--- title: "Propagating Uncertainty Through a Calculation" description: "A result is usually calculated from several measurements, each with its own range. The rules for combining them show which measurement is worth improving and which is already good enough." canonical: https://lightmysky.com/learn/science/propagating-uncertainty-through-a-calculation-mt_xssxg6NqQM source: https://lightmysky.com/learn/science/propagating-uncertainty-through-a-calculation-mt_xssxg6NqQM.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Propagating Uncertainty Through a Calculation A result is usually calculated from several measurements, each with its own range. The rules for combining them show which measurement is worth improving and which is already good enough. Subject: Science · Area: Chemistry · Ages 18 to 19 Page: https://lightmysky.com/learn/science/propagating-uncertainty-through-a-calculation-mt_xssxg6NqQM ## Ready when they can - Combines absolute uncertainties for a sum or difference and relative ones for a product or quotient - Handles a power correctly and explains why the exponent multiplies the relative uncertainty - Identifies the dominant contribution in a multi-step calculation and says what to improve first - Explains why a small difference between two large numbers carries a large relative uncertainty ## Lesson: How uncertainty travels through your math Some things cannot be measured directly. You cannot lay a ruler on the speed of a reaction or pour out the density of a coin, so you measure what you can reach, like mass and volume, and calculate the rest. Every range you feed into a formula moves through it into the answer, the way wobble in your angle readings follows along into the refraction index. Combine with the matching rule. For a sum or a difference, add the absolute ranges. For a product or a quotient, add the percentage ranges. For a power, multiply the percentage range by the exponent, so a value that wobbles by 2 percent carries about 4 percent when squared. **Example.** Take 10.24 minus 10.11, which is 0.13, while each input wobbles by about 0.01. The answer is tiny next to the inputs, so its percentage wobble is huge. That is the pattern to watch: a small difference between two large numbers always carries a large relative uncertainty. **Tip.** Ask which input moves your answer most, and improve that one first, the way you would ask how much the index shifts when your angles wobble by two degrees. Then round the final answer to stop where your least precise measurement stops. **Recap.** Carry every range through the formula with its matching rule, and spend your effort on the input that moves the answer most. ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Uncertainty in a Measurement and Where It Comes From](https://lightmysky.com/learn/science/uncertainty-in-a-measurement-and-where-it-comes-from-mt_JXw4D9DReI) - [The Derivative as a Rate of Change](https://lightmysky.com/learn/mathematics/the-derivative-as-a-rate-of-change-mt_RqyLE3jrAW) ## Opens up - [Significance Tests for Chemical Data](https://lightmysky.com/learn/science/significance-tests-for-chemical-data-mt_2-SDgPjOQk) - [Dimensional Analysis, Scaling and the Order-of-Magnitude Estimate](https://lightmysky.com/learn/science/dimensional-analysis-scaling-and-the-order-of-magnitude-estimate-mt_jXnXRMKnC3)