Error Propagation Calculator - Sum, Product, and Power Rules

An error propagation calculator combines the 1-sigma standard uncertainties on two or more measured values into a single propagated uncertainty on a derived quantity. Pick the formula mode, type in each value with its uncertainty, and read the propagated absolute and relative uncertainty on the result. It is the workhorse tool for any lab report that combines raw measurements into a final number with a defensible plus-or-minus interval.

• • Combine measurements: Two or more measured values feed a derived quantity and you need a single plus-or-minus to defend.
• • Power or product rules: Use the power rule for kinetic energy, the product rule for density, or the quotient rule for a ratio.
• • Calibration constant: Turn a calibration constant with known tolerance plus a measured reading into a propagated uncertainty.
• • Compare two rules: Switch between linear and product rules on the same dataset to see which dominates.

Error propagation answers a different question than the absolute uncertainty on a single measurement. The right way to get the uncertainty on a derived quantity is to apply the first-order propagation rule to the partial derivatives of the formula, and the error propagation calculator does exactly that for the three closed-form rules that cover most lab formulas.

When the first step is to turn repeated stopwatch or balance readings into a single sigma_x, Absolute Uncertainty Calculator gives you the half-range, resolution, and standard-error modes that feed the error propagation step.

Pick a formula mode and the calculator applies the matching first-order propagation rule. For a linear combination it adds squared weighted uncertainties; for a product or quotient it adds squared relative uncertainties; for a power it scales the input uncertainty by the absolute value of the derivative.

• f: The derived quantity you are computing. Shown as the result row in the output panel.
• x, y: Measured values. Each carries a 1-sigma standard uncertainty sigma_x or sigma_y in the same unit.
• sigma_x, sigma_y: 1-sigma standard uncertainties on the measured values, taken from your instrument or from an absolute-uncertainty step.
• a, b, c: Coefficients in the linear formula f = a*x + b*y + c. The constant c may itself have an uncertainty sigma_c.
• n: Exponent in f = x^n. The first-order rule gives sigma_f = |n| * |x|^(n-1) * sigma_x.
• sigma_f: Propagated 1-sigma absolute uncertainty on f, reported in the same unit as the inputs.

The first-order propagation rule assumes the input uncertainties are small relative to the values themselves and that the formula is well approximated by its tangent line over the range covered by the uncertainties. Both assumptions hold for the typical lab-report case where sigma_x is one to ten percent of x.

According to NIST/SEMATECH e-Handbook of Statistical Methods, propagation of error for independent inputs gives sigma_f^2 = sum (partial f / partial x_i)^2 * sigma_xi^2

When the underlying dataset of readings is what you actually have and sigma_x is the sample standard deviation rather than a quoted instrument tolerance, Standard Deviation Calculator returns the same sigma value from the comma-separated input format.

Four ideas cover almost every error-propagation question in an undergraduate lab course.

All four rules are special cases of the master Taylor propagation equation. The partial derivative of f with respect to each input tells you exactly how much that input contributes to sigma_f, and when one term in the variance sum dominates the others you can usually ignore them for a quick estimate.

For the product and quotient rules it is often easier to think in percent uncertainty than absolute, and Relative Standard Deviation Calculator returns the percent coefficient of variation from the same readings list so the two views stay in sync.

Four steps take you from a formula and a set of measured values to a defensible plus-or-minus on the result.

1. 1 Pick the formula mode: Choose linear for f = a*x + b*y + c, product or quotient for f = x*y or f = x/y, and power for f = x^n.
2. 2 Enter the measured values: Type x and y (or just x for power mode) in the same unit. Mixing units is the most common source of a wrong answer.
3. 3 Enter the 1-sigma uncertainties: Type sigma_x and sigma_y in the same unit as the values. Set an uncertainty to 0 for an exact value.
4. 4 Read the result: The output panel shows f, sigma_f, the percent uncertainty, and the reported result already rounded to the right significant figures.

Example: a density measurement with mass m = 23.4 g, sigma_m = 0.05 g and volume V = 10.0 mL, sigma_V = 0.05 mL. Use product mode with the divide operator. The result is rho = 2.34 g/mL with sigma_rho = 2.34 * sqrt((0.05/23.4)^2 + (0.05/10.0)^2) = 0.04. Report 2.34 +/- 0.04 g/mL.

When the final step of the report is to compare the propagated result to a known textbook value, Percent Error Calculator turns the same f and sigma_f into a percent-error number for the discussion section.

A single tool that covers the three propagation rules most lab reports actually need.

• • Three rules in one tool: Linear, product or quotient, and power rules are the three recipes a typical lab report needs.
• • Absolute and relative uncertainty: The output panel reports sigma_f and the percent uncertainty together.
• • Constants with uncertainty: The linear mode lets you set a constant c with its own sigma_c, the right way to handle a calibration constant.
• • Negative and fractional exponents: The power mode supports n = 0, n < 0, and fractional n, and warns when the formula becomes undefined.
• • Formatted for the report: The reported result row rounds f and sigma_f to the same significant figures, ready to paste.

These five benefits are the reasons the error propagation rule deserves its own tool rather than a spreadsheet cell. The most common error in student lab reports is applying the sum rule to a product, and switching modes here keeps the rule that matches the formula.

When the lab report needs an interval rather than a plus-or-minus, Confidence Interval Calculator takes the same sigma_f and turns it into a lower and upper bound at any confidence level for a normal distribution.

Five factors drive the size of the propagated uncertainty, plus two limitations to know before you defend the number.

• • The first-order Taylor approximation breaks down for strongly non-linear functions like f = sin(x) or f = exp(x) when sigma_x is more than about ten percent of x. Use a Monte Carlo propagation or expand to second order in that case.
• • The independent-input assumption fails when x and y are correlated, for example when both come from the same calibration or share a common systematic effect. In that case add the covariance term 2 * rho * (df/dx) * (df/dy) * sigma_x * sigma_y to the variance sum.

If the propagated relative uncertainty comes out larger than about ten percent, stop and reconsider whether the first-order rule is really the right one. The same five factors that drive the size of sigma_f also drive the size of the second-order correction. For most error propagation calculator users the first-order rule is the right one.

According to JCGM 100:2008 (GUM), for uncorrelated input quantities, the combined standard uncertainty is u_c^2(y) = sum (partial y / partial x_i)^2 * u^2(x_i)

When a measured value is far from a known constant and you want to know how many sigma_f away it sits, Z-Score Calculator turns the same f and sigma_f into a standardized z-score for the discussion.