---
title: "Combining Uncertainties and Reading a Result off a Graph"
description: "Uncertainties in a sum add in absolute terms and uncertainties in a product add as percentages, so the dominant one usually decides the answer. Plotted with error bars, the steepest and shallowest lin"
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source: https://lightmysky.com/learn/science/combining-uncertainties-and-reading-a-result-off-a-graph-mt_kn8RhTDEOK.md
retrieved: 2026-09-12
---

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# Combining Uncertainties and Reading a Result off a Graph

Uncertainties in a sum add in absolute terms and uncertainties in a product add as percentages, so the dominant one usually decides the answer. Plotted with error bars, the steepest and shallowest lines that still pass through them give the uncertainty in a gradient.

Subject: Science · Area: Scientific Inquiry · Ages 17 to 18
Page: https://lightmysky.com/learn/science/combining-uncertainties-and-reading-a-result-off-a-graph-mt_kn8RhTDEOK

## Ready when they can

- Combines uncertainties through a multi-step calculation and names which measurement dominates the result
- Draws error bars, then a best line and a worst acceptable line, and gets an uncertainty in the gradient from the pair
- Decides whether two published values agree, using their stated uncertainties rather than their central values

## Lesson: Honest numbers with error bars

Every measurement carries uncertainty, and a result without a stated uncertainty is unfinished. Accuracy means closeness to the true value, while precision means closeness of repeats to each other. Absolute error is plus or minus so much, and fractional error divides that by the value.

Uncertainties travel through calculations by fixed rules. Absolute errors add for sums, and fractional errors add for products, with quadrature as the refined rule for small errors. The input with the largest fractional error dominates the final spread, so shrink that measurement first. Answers inherit the coarsest input, which is the weakest link principle behind significant figures.

**Example.** On a graph, draw error bars, then a best line plus the steepest and shallowest lines the bars still allow. Half the spread between those two slopes is the gradient uncertainty. Two published values agree when the gap between their centres fits inside the combined uncertainty, which overlapping bars usually show.

**Tip.** Keep one extra digit while working, then round the answer to match the weakest input. Never invent certainty the measurements cannot support.

**Recap.** Track how each error travels, let the weakest input set the figures, and quote the spread.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [Uncertainty, Precision and Significant Figures in a Measurement](https://lightmysky.com/learn/science/uncertainty-precision-and-significant-figures-in-a-measurement-mt_HPc3OahYoV)
- [Acceleration from Velocity-Time Graphs](https://lightmysky.com/learn/science/acceleration-from-velocity-time-graphs-mt_nmvfbSgnOn)

## Opens up

- [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)
- [Systematic Error, Controls and the Design of a Measurement](https://lightmysky.com/learn/science/systematic-error-controls-and-the-design-of-a-measurement-mt_rGjB_JcsLJ)