel) at the Whampoa Militay Academy. One had to get used to this as one could get thrown off.

It was really quite funny.  One day I heard my students them talking about all the animals their surnames represented: there was Ma, Yang, and Liu, whom they called Niu ("cow"). 

My father in law called milk (niunai) liulei. So your example hits home.

Here are some information about the distribution of Mandarin tone (please see https://www.quora.com/What-is-the-distribution-of-tones-in-Mandarin-Chinese).

 I thought one of the issues related to this posted question would be the accurate coding of tone changes, for example, when followed by a 4th tone, 不 (bù) changes to 2nd tone (bú).

By the way, I am wondering what would be the pedagogical implication of conducting research in the distribution of tone for Mandarin education either as the first and second language.    

Jia Li, thanks much. Kevin Lin's response on the quora link you gave leads me in a good direction I think.----As to bu2 versus bu4 thing: thanks. I agree, the  situation of the bu in combination with the following syllable's tone [which tells one whether to use bu2 or bu4, generally] should be appropriate for Bayesian; I hadn't considered that before, but it makes sense to use conditional probabilities in that case.------As far as developing pedagogical knowledge out of this kind of thing, I believe that eventually it could be done. Linguistics and statistics/probability work together. 

Hi Jim, Thanks for the response and I look forward to reading your work in the near future.

Thanks, I have other questions posted on Mandarin. One is on Tone sandhi. I think the bu2/bu2 thing is a kind of Tone sandhi but I do not remember whether it is actually called Tone sandhi.

I think everyone would understand if you used "tone sandhi". 

Thanks but Tone sandhi is often used for the Tone 3 to to Tone 2 phenomenon. I agree that the Bu2/Bu4 thing is similar--I'll use the term here too.