Absolute Uncertainty Calculator
Enter a measured value and its percentage uncertainty to find the absolute uncertainty instantly. Switch to the relative-uncertainty or repeated-measurements mode for other common lab scenarios. The steps panel shows the full working so you can check the arithmetic and adapt it to your own experiment.
What is absolute uncertainty?
Absolute uncertainty is the margin of error of a measurement, expressed in the same units as the measured quantity. If you measure a length as 25.0 cm with an absolute uncertainty of 0.5 cm, you write it as 25.0 ± 0.5 cm, meaning the true value is estimated to lie somewhere between 24.5 cm and 25.5 cm. It differs from relative uncertainty, which is the ratio of absolute uncertainty to the measured value (0.5 / 25.0 = 0.02, or 2%), and from percentage uncertainty, which is simply relative uncertainty multiplied by 100 (2%). The three forms are interconvertible and this calculator handles all of them.
How to calculate absolute uncertainty
The method depends on how your uncertainty is specified. If you have a percentage uncertainty, multiply the measured value by that percentage divided by 100: absolute uncertainty = (percentage uncertainty / 100) x measured value. If you have a relative uncertainty (a decimal fraction), multiply it directly by the absolute value of the measurement: absolute uncertainty = relative uncertainty x |measured value|. If you performed repeated measurements, compute the mean, find the sample standard deviation (using n-1 in the denominator), then divide by the square root of the number of readings to get the standard error of the mean, which becomes the absolute uncertainty on the mean. For a single reading from a scale-based instrument, the absolute uncertainty is typically taken as half the smallest graduation.
Uncertainty propagation: combining measurements
When you derive a result from two or more measured quantities, the uncertainties propagate into the final answer. For addition and subtraction, add the absolute uncertainties in quadrature: if z = x + y or z = x - y, then delta_z = sqrt(delta_x^2 + delta_y^2). For multiplication and division, add the relative uncertainties in quadrature: if z = x * y or z = x / y, then (delta_z / |z|) = sqrt((delta_x / |x|)^2 + (delta_y / |y|)^2), then convert back to absolute: delta_z = (delta_z / |z|) x |z|. These quadrature rules (also called the law of error propagation) assume the measurements are independent and that the uncertainties represent one standard deviation. Enable the propagation helper in this calculator to see the arithmetic for your specific values.
Significant figures and rounding conventions
Absolute uncertainty is conventionally rounded to one or two significant figures, and the measured value is then rounded to the same number of decimal places as the uncertainty. For example, if your calculation gives 12.4872 ± 0.0347, you round the uncertainty to 0.035 (two significant figures) and express the result as 12.487 ± 0.035. If the first significant digit of the uncertainty is 1 or 2, using two significant figures is recommended to avoid overstating the precision. This calculator displays four decimal places as a working guide; apply the rounding convention yourself when writing up a lab report.
Common instrument absolute uncertainties
| Instrument | Typical absolute uncertainty | Notes |
|---|---|---|
| 30 cm ruler | ± 1 mm | Half smallest graduation |
| Meter stick | ± 1 mm | Half smallest graduation |
| Vernier caliper | ± 0.05 mm | Typical analog vernier |
| Digital caliper | ± 0.01 mm | Manufacturer spec |
| Micrometer screw gauge | ± 0.005 mm | Half smallest graduation |
| Analytical balance (0.001 g) | ± 0.0005 g | Half last digit |
| Digital thermometer (0.1 °C) | ± 0.05 °C | Half last digit |
| Mercury thermometer | ± 0.5 °C | Half smallest graduation |
| Stopwatch (0.01 s) | ± 0.005 s | Plus human reaction ~0.15 s |
| Digital multimeter (voltage) | ± 0.5% + 1 digit | Typical DMM spec |
| Burette (50 mL) | ± 0.05 mL | Half smallest graduation |
| Measuring cylinder (100 mL) | ± 0.5 mL | Half smallest graduation |
Typical absolute uncertainties quoted by manufacturers. Actual values depend on the specific instrument and calibration. Use the half the smallest scale division as a starting estimate.
Frequently asked questions
What is the difference between absolute uncertainty and relative uncertainty?
Absolute uncertainty has the same units as the measurement (for example, ± 0.5 cm for a length). Relative uncertainty is the ratio of absolute uncertainty to the measured value, so it is dimensionless (for example, 0.02 for the same 25.0 cm ± 0.5 cm measurement). Percentage uncertainty is relative uncertainty expressed as a percent (2% in that example). Absolute uncertainty tells you the size of the margin of error in real units; relative uncertainty tells you what fraction of the measured value that error represents.
How do I find absolute uncertainty from a single instrument reading?
For an analog instrument (ruler, thermometer, burette), take half the smallest scale graduation. A ruler graduated in mm has ± 0.5 mm; a burette graduated in 0.1 mL has ± 0.05 mL. For a digital instrument, take half the last displayed digit: a digital balance that reads to 0.001 g has ± 0.0005 g. Some manufacturers quote a different value in the specification sheet, which should be used instead if available.
Why do I add uncertainties in quadrature rather than just adding them?
Adding uncertainties in quadrature (sqrt of the sum of squares) is the standard approach when the uncertainties are independent and random. The straight sum is overly conservative because random errors are as likely to partially cancel as to accumulate. The quadrature rule comes from the statistical law that the variance of a sum of independent random variables is the sum of their variances, and standard deviation is the square root of variance. If errors might be systematic (always in the same direction) you would add them directly as a worst-case bound.
Can absolute uncertainty be larger than the measured value?
Yes, though it is unusual in practice. If your measurement is very small compared to the precision of your instrument, the absolute uncertainty can equal or exceed the measured value. This means a percentage uncertainty of 100% or more and usually signals that the instrument is not appropriate for the task. An example is trying to measure a 0.3 mm gap with a ruler graduated in 1 mm.
What is standard error of the mean and when should I use it?
The standard error of the mean (SEM) is the sample standard deviation divided by the square root of the number of readings. It estimates how far the mean of your sample is likely to be from the true population mean, so it is the appropriate absolute uncertainty to quote when your result is the average of repeated independent measurements. The sample standard deviation itself, by contrast, describes the spread of individual readings and is more appropriate if you need to predict a single new measurement.
How do I propagate uncertainty through a power or square root?
For a quantity raised to a power n (z = x^n), the relative uncertainty of the result is |n| times the relative uncertainty of x: delta_z / |z| = |n| x (delta_x / |x|). For a square root (n = 0.5), the relative uncertainty halves. For example, if the diameter of a circle has a 2% relative uncertainty, the area (which scales as diameter^2, so n = 2) has a 4% relative uncertainty.