Body Mass Scale: The Number Is Noise, the Trend Is Data
A body mass scale is the cheapest instrument in the house and the one you will actually use every day, which makes it the most valuable one — but only if you read it as a series. A single weigh-in carries a wobble of roughly half a kilogram that has nothing to do with fat, and a body mass index computed from that one reading inherits all of it. Fit a line through thirty of them instead and the same wobble collapses into a rate you can defend. This page computes that trade-off exactly: how wide one reading is, how many days a given rate needs before it is real, why using every day beats comparing the endpoints by a factor of √(N/6), what the display graduation is actually worth, and how many body-fat points one kilogram on the dial can mean. Nothing here was measured on a device and no product is named or ranked; every figure is arithmetic performed on this page.
One weigh-in is ±0.98 kg wide at 95%, which is ±0.32 BMI points at your height. Average 7 consecutive days and the same wobble shrinks to ±0.37 kg — ±0.12 BMI points. The nearest category line is BMI 25, sitting 1.44 kg away, so the noise on your mean cannot move you across it.
Fitting a line through 28 daily readings instead of comparing two of them gives a slope whose standard error is 0.01172 kg/day. The smallest weekly rate that many days can call real is 0.161 kg/week. Your 0.40 kg/week clears that — the trend is visible in the data you have. Using every day rather than the first and last makes the estimate 2.16× tighter — the same information as two readings 60 days apart.
A change is only real once it exceeds 0.52 kg on your 7-day mean; at 0.40 kg/week that takes 9.2 days of genuine loss before the scale can confirm it. Over your 28-day window the entered rate amounts to 1.60 kg, which is worth somewhere between -0.45 and +1.60 body-fat points depending on what the mass was — a 2.05-point spread the scale cannot resolve, and one that does not depend on your body fat at all.
The graduation. A 0.1 kg display adds 0.0008 to the variance of each reading, widening your effective scatter from 0.5 to 0.501 kg — 0.2% wider, which costs 0.1% more days of data. The smoothing. An exponentially weighted average at alpha = 0.2 behaves identically to a plain 9.00-day mean: both cut the scatter to 0.333σ and both lag the truth by 4.00 days.
SE(mean of n days) = σ ÷ √n and SE(slope over N days) = σ·√(12 ÷ N(N² − 1))
regression beats two readings by √((N² − 1) ÷ 6N) ≈ √(N ÷ 6)
EWMA with constant α ≡ simple mean of (2 − α) ÷ α days
ΔBF (points) = 100·(1 − BF)·ΔW ÷ W if it was fat, −100·BF·ΔW ÷ W if it was notThe first line is the definition of body mass index and its exact inverse. The second is the two standard errors that decide everything below: averaging shrinks noise by the square root of the count, and the slope of a line fitted to N equally spaced days has a standard error of σ divided by the square root of the sum of squared deviations, which for consecutive integers is exactly N(N² − 1)/12. The third is the ratio of that to the standard error of a two-point comparison, √2·σ/N. The fourth is not an approximation — an exponentially weighted average and a simple moving average of (2 − α)/α terms have both the same variance and the same mean age, which is why neither is faster than the other. The fifth is the derivative of the two-compartment identity taken two ways, and its width, 100·ΔW/W, does not contain BF at all. Every table below is this page evaluating these five lines. No device was measured and nothing is quoted from a manufacturer.
The short answer
A body mass scale measures one quantity — the force your body puts on a load cell, reported as mass — and everything else on the display is computed from that number plus a height you typed in once. Its single-reading accuracy matters far less than most buyers think, because the day-to-day variation in your own body is larger than the error in almost any load cell. What matters is whether you can see a rate, and that is a question about how many readings you have, not about how good the scale is.
- One reading is not a measurement, it is a sample.At ±0.5 kg of daily scatter, a single weigh-in is ±0.98 kg wide at 95% confidence — about 1.3% of a 78 kg body, and ±0.32 BMI points at 1.75 m.
- Seven days buys you a factor of 2.65.The mean of seven readings is ±0.37 kg wide. That is the single cheapest improvement available to you, and it costs nothing but consistency.
- A rate needs surprisingly few days if you use every reading.At ±0.5 kg of scatter, a loss of 0.5 kg per week separates from zero in 14 days of daily weighing. The same conclusion drawn from two readings 14 days apart would take 21 days.
- The display graduation is nearly irrelevant. A 1 kg graduation costs you 10% more days of data than a 0.1 kg one. Rounding averages away at exactly the same rate as the noise does.
- Mass is not fat.One kilogram on the dial is worth anywhere from −0.28 to +1.00 body-fat points on a 78 kg body at 22% fat. The scale cannot tell you which, and the ambiguity is 1.28 points wide regardless of how lean you are.
How wide is one weigh-in, and what a mean buys you
Take a body at 78 kg with a day-to-day scatter of ±0.5 kg — a figure you should replace with your own, measured by stepping on and off five times in one minute. The 95% interval on a single reading is ±1.96σ. Every additional day you fold into the mean divides that by √n, and the minimum change you can call real between two such means carries an extra √2 because both ends are noisy.
Read the last column as a clock: at a true loss of 0.5 kg per week, a single reading needs 19.4 days of real change before it can be called real, a 7-day mean needs 7.3 days, and a 28-day mean needs 3.7 days. This is the whole argument for daily weighing. Not because the readings are good — they are not — but because the error shrinks with the square root of how many you have while the signal grows linearly with time.
The same numbers land differently on different frames, because a BMI point is H² kilograms and H² grows fast. At 1.55 m a BMI point is only 2.40 kg, so a ±0.98 kg reading is 0.41 BMI points; at 1.90 m it takes 3.61 kg to move one point and the same reading is worth 0.27. The table below is the single-reading 95% band in BMI points, alongside the width of the whole overweight band (BMI 25 to 30) at each height.
A single reading's band is between 10.9% and 16.3% of the width of the overweight category depending on height — large enough to matter near a line, small enough to ignore in the middle of one. A 7-day mean's band is between 4.1% and 6.2% of it, which is why a weekly average is the right unit for deciding which category you are in, and a single reading is not.
How many days before the trend is real
The rate is a slope, not a difference. Fit a straight line by least squares to N daily readings taken on consecutive days and the standard error of that slope is σ·√(12 / N(N² − 1)), because the sum of squared deviations of the integers 0 to N − 1 about their mean is exactly N(N² − 1)/12. Note the N¹·&sup5; in the denominator: the error falls faster than √N, because spreading the same number of readings over a longer window lengthens the lever arm as well as increasing the count. That is the single most useful fact about daily weighing.
The table below is the number of consecutive daily weigh-ins needed before a rate of the given size separates from zero at 95% confidence, at five levels of daily scatter. All figures are in kilograms per week.
Read it the other way round and you get the smallest rate a given stretch of data can see. At ±0.5 kg of scatter: 7 days resolves 1.296 kg/week, 14 days resolves 0.455, 21 days resolves 0.247, 28 days resolves 0.160, 56 days resolves 0.057 and 84 days resolves 0.031. The window does not have to be long — it has to be longer than the table says, and the table is driven almost entirely by your own scatter, which is the one number on this page you should measure rather than assume.
There is a second reading of the same arithmetic that surprises people: the total change these thresholds represent falls as the window lengthens. Detecting 1.296 kg/week over 7 days means detecting 1.30 kg of total change; detecting 0.160 kg/week over 28 days means detecting 0.64 kg. A longer window does not need a bigger result — it needs a smaller one, because the slope has more leverage.
Why every day beats the two readings you would have compared
The natural way to use a body mass scale is to weigh once, wait a month, weigh again and divide. That estimate has a standard error of √2·σ/N. The regression on all N daily readings has σ·√(12 / N(N² − 1)). Their ratio simplifies cleanly to √((N² − 1) / 6N), which is √(N/6) to within a fraction of a percent once N is past a fortnight.
The third column is the practical version: one month of daily weighing carries the same information as a single comparison spread over two months, and three months of daily weighing carries as much as a comparison spread over nearly a year. The extra information does not come from the scale being better. It comes from the middle of the series, which the two-reading method throws away.
Here is the same thing on a worked series. The 28 readings below were generated on this page from a stated model — a true loss of 0.400 kg per week from a start of 78.0 kg, with ±0.5 kg of independent daily scatter added. They are not measurements of anybody; they exist so the two methods can be compared on identical data. The last two columns are a 7-day simple moving average and an exponentially weighted average at α = 0.2.
Now read it both ways. Regression on all 28 daysgives −0.394 kg/week with a standard error of 0.071 kg/week — a 95% interval of −0.533 to −0.256, comfortably clear of zero, and comfortably containing the true 0.400. The residual scatter it infers is 0.432 kg, close to the 0.500 that was put in. The first and last readinggive (77.3 − 77.9) / 27 × 7 = −0.156 kg/week, with a standard error of 0.183 kg/week. That is not distinguishable from zero. Same scale, same 28 days, same body: one method says the loss is real and measured, the other says nothing happened. The difference is not luck — the endpoint pair happened to land on a high reading at the start and a high reading at the end, and a two-point estimate has no way to know that.
An exponential average is not faster than a plain one
Most scale apps show a smoothed line, and the smoothing is usually an exponentially weighted moving average: today's smoothed value is α times today's reading plus (1 − α) times yesterday's smoothed value. It feels like it should respond faster than a plain average of the last n days, because it never fully forgets anything. It does not.
In steady state an EWMA with constant α has variance σ²·α/(2 − α), and the mean age of the data inside it — how far back, on average, the information it carries comes from — is (1 − α)/α days. A simple mean of n consecutive days has variance σ²/n and mean age (n − 1)/2. Set the variances equal and n = (2 − α)/α; substitute that into the mean age and you get ((2 − α)/α − 1)/2 = (1 − α)/α. Identical.
For any smoothing you can choose, there is a plain average with exactly the same noise and exactly the same delay, and there is no way to get one without the other. If your app lets you set α = 0.2, it is showing you a 9-day average that lags four days behind; if you want it to catch up in two days, you have to accept 0.42σ of scatter instead of 0.33σ. The trade-off is fixed, and it is the same trade-off whichever algorithm draws the line.
The display graduation is worth about ten percent
Scales are sold on resolution — 0.1 kg, 0.05 kg, "100 g precision". Rounding to a graduation of u contributes u²/12 to the variance of each reading, which is a standard error of u/√12: 0.289 kg for a 1 kg display, 0.029 kg for a 0.1 kg one. The question is what that does to a trend, and the answer follows from the fact that the number of days needed scales as σ²/³.
All of this is computed at ±0.5 kg of bodily scatter, and that is the point: the graduation is being compared against a noise source that is larger than it. At a 0.1 kg display the rounding accounts for 0.33% of the variance of a reading; at a 1 kg display it accounts for 25%, and even then the penalty on your trend is only 10% more days. Resolution matters for a single weigh-in and almost vanishes in a series, because rounding error averages away at exactly the same √n rate as everything else.
What one kilogram is worth in body-fat points
A body mass scale reports mass, and mass is not a compartment. If a change of ΔW is all fat, then fat mass rises by ΔW and the percentage becomes (F + ΔW)/(W + ΔW); differentiating at ΔW = 0 gives 100·(1 − BF)/W points per kilogram. If the same change is lean or water, fat mass is unchanged and the percentage becomes F/(W + ΔW), giving −100·BF/W. On a 78 kg body at 22% fat those are +1.00 and −0.28 points per kilogram — opposite in sign and 3.5× apart in size.
The interesting column is the fourth. The width of the ambiguity is 100·ΔW/W × (1 − BF) + 100·ΔW/W × BF = 100·ΔW/W exactly — the BF terms cancel, so the uncertainty a mass change carries about your body fat does not depend on how much fat you have. It depends only on the size of the change and your size. On the 78 kg body: 0.5 kg is 0.64 points of ambiguity, 1 kg is 1.28, 2 kg is 2.56, 3 kg is 3.85 and 5 kg is 6.41. Compare that with the trend precision computed earlier — 28 days at ±0.5 kg of scatter resolves 0.160 kg/week, or 0.64 kg over the window, which is itself 0.82 points of ambiguity. The mass series and the body-fat question run out of resolution at about the same place, and no improvement in the scale can fix that, because the missing information is not in the weight.
This is also why the body-fat number printed by a scale and the body-fat number implied by your weight can move in opposite directions without either being broken: they are answering different questions about the same kilogram. If you want the compartments rather than the total, the mass series is the wrong instrument, and the two-compartment identity is the honest way to see what the total can and cannot be split into.
How to run a body mass scale so the arithmetic above applies
- Measure your own σ before trusting any table.Step on and off five times in one minute, same spot, same posture. The spread of those five is the instrument's contribution; weigh once a day for a fortnight at a fixed time and take the spread of the 7-day means to get the bodily part. Every number on this page scales as σ or σ²/³, so getting σ wrong by a factor of two moves the required days by 1.59×.
- Fix the clock, not the clothing. The model above assumes the daily deviations are independent with a constant spread. Anything that makes them drift together — a weekly pattern, a change in routine partway through — breaks the independence assumption, and the standard errors will be optimistic.
- Never move the scale. A level floor and the same spot remove a source of error that the model has no term for. If you must move it, treat the series as two series.
- Do not chase the daily number.At ±0.5 kg, a single reading is ±0.98 kg wide. A 0.3 kg jump between two mornings is not information, and no amount of staring at it will make it information.
- Judge on the slope, and only after the table says you can.At ±0.5 kg and a real rate of 0.5 kg/week, 14 days. Before that, a flat line means nothing.
Where this page's arithmetic stops being true
- No device was measured.Nothing here is a claim about any scale's accuracy, and no product is named, ranked or reviewed. The scatter figures are placeholders for you to replace.
- Independence is assumed, not established.Every standard error on this page assumes successive daily deviations are independent draws with a constant spread. Real bodies have patterns — weekly, monthly, seasonal — and correlated errors shrink more slowly than √n. If yours do, all the day-counts here are too optimistic.
- A straight line is assumed. The regression asks whether a constant rate fits. Real mass change is not linear — early losses are usually faster — and a curved series will show a slope that is a weighted average of the rates over the window, not the rate now.
- The two-compartment conversion is a model.Dividing mass into fat and fat-free is one level of a model that has more levels, and the −100·BF/W branch is the limit of "none of it was fat", which is not the same as "it was water".
- WHO categories are screening bands, not diagnoses. A BMI line crossed by a noisy mean is a reason to look closer, not a finding.
- The worked series was generated here. The 28 readings are drawn from a stated model with a stated seed so that two estimation methods could be compared on identical data. They are not measurements of any person.
- None of this is medical advice. It is arithmetic about what a series of numbers can support. See our disclaimer.
Questions people ask about body mass scales
Is a body mass scale accurate enough to be worth using every day?
The question practically answers itself once the numbers are separated. At ±0.5 kg of daily scatter a single reading is ±0.98 kg wide — useless for a daily decision and fine for a series. Seven days later the mean is ±0.37 kg wide and a month of daily readings resolves a rate of 0.160 kg/week. Accuracy in the sense of agreeing with a laboratory standard is a different question, and it is not the one that decides whether the instrument is useful.
Does a 0.1 kg display matter more than a 0.5 kg one?
Not for a trend. Rounding to 0.5 kg contributes 0.144 kg of standard error, which against ±0.5 kg of bodily scatter widens your effective noise by 4.1% and costs 2.7% more days of data. Even a 1 kg graduation — 0.289 kg of rounding error, 25% of the variance of a single reading — costs only 10.1% more days, because rounding averages away at the same √n rate as the noise it is added to.
How many days of flat data before I am really plateaued?
Whatever the day-count table says for the rate you care about. At ±0.5 kg of scatter and a rate of 0.25 kg/week, 21 days of daily weighing — below that, "no change" and "change too small to see" look identical. The general rule is that the smallest rate N days can detect is 1.96·σ·√(12 / N(N² − 1))·7 per week, and a plateau is only a claim about a rate smaller than that. Our weight loss percentage calculator treats the same question from the weekly-average side.
Why does the scale drop two kilograms overnight?
This page does not model physiology and will not guess at a cause. What it can tell you is the size of the event: at ±0.5 kg of scatter, a 2 kg move between two single readings is larger than the 1.386 kg minimum detectable change, so it is not explainable by the noise alone — but a single reading is ±0.98 kg wide, so the magnitude of that 2 kg is uncertain by about half of itself. Two 7-day means 2 kg apart is a much stronger statement than two mornings 2 kg apart.
Should I weigh once a week instead of every day?
Only if you would otherwise not weigh at all. Five weekly readings across 28 days resolve 0.310 kg/week against the 0.160 kg/week that 28 daily readings resolve over the same span — so a fifth of the readings costs you 1.93× the precision, not five times. The loss is sub-proportional because spreading readings further apart lengthens the lever arm as well as thinning the count, and that partly pays for the missing days. Daily still wins, and it wins for free: the extra information costs you twenty seconds a morning and nothing else.
Can a body mass scale tell me my body fat percentage?
Not on its own. Mass is one number and body fat needs a second, independent one; the ambiguity in converting between them is 100·ΔW/W points, which is 1.28 points per kilogram on a 78 kg body regardless of how lean you are. Scales that print a percentage are measuring an electrical property, not fat — see what a smart scale really measures for that side of it, and what the spec sheet is worth for the hardware side.
My scale and my doctor's scale disagreed by a kilogram. Which is right?
This page cannot say, because it measured neither. What it can say is that a constant offset cancels out of every rate and every change on this page — only the slope matters for a trend — and that a gain error does not. If you want to find out which kind yours has, five tests for the one you already own separates them with computed thresholds.
Related pages on this site
More on this site: body fat calculator · fat percentage across three methods · best way to measure body fat · body fat percentage chart
Not medical advice. Every figure on this page is arithmetic performed here on the BMI definition, on the standard errors of a mean and of a least-squares slope, and on a stated two-compartment model. No scale was measured and no product is ranked. See our disclaimer.