This is a topic which somehow still confuses people and there is a lot of seemingly contradicting advice. The main problem is that bodyweight fluctuates daily, but the best solution people came up with is doing weekly averages or, worse, only to step on the scale once a week. I give a tl;dr first, which might be sufficient for people who just lacked the right idea. Afterwards I will briefly remind you of the main issues with weight tracking and then go on to describing how I actually solve the problem.
tl;dr: Weigh in daily and use exponential moving average as it reflects the biology of tissue turnover reasonably well and does not mess up your psychology on a diet. Bonus: measure difference from today's EMA and last weeks EMA.
When you try to change your bodyweight into any direction, you embark on a very quantifiable progress. The more data you can get, the more accurate is the picture you can paint. But on the other hand, intraday fluctuations are usually  not interesting to us. Thus you weigh in daily at the time with the least amount of confounding factors, which is after waking (if you have a scale) or at least at the beginning of each gym session (if you do not have a scale at home). Given that scales are cheap and I've never seen an adult's home without one, unless you are a student you probably do not have to resort to the latter.
Now some people will be thrown off when the weight fluctuates up one day, but this is just the nature of the beast and it can take some time until everyone trusts the progress to the same degree. Furthermore, when does a fluctuation stop being a fluctuation? And is - weigh ins for competition aside - the daily weight a thing that matters anyway? While fat loss can happen rather rapidly and it's possible to lose a hundreds of grams of fat a day, productive tissue gain is lost in the statistical noise of bodyweight fluctuations, scale-inaccuracy and can only be observed over longer periods of time, ranging from several days in people in early stages of their career, to weeks and months in more advanced trainees.
Problems of simple averages
The simple trick of using weekly averages instead of daily numbers does not really solves this issue, it only takes a bit of the edge off. When taking weekly averages, each fluctuation only weighs in with 1/7 of its strength. But 1/7 of a kilo is still enough to throw you off mentally and what might be even worse, now this 1/7 of a fluctuation haunts you for the next week!
The practical problems aside, there is also a theoretical issue. Anecdotal evidence, mechanistic reasoning and the general theory of complex systems suggest that there should be a certain momentum to tissue change. Even if you were to generate a huge deficit on one day, large parts of it would be served from glycogen stores and only downstream this would manifest in actual fat loss. The other way around, if you are in a hypertrophy phase, shifts in water and glycogen may elevate your bodyweight immediately, but it will take weeks until actual contractile tissue is accrued.
The exponential moving average
So it appears sensible to use a weighted average over the past few days, but instead of using 1/7 as weight for seven data points, you use something smoother. Like a smoothing average. Luckily people have already thought about several such things and one of these smoothing moving averages is of particular ease of use, namely the exponential moving average or for short, EMA. It can be calculated as follows:
As you can see, the two coefficients sum up to one, so it is indeed an average. If you expand the recursive formular for a couple of days you see why it is called EMA, because the weights then decay exponentially. There is a choice to be made, namely about the value of α. The closer it is to zero, the more weight you give to your previous days and the stronger the smoothing out of the curve becomes. Something between 0.01 and 0.1 will probably be fine, depending on how much it messes with your head if the number goes up a bit.
I think 0.1 is a good value for dieting purpose, i.e. each new datapoint is 90% your previous EMA and 10% today's weight. This is already smoothing out quite nicely, yet still allows for detection of changes in progress. Intuitively, it is a smoother version of a 10 day average.
If you really struggle with the ups and downs, another trick would be to measure the difference of the average between today and a week back. This is a nice gauge for progress, as it is both resistant to ups and downs and also gives you the rate of weight loss or gain in pretty much the most accurate way - which is what you are interested in after all.
In conclusion I find the exponential moving average taken from daily weigh ins much more useful for tracking and evaluating bodyweight changes. In particular, if one day or so you miss your weigh in, you can just copy the previous day's weight without doing much damage to the validity of your tracking.
In general the EMA works well for measures that are done frequently but fluctuate, e.g. estimated 1RM. However, tape measurements, caliper site measurements or tested 1RM are usally measured infrequently and are less noisy than weight fluctuations.
|||in some situations it is interesting, like if you are prone to holding a lot of water and you have to make weight later in the day on a specific date|