# How can you accurately determine your own mass?

I'm trying to figure out whether it is possible for an individual to accurately determine his/her own mass, to within 100g, using equipment that is readily accessible or can be purchased at a reasonable price.

It may be tempting to say "go to the supermarket and buy a bathroom scale" -- however as an anecdotal point of reference I can say that I own 3 and they give me readings which differ by several kilograms. Also, with many digital scales I've used, the readings change if I place them in different locations on the ground, as well as if I change the battery.

Is there a practical yet effective way for an individual to determine their mass, with confidence that the results are accurate?

Quick summary of suggestions offered in the comments:

• Use the scale at the doctor's office (or get a better scale from somewhere) ==> That's cheating, the problem is to find it by yourself at home.
• Take measurements from 3 scales and calculate the mean ==> That's wrong, what if all 3 scales are incorrect? It doesn't matter how many samples you take, theoretically we don't know the relationship between the real weight and the shown weight, nor do we know the distribution in the case of inaccuracies. (It could be the case that old scales will always show a higher value, or that certain manufacturers skew their results down to make overweight people happier.)
• Use known weights (such as weightlifting plates or known volumes of water) and determine a calibration curve ==> The problem here is twofold. First of all, it is not so easy to find objects with accurate weights. Weightlifting plates can be a few percent off. Also, the error will accumulate. For example, if you calibrate with water and you want your curve to go up to 80kg, you need 80 liters of water. If every liter is measured with 1% error, you can end up with anything between 79.2 and 80.8kg at the end.
• "equipment that is accessible"... the scales at the doctor's office don't count, I presume? Whatever method you use, the answer always comes back to "calibration". You need a standard that you trust - whether it be mass, frequency, force, ... - and build an experiment based on that. If you have some weightlifting weights you can create a calibration curve for your scales (at a particular location, flat surface). But now you need to know if your weights are accurate. Maybe you can use equivalent volume of water - if you have a means of measuring that... – Floris Sep 22 '14 at 15:50
• @Floris To be honest, I'm asking from more of a theoretical standpoint. I know that it is possible to buy or rent or otherwise access highly accurate equipment, however I'm curious as to whether or not it's possible to do it from home. – CaptainCodeman Sep 22 '14 at 15:52
• Yes it is possible. But you need to have some standard that you trust. Essentially, you need a way to trace your measurement to the definition of the kg with better than 0.1% accuracy. That is quite a challenge. Differential measurements are always easier than absolute ones. – Floris Sep 22 '14 at 15:53
• 100 gram in 100 kg is one part in 1000 - that is 1 mm in one meter. You would definitely need "point attachment" (or knife edge) to both sides of your beam and you would want to confirm they are equal length by swapping the weights and confirming you are still in balance. At this precision it is not an easy thing to do. Calibration weights can be bought - but a 10 pound class 6 weight (class 6 = 500 mg tolerance) costs 116 dollars so is already out of range. Interesting challenge. – Floris Sep 22 '14 at 16:22
• As always, buy a nice barometer, call the doctor's office, and say "I'll give you this nice barometer if you bring your scale to my house so I can weigh myself" :-) (If this makes no sense, you aren't a physicist. Sorry) – Carl Witthoft Sep 22 '14 at 19:45

I'm not sure that it makes sense to try to measure your body weight to a precision of 100 g. For example I was just thirsty and drank a 20 ounce bottle of water, which transferred about 600 g extra mass to my stomach. Even just breathing changes your mass: if you take ten half-liter breaths per minute and your exhalations contain 5% carbon dioxide by volume, that's a mass loss of tens of grams of carbon per hour. (Moisture is probably a bigger effect there, too.) To measure a 50–100 kg mass to a precision of 0.1 kg is a fractional uncertainty of $10^{-4}$, which is about two orders of magnitude better precision than most college-course laboratory experiments. Furthermore you would expect to see changes of several hundreds of grams over the course of the day (which would be interesting, which is maybe why you're asking).

You won't in general find $10^{-4}$ precision in cheap consumer electronics. I'd expect a bathroom scale to have an absolute precision of 1%–5%, or one to seven pounds for a 150 pound person, with poorer quality loosely associated with cheaper scales.

If what you want is a well-calibrated absolute weight with three or four significant figures, the best setup for you is going to be a cantilever system with well-calibrated reference weights. That's what's at your doctor's office — sorry. If it weren't the most cost-effective way to get a reasonably accurate weight, then doctors would buy something else.

If, on the other hand, you're interested in seeing kilogram-level changes in your weight with sub-kilogram precision, you might not need the absolute weight after all. If you can convince yourself that your scale is linear for small deviations from your weight, then maybe you can take your scale's last two digits, the kilogram and decigram digits, at face value. Here's one way you could test that:

1. Get several similar-but-different sized weights, about the size of the mass differences you're hoping to measure. Brick pieces might work. Label them somehow: A, B, C, etc.

2. Put a base load on the scale so that it reads somewhere near the value that you're interested in. For instance, if you're weighing yourself, you could stand on the scale and have someone help you with the next steps.

3. One at a time, add your test weights to the scale. Each one will increment the reading on the scale by some amount. You'll make a data table like

reading  load
-------  -----
80.0 kg  just you
81.2 kg  you + A
82.2 kg  you + A + B


and so on. From this you can find the mass of each little weight. (This is how veterinarians weigh stubborn cats, but they do it one cat at a time).

4. Now repeat the measurements with the same base, but with the other weights in a different order:

reading  load
-------  ----
95.2 kg  you plus all your weights
80.2 kg  just you
81.1 kg  you + B
82.3 kg  you + B + A


There are a couple of things that you might learn from this procedure:

• One is the random error inherent in each scale. For instance, I made the two "just you" weights different in the last digit. That's not unreasonable: essentially all digital readouts have what's called a "Schmidt trigger" that puts some hysteresis in the last digit, so that it doesn't flicker between adjacent values; however that means that the uncertainty in the last digit of a digital readout is at least $\pm1$ in the final digit.

• You might also find that the same brick fragment C reliably takes the scale from 80.0 kg to 82.0 kg with one base load, but from 95.0 kg to 97.2 kg with another base load. That would mean that your scale is "nonlinear," since the same increase in signal gives a different increase in output starting from a different place. You'd have to decide how much this bothers you, if you find it.

This technique doesn't address the question of stability: presumably you're interested in measuring your weight over many days. I'd suggest essentially the same test for measuring the stability of your scale(s): find an inert weight that's close enough to your body weight that you expect the scale to be linear for nearby values, and compare your weight on the scale to that rather than simply to the reading of the scale. Depending on the precision you're interested in, you may still have some strange stuff happen. For instance some electronics will respond differently in humid weather than in dry weather; also some weights will absorb moisture from the air and have different masses in humid weather than in dry weather.

As the saying goes: quick, cheap, or correct, pick any two. You're not going to be able to get the precision that you want without some expenditure of money or time, but you can perhaps get the result that you want a little easier.

• I'd give an extra +1 for the "pick any two" part if I could! – Carl Witthoft Sep 23 '14 at 11:56

You must narrow your condition. Try measure yourself on several equipment from the same company. Choose the best spot you think that is the perfect place to measure (flat ground, nice temperature, without wind). Yes, do all the measurement on the exactly same condition. Do about 10 measurement for each equipment. By statistics, this condition will minimize your measurement error for sure!