Do you know how good you are at tai chi? If you have only been doing it for a for a few months it's pretty obvious you're only a beginner. Likewise, if you've practiced every day for 40 years you're going to be at a pretty high level  but most of us are somewhere in the middle.
Working out how good you are is not straightforward. Unlike external martial arts (like karate) there is no belt system to gauge progress [irrespective that most belt systems are primarily about
revenue generation rather than true skill level, but let's not go there]. Although push hands can theoretically be used to gauge how good you are compared to your peers, it takes quite a high level of skill before this is actually a meaningful measure of tai chi skill, rather than just a form of "wrestling" and a result of hard strength rather than technique.
What I am going to do here therefore is set out a formula that you can use to calculate your approximate level of tai chi skill.
This calculation came about through two experiences that I had recently. First of all reading the
The five levels of taijiquan book by Master Jan Silberstorff. This book defines some approximate time periods needed to achieve specific levels of tai chi skill. It gives some constraints which (combined with some assumptions) can be used to construct a formula for calculating your level of tai chi skill.
The second experience was attending a recent Easter training camp with
Master Wang Hai Jun. Master Wang frequently speaks about the amount of training and effort required to progress in tai chi. Again he gave some specific numbers, combined with traditional teachings and sayings, that can be used to derive a tai chi skill calculation formula.
Plug in some numbers and out pops your approximate tai chi skill level  at least that's the idea. The premise of this approach is that "
there are no secrets" as Cheng Man Ch'ing famously said. There are no secret techniques or shortcuts, the only way to get good is to practice. A lot.
Show your working
So, let's begin with the derivation, like any good maths student should.
[For those of you who are not mathematically minded, or are simply not interested in why I have done what I have done, I suggest you skip over these explanations and graphs, to the end of this article and the "results" (such as they are)].
The first postulate that I'm going to make is that learning tai chi is a cumulative process. The more you practice, the better you get. I'm also going to assume that your skill level increases in a linear fashion. If one person practices twice as much as another person, they will be twice as good  which I think is a reasonable simplification to begin with. Basically, how good you are is directly proportional to your total practice time.
Now in order to gauge skill we need some objective measure of performance. Master Silberstorff's book expands upon the wellknown
five levels of tai chi skill outlined by Chen Wang Xiao. The result of my formula is therefore going to tell you what level you are using this yard stick. The only change is that (without loss of generality) I extend this concept from these discrete levels, to a continuous function so that more than likely you're going to end up being level 1.34 (or something).
The other thing I am going to define is that it is exponentially more difficult to go up the levels as you progress. This is a bit of an arbitrary assumption, but I can't really find any quantitative information on how much better someone at level 1 is compared to level 2 (say). So I'm going to define that each level results in a twofold skill level increase. Hence someone at level 4, is twice as good (whatever "good" means) as someone at level 3. And someone at level 3, is twice as good as someone at level 2. Level 2 is twice level 1, and so by implication someone at level 4 is eight times "better" than someone at level 1.
We can now put the bare bones of our formula together. As this is cumulative and linear, we can say that our tai chi skill is purely a function of how many hours of training we have put in. All I need is to introduce some constant that allows the mapping from number of hours training to skill level. Something like:
Skill level = constant * number of hours training
Now I use the five levels book, to determine what this constant should be. The book gives the following two reference points. That a practitioner should be able to reach level (the top of) 1 after one year of training and (the top of) level 3 after seven years. The crucial point though is that this is for a student in the traditional model i.e. a young chinese disciple who gives up his life to live with his master as his apprentice, training extremely hard every day. [Note that it is one year to level 1, a further two years to get to level 2, and then another four years to get to level 3 = 7 years in total, which does fit with my exponential increase in skill level assumption that I made above, if you assume the student progresses at a continuous rate].
An assumpton is now required here. I'm going to say that this dedicated disciple is probably going to train on average about four
quality hours per day. This is unlikely to be in one go, so if you allow time for warmups and occasional rests on top of that, you're more than likely talking about 5 to 6 hours of actual training time per day. I think that this is a reasonable longterm sustainable level (if it is your primary "job") but I'm happy to be corrected and would not be surprised if it was more.
This now allows us to calculate the constant which turns out to be 0.000018265. Below is the plot of this simple function on a graph of time versus skill level for this traditional student. We can see that the student is approximately at level 1 after the first year and approximately at level 3 after the seventh year. Note that the both the scales are logarithmic.

Time vs skill level for a traditional student practising for 4 hours every day 
Even at this stage, this simple formula now allows us to do a bit of experimenting. Imagine (and this may not be too hard) that you are not this traditional disciple, but are only able to practice a bit each day. The following graph is a demonstration of how long it will take you to reach the top of level 1 given a different amount of practice each day. Note that I express the amount time for daily practice in the number of Laojia's (the chen style 74movement long form) performed per day. This is the way the Master Wang Hai Jun expresses training.
A Laojia takes about 15 minutes to do, so 15 laojia's represents about ~4 hours of training/day (but remember you need to add warmup time on to that). This graph shows that if you only do about a quarter of an hour of training each day, it's going to take you about 5,500 days (~15 years) to reach the top of level 1. If you do somewhere between 1 to 1.5 hours per day you'll be there in about 3 years. Our disciple does 4 hours a day and gets there in 1 year.

Number of days to reach the bottom of level 2, given different amounts of daily regular practice. 
The uphill battle
Now we need to add in the second major component of the skill calculation formula. Master Wang describes learning tai chi as like riding a bicycle uphill  if you stop pedalling you go backwards. He also talks about the commonly used traditional saying that "one day missed is 10 days back". In other words you need to train absolutely every single day, otherwise you are seriously hurting your efforts. Tai chi skill is a cumulative thing, but every day you don't practice, you have to add "negative skill" to your current level.
Working out how to represent this analytically turns out to be a bit tricky. After all, what if you missed two days of practice in in a row? Are you now 20 days back? 11 days back? What about if you missed 10 days in a row? Equally, if you happen to have trained twice as long as normal for the previous 10 days, surely if you miss a day it's not going to be as bad as if you had not trained twice as hard?
The "one day missed is 10 days back" is a reference point, but it is only saying as Master Wang acknowledges and goes on to elaborate. He says that once you have reached a higher level, this fact is less of an issue and you can take the occasional day off without significant detriment. What this means, is that the amount of degradation in your tai chi skill, if you don't practice for a day, is dependent on your current skill level.
For simplicity of my formula I choose to make the degradation inversely proportional to your current skill level. The higher your level of tai chi, the less skill you will "lose" if you miss a day's training. Tai chi is like cycling up a hill, but the higher up the hill you get, the flatter the gradient becomes. Intuitively this makes sense, someone who has been practising for decades could miss a month or two (say through injury), but would be able to recover reasonably quickly. Someone who goes along to a few classes and then has several months break, is likely to find themselves back at square one.
Skill level change per day with no training = constant / current skill level
In order to work out what this second constant should be I make assumption that this "one day missed is 10 days back" saying is really aimed at someone who is just starting to learn. It is designed as encouragement to keep consistency in the early phases. I arbitarily choose this to be someone who is at level 0.5.
To quantify "10 days back", I will assume that student is following the advice of their teacher. Master Wang says that in order to make
progress in tai chi you need to be doing at least five laojia's a day. However to only
maintain your level (i.e. for health) he says you need to be doing two or three (i.e. ~3045mins) per day. Hence, missing a day when you are at level 0.5 is equivalent to losing 25 laojia's worth of training. Using this constraint, it turns out that this degradation constant is 0.0014.

Impact of taking days off for people in the early years of their tai chi training. 
The above graph shows the effect of missing days training for someone
who is in the relatively early phases of learning tai chi. Each of these
four curves represent the same total amount of training per week (one
laojia per day). However in each of these cases the distribution of
training and rest days is altered. The graph clearly shows that unless
you train each and every single day, you never really get off the
ground (if you only do 15mins per day average). I believe this is what the traditional wisdom is trying to say.
Other graphs show that if you start at much higher level, say at level 3,
your good work is not undone so rapidly, and you can maintain your level
with days off. So that's a bit of a reward to look forward to after
earning it with your initial years of dedication and hard work.
The formula
So here we go then with the tai chi level skill calculation formula. I have set it up to be a recurrence relation, so your level at (the end of)
day n depends on your level at
day n1 and the amount of practice you did on
day n. My units for "
the amount of practice",
p, is a
points system I previously thought up for my particular style. However, for use in a different style 15 minutes is effectively 10 points. But remember, you can't include warmups etc. You only get points for concentrated effort, not for time spent standing around in your dojo. The reason I prefer points to time, is to reinforce this distinction.
So we actually have quite a simple formula. I'm sure the constants could use some tweaking, but the aim of this is to give a first order approximation to your skill level. A sort of mathematical rule of thumb.
My temptation would be to spend a long time trying to refine and extend the formula to include secondorder effects, but I'll resist as I don't think that's actually a very fruitful thing to be doing. This is supposed to be a guide, not an accurate measure. Everybody's situation is different, and so for any particular individual the formula is going to be more or less correct, but on average across everone, it should be a good approximation. As an aside however here are some thoughts on refinements and secondorder effects:
 Natural talent: Some people are going to be naturally more suited to tai chi than others and are going to improve faster. Equally other people might lose tai chi skill on days off at different rates. It's a fact of life and nothing can account for that.
 Your teacher: How good your instruction is will make a big difference, as will how often you attend class. Mathematically I can imagine this is as an additional term in the function related to the difference between your skill level and your teacher's skill level. When the difference is big (your teacher is at much higher level than you), you will improve more rapidly.
 Intensity: There is no doubt that it is not quite a linear cumulative process. Training for 20 hours over five days is better than training for 20 hours over 20 days. In formula terms, I imagine representing this as a derivative  a feedback so that the steeper the gradient of the recent past, the greater the improvement per "practice point".
A lookup guide to your tai chi skill level (a.k.a "the results")
So we're done with the maths now.
[Welcome back all those of you who jumped ahead!].
Having developed a formula we can now put it to work. Of course you can use the formula yourself for your own circumstances to calculate your skill level estimate, but at this point it's probably sufficient to provide a few examples. Hopefully, one of these is reasonably close to what you actually do, and you can just use these guides and a bit of a fudge, to guess what level you might be.
Below is a table that sets out the amount of training time needed to reach level 1.0 and level 3.0 given a daily training regime. Remember though this is a rigorous daily training regime which means training
each and every single day, without exception.
Training regime  Time to Level 1.0 (years)  Time to level 3.0 (years) 
15 mins per day every day  15  127 
30 mins per day every day  8  64 
1 hour per day every day  4  32 
2 hour per day every day  2  16 
4 hour per day every day
(the traditional chinese disicple)  1  7 
The problem is that most of us mere mortals find it difficult to train every day without exception. The next table therefore shows some more realistic training regimes, for those of you who are only doing tai chi as a "hobby" rather than your life's purpose.
Training regime  Time to Level 1.0 (years) 
1 hour class a week  Never 
1 hour class a week and 20 mins practice by myself a couple of times in between  Never 
1 hour every other day  Never 
1 hour per day but take Sunday off  Never 
Bummer  what a kick in the teeth! Now we can see why
Master Silberstorff says that 99% of tai chi practitioners are below level 1.0. By Western standards, doing an hour every other day would seem quite impressive in the context of "going to the gym". But we see from the table that even doing this supposedly prestigious amount is still never going to get you any
real tai chi skill. Health benefits  yes, real tai chi  no.
But do not despair, the formula shows us an important aspect of training: the detriment of missing days is more pronounced at lower skill levels than higher. In practical terms, what this means is that when you're just starting, there is a crucial and necessary intensity/continuity period that is required. Once you get over this hurdle, you can slack off a bit (if you wish) and allow yourself a more interrupted training regime. If you slack off, you hamper your progress to the next level of course, but at least you won't be going backwards anymore.
The next table shows some examples with an initial period followed by a consistent period of doing one hour a day but taking one day off a week. The point of this table is to show how much and how long you need this initial period to be. There is a threshold where the amount of training outweighs the negative effects of number of rest days. If you are above this threshold you make progress and get there eventually, if you are below, your good work is always being undone and you never make any "real" progress.
Training regime  Time to Level 1.0 (years) 
1.5hr/day for the first year. Then 1hr/day, 6days/week.  4 
1hr/day for the two years. Then 1hr/day, 6days/week.  6 
30mins/day for first three years. Then 1hr/day, 6days/week.  9 
1.5hr class once per week. Plus: 30mins/day for one year, and then 1hr/day, 6days/week.  Never 
1.5hr class once per week. Plus: 30mins/day for two years, and then 1hr/day, 6days/week.  Never 
1.5hr class once per week. Plus: 30mins/day for three years, and then 1hr/day, 6days/week.  6 
1.5hr class once per week. Plus: 1hr/day for one year, and then 1hr/day, 6days/week.  5 
1.5hr class twice per week. Plus: 30mins/day for one year, and then 1hr/day, 6days/week.  Never 
1.5hr class twice per week. Plus: 30mins/day for one year, and then 1hr/day, 6days/week.  4 
1.5hr class twice per week. Plus: 1hr/day, 6days/week, from the start  Never 
Effectively what this table shows is that it is the daily consistency in the initial phase that is important. It looks roughly like you need to put in about ~700hours of training without days off, to get over the threshold. The exact numbers here shouldn't be taken as gospel truth remember, this is only an approximate formula to give you a rough feel for things. The key message is:
If you want to learn tai chi seriously, you need to make a commitment to start off practicing every single day for a number of years.
The longer you can spend each day practising, the shorter the overall initial period will have to be before you're allowed to start taking occasional days off. But beware of starting to slacking off to early, as the "one day missed, 10 days back" effect will really start to hurt you, and you wouldn't want to waste all your earlier efforts.
So there you have it. The answer is simple, just as your teacher said. You need to practice every day.
The two paths
I can already here the cries.... "
but I've been going to weekly classes for years, without practicing at home, and it's obvious I'm better".... Although I accept that this formula may well be too simplistic, such a sentiment would be to misunderstand the two paths of tai chi. The path I have been talking about here is the
tai chi as a martial art path.
There is a second path however which is the health path, and is frequently referred to as the gentle dance path. On this path, what you're really learning is chi gung not tai chi. These skill levels and training requirements that I have been talking about, are about gaining a deep understanding, rather than how gracefully you can wave your arms around. With chi gung you can be very graceful (and healthy), but still be at a low level of tai chi understanding.
Any amount of tai chi will give you health benefits, but after all, you can get health benefits from going to dancing classes, or the gym, or sports.... If all you want is health, balance, a good way to manage stress, these levels are irrelevant  you enjoy it, it makes you a bit healthier, that's all that matters. It is important to acknowledge this fact and be aware that you are on this path, which after all is an equally valid one.
Why is that important? Because on the health path you are not learning a martial art any more, and so you should not expect to
ever gain any meaningful self defence skills or any of the really deep physiological, structural and awareness changes that transform your body. I'm sure a large proportion of practitioners, probably the vast majority, are entirely happy with that. But if you want to be in the 1% of people who are not, this formula shows you how to get there. Make sure you choose your path, because if you haven't choosen, you're almost certainly on the health one.
For me at least, this exercise has motivated and inspired me to redouble my training efforts as I want to learn tai chi, the martial art. Seeing those stark realities in black, white and mathematics has helped me to feel that I am on that path, and that every day's training really does count. I hope it might do the same for you.