Uncertainty Calculator

Q: What is the difference between error and uncertainty?

Error is how far your measured value sits from the true value. You can almost never know it, because you do not know the true value. Uncertainty is the size of the range that probably holds the true value. You can always estimate it. That is why lab reports state uncertainty, not error.

Q: What is the uncertainty of a ruler or an analog scale?

A common rule is half of the smallest division . A ruler marked every 1 mm gives ±0.5 mm per reading. If you measure a length using both ends, that doubt applies twice, so combine the two in quadrature. For a digital display, use the last digit. A readout of 12.34 g usually gets a half-digit half-width of 0.005 g, treated as rectangular.

Q: What is the difference between standard uncertainty and expanded uncertainty?

Standard uncertainty (u) is one standard deviation. It covers about 68% of likely values. Expanded uncertainty (U) is u times a coverage factor k, written U = k·u. Most reports use k = 2 for about 95% coverage. Always say which one you are quoting, and state the value of k.

Q: What coverage factor should I use for 95% or 99% confidence?

For a normal distribution with plenty of data: k = 1 → about 68% k = 2 → about 95% k = 2.58 → about 99% k = 3 → about 99.7% With only a few readings, use a t-value instead of k, since the tails are fatter.

Q: How do you calculate percentage uncertainty?

Divide the uncertainty by the value and multiply by 100: % uncertainty = (u / |x|) × 100 For 12.0 ± 0.1 cm, that is (0.1 / 12.0) × 100 = 0.83%. Percent uncertainty is handy because you can add percents in quadrature when you multiply or divide values.

Q: How do you propagate uncertainty through a logarithm or an exponential?

Use the slope of the function: y = ln(x): u(y) = u(x) / x. The answer's absolute uncertainty equals x's relative uncertainty. y = log₁₀(x): u(y) = u(x) / (x · 2.303) y = e^x: u(y) = e^x · u(x), so the relative uncertainty is u(x) itself. This is why a pH of 7.00 ± 0.01 means the hydrogen ion concentration is uncertain by about 2.3%.

Uncertainty Calculator

Introduction

Every measurement has some doubt in it. A ruler, a scale, or a timer can only get so close to the true value. That small doubt is called uncertainty. When you use those measured numbers in a math formula, the doubt carries through to your answer. This uncertainty calculator works out how much.

The tool has three modes:

Expression Mode. Type a formula like (12±0.1) * (x + y). The calculator finds the partial derivatives, combines them in quadrature, and shows each step.
Step Calculator. Work with two numbers at a time. Add, multiply, take a square root, and more. Pick standard error or worst-case error, and store results in memory.
Monte Carlo. Run up to 2 million random trials. Choose normal, uniform, triangular, or t-distributions for your inputs. You can add correlations and copulas too.

Each mode returns a result with a ± value, a step-by-step solution, and a chart showing which input adds the most doubt. Results are rounded to the right number of significant figures, so you can copy them straight into a lab report.

This is useful for physics and chemistry labs, engineering work, and calibration jobs that follow the GUM method. Start with a preset example, or type in your own numbers.

How to use our Uncertainty Calculator

Type your formula and the value and uncertainty of each quantity. The calculator gives you the result, its uncertainty, the partial derivatives, step-by-step math, and a chart. Pick one of the three tabs: Expression Mode, Step Calculator, or Monte Carlo.

Expression Mode

Expression: Type your formula, like (12±0.1) * (x + y). Type +- and it turns into ±, or click the ± button. Use a period for decimals.

Insert function / constant: Click this button to add sqrt, log, sin, pi, and more without typing them.

Value of each variable: Enter the measured number for every named variable, such as x or y. A box appears for each one you type.

± uncertainty of each variable: Enter how much that value can be off by. It must be zero or higher.

Calculate: Press this to see the formula, the partial derivatives, the final answer, and the variance chart. Press Clear to start over.

Step Calculator

Propagation method: Pick Standard Error to add the errors in quadrature, or Maximum Error for the worst case.

X and ± dX: Enter the first value and its uncertainty. Blank boxes count as zero.

Y and ± dY: Enter the second value and its uncertainty. Only needed for two-number operations.

Binary operations: Click a button like X + Y, X × Y, or X ^ Y to use both numbers.

Functions: Click a button like sqrt(X), ln(X), or sin(X) to use X alone.

Number format: Choose Standard or Scientific for how the answers are shown.

Memory buttons: Use Z → Mem, Z → X, Mem → Y, or Swap X↔Y to move results into new steps.

Monte Carlo

Preset example: Pick a ready-made example, like Voltage Divider, to fill everything in for you.

Number of input quantities: Choose how many quantities your model has, from 1 to 15.

Name: Give each quantity a short name, like R1 or Vin. Start with a letter.

Distribution: Pick the shape of each quantity: Normal, Uniform, Triangular, t, or Rectangular.

Nominal value: Enter the best estimate for that quantity.

Standard uncertainty u / half-width a: Enter the spread for that quantity. It must be zero or higher.

Additional parameter: Enter degrees of freedom for the t-distribution, or the upper half-width for a triangular shape.

Output expression: Write the formula for y using the exact names from the table, like Vin * R2 / (R1 + R2).

Include input correlations: Tick this if your quantities are linked, then fill the matrix with values between −1 and 1.

Copula type and θ: Under Advanced Dependency Modeling, pick Gaussian, Clayton, Gumbel, or Frank, and set θ for the last three.

Number of realizations: Enter how many random trials to run, from 1,000 to 2,000,000. More trials means a smoother result.

Random seed: Enter a whole number to get the same result each run, or leave it blank for random.

Coverage probability (%): Enter the confidence level for your interval, such as 95.

Symmetrical coverage interval: Leave it ticked for an even split on both sides, or untick it for the shortest interval.

Run Simulation: Press this to get the mean, standard uncertainty, coverage interval, histogram, and steps. Press Reset to start fresh.

What Is Uncertainty Propagation?

Every measurement has a little bit of doubt in it. A ruler, a scale, or a thermometer can only get so close to the true value. That doubt is called uncertainty, and we write it with a ± sign. For example, 12 ± 0.1 cm means the real length is most likely between 11.9 and 12.1 cm.

When you use those measured numbers in a math formula, the doubt travels with them into your answer. Working out how much doubt ends up in the final answer is called uncertainty propagation, or error propagation.

How the Math Works

The most common method is the quadrature rule from the GUM (Guide to the Expression of Uncertainty in Measurement). You find how much the answer changes when each input changes a little. That change rate is the partial derivative, also called the sensitivity coefficient. Then you multiply each one by its uncertainty, square the results, add them, and take the square root:

δf = √[ (∂f/∂x · δx)² + (∂f/∂y · δy)² + … ]

Squaring and adding works because random errors do not all push the answer the same way at the same time. Some cancel out. If you want the worst case instead, you just add the contributions without squaring. That gives a bigger, safer number.

Simple Rules of Thumb

Adding or subtracting: add the absolute uncertainties in quadrature.
Multiplying or dividing: add the percent (relative) uncertainties in quadrature.
Powers: multiply the percent uncertainty by the exponent. Squaring a value doubles its percent uncertainty.

The Monte Carlo Method

Sometimes a formula is curvy, or the inputs are not shaped like a normal bell curve. Then the simple formula can be off. The Monte Carlo method fixes this. It picks thousands (or millions) of random values for each input, runs them all through the formula, and looks at the spread of answers. The average is your result, the standard deviation is your uncertainty, and the middle 95% of answers is your coverage interval.

Monte Carlo also handles inputs that are linked together. If two measurements come from the same instrument, their errors move together. That link is called correlation, and ignoring it can make your uncertainty too small or too big.

Common Distributions

Normal: the bell curve. Use it for repeated readings and most lab data.
Uniform (rectangular): any value in a range is equally likely. Good for digital readouts and tolerance limits.
Triangular: values near the middle are more likely than values at the edges.
t-distribution: like the bell curve, but with fatter tails. Use it when you only have a few readings.

Writing Your Answer Correctly

Round the uncertainty to one or two significant figures first. Then round the result to the same decimal place. Write 9.82 ± 0.05 m/s², not 9.8173 ± 0.0492. Also state your coverage factor. A factor of k = 2 covers about 95% of the possible values for a normal distribution.

Where It Is Used

Uncertainty propagation shows up in physics labs, chemistry, engineering design, calibration certificates, medical testing, and quality control. Any time you report a measured number, the uncertainty tells people how much they can trust it.

Formulas used

General uncertainty propagation (quadrature, independent inputs)

\delta f = \sqrt{\sum_{i=1}^{n}\left(\frac{\partial f}{\partial q_i}\,\delta q_i\right)^{2}}

Maximum (worst-case) error propagation

\delta Z = \left|\frac{\partial Z}{\partial X}\right|\delta X + \left|\frac{\partial Z}{\partial Y}\right|\delta Y

Variance contribution share of each input quantity

s_i = \frac{\left(\frac{\partial f}{\partial q_i}\,\delta q_i\right)^{2}}{\sum_{j}\left(\frac{\partial f}{\partial q_j}\,\delta q_j\right)^{2}}\times 100\%

Relative uncertainty and expanded uncertainty (k = 2)

\text{rel} = \left|\frac{\delta f}{f}\right|\times 100\%,\qquad U_{95} = 2\,\delta f

Monte Carlo mean of the output quantity

\bar{y} = \frac{1}{M}\sum_{j=1}^{M} y_j,\qquad y_j = f\!\left(x_{1,j},\dots,x_{n,j}\right)

Monte Carlo standard uncertainty of the output

u(y) = \sqrt{\frac{1}{M-1}\sum_{j=1}^{M}\left(y_j-\bar{y}\right)^{2}}

Symmetrical coverage interval at probability p

\left[\,y_{(1-p)/2},\; y_{(1+p)/2}\,\right],\qquad \text{shortest: } \min_{r}\left(y_{r+q}-y_{r}\right),\; q=\lfloor pM \rfloor

Gaussian copula sampling of correlated inputs

\mathbf{z} = \mathbf{L}\mathbf{w},\quad \mathbf{L}\mathbf{L}^{\mathsf{T}} = \boldsymbol{\rho},\quad x_i = F_i^{-1}\!\left(\Phi(z_i)\right)

Frequently asked questions

What is the difference between error and uncertainty?

Error is how far your measured value sits from the true value. You can almost never know it, because you do not know the true value.

Uncertainty is the size of the range that probably holds the true value. You can always estimate it. That is why lab reports state uncertainty, not error.

What is the difference between Type A and Type B uncertainty?

Both are standard uncertainties. They only differ in how you get them.

Type A: found from statistics on repeated readings, such as the standard deviation of 10 trials.
Type B: found from other information, such as a calibration certificate, a spec sheet, a tolerance, or the resolution of a display.

Once you have them, you combine both kinds the same way, in quadrature.

How do you find the uncertainty of repeated measurements?

Take the average of your readings. Then find the standard deviation s. The uncertainty of the average is the standard error:

u = s / √n

Here n is how many readings you took. So 9 readings cut the uncertainty by 3, not by 9. Taking more readings helps, but slowly.

What is the uncertainty of a ruler or an analog scale?

A common rule is half of the smallest division. A ruler marked every 1 mm gives ±0.5 mm per reading. If you measure a length using both ends, that doubt applies twice, so combine the two in quadrature.

For a digital display, use the last digit. A readout of 12.34 g usually gets a half-digit half-width of 0.005 g, treated as rectangular.

How do you turn a tolerance into a standard uncertainty?

A tolerance like ±0.02 is a half-width, not a standard uncertainty. Divide it by a number that depends on the shape you assume:

Rectangular (any value equally likely): u = a / √3 ≈ a / 1.732
Triangular (middle more likely): u = a / √6 ≈ a / 2.449
Normal with k = 2 stated: u = a / 2

So ±0.02 rectangular gives u = 0.0115.

What is the difference between standard uncertainty and expanded uncertainty?

Standard uncertainty (u) is one standard deviation. It covers about 68% of likely values.

Expanded uncertainty (U) is u times a coverage factor k, written U = k·u. Most reports use k = 2 for about 95% coverage.

Always say which one you are quoting, and state the value of k.

What coverage factor should I use for 95% or 99% confidence?

For a normal distribution with plenty of data:

k = 1 → about 68%
k = 2 → about 95%
k = 2.58 → about 99%
k = 3 → about 99.7%

With only a few readings, use a t-value instead of k, since the tails are fatter.

How do you calculate percentage uncertainty?

Divide the uncertainty by the value and multiply by 100:

% uncertainty = (u / |x|) × 100

For 12.0 ± 0.1 cm, that is (0.1 / 12.0) × 100 = 0.83%. Percent uncertainty is handy because you can add percents in quadrature when you multiply or divide values.

How do you propagate uncertainty through a logarithm or an exponential?

Use the slope of the function:

y = ln(x): u(y) = u(x) / x. The answer's absolute uncertainty equals x's relative uncertainty.
y = log₁₀(x): u(y) = u(x) / (x · 2.303)
y = e^x: u(y) = e^x · u(x), so the relative uncertainty is u(x) itself.

This is why a pH of 7.00 ± 0.01 means the hydrogen ion concentration is uncertain by about 2.3%.

Why does subtracting two close numbers give a huge percentage uncertainty?

The absolute uncertainties still add in quadrature, but the result gets small. So the ratio explodes.

Example: 100.0 ± 0.1 minus 99.0 ± 0.1 gives 1.0 ± 0.14. That is 14% doubt from inputs that were only 0.1% uncertain each.

If you can, measure the small difference directly instead of subtracting two big numbers.

How does correlation between inputs change the combined uncertainty?

Correlated inputs add an extra covariance term:

u²(y) = (c₁u₁)² + (c₂u₂)² + 2·c₁c₂·u₁u₂·r

Here r is the correlation coefficient, from −1 to 1. A positive r makes the uncertainty bigger when you add values, and smaller when you subtract them. Errors that come from the same instrument or the same calibration are usually correlated, so ignoring them gives the wrong answer.

How many Monte Carlo trials are needed for a reliable result?

The GUM supplement suggests at least 10⁶ trials for a 95% coverage interval. That usually gives about 2 or 3 stable significant figures.

Fewer trials, like 10,000, are fine for a quick check but the interval edges will wobble. Run the same model twice with different seeds; if the answers match to the digits you plan to report, you have run enough.

What are degrees of freedom in uncertainty analysis?

Degrees of freedom show how well you know an uncertainty. For n repeated readings, ν = n − 1. So 5 readings give ν = 4.

Low degrees of freedom mean a shaky estimate, so you need a bigger coverage factor. A Type B value taken as exact gets ν = ∞. The Welch-Satterthwaite formula mixes them into one effective value for the final result.

What is the difference between accuracy and precision?

Accuracy is how close your readings sit to the true value. Precision is how close your readings sit to each other.

A scale that always reads 2 g too high is precise but not accurate. Repeating a measurement fixes poor precision, but only calibration fixes poor accuracy. Uncertainty statements should cover both.

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