How to add absolute uncertainties PLEASE HELP!

Question

Instrument	Type of instrument	Limiting of reading	Absolute Uncertainty	Total absolute uncertainty for experiment
Electronic Scale	Digital	0.1g	+-0.1g	+-0.6
Stopwatch	Digital	0.01s	Average around +-0.25s
Measuring Cylinder	Analogue	1.0ml	+-0.5ml
Lab quest thermometer	Digital	0.1°C	+-0.1°C

How do I find the total uncertainties?

i used this formula, but I'm not sure if it is right;

+- square root of uncertainties squared and added to eachother

Answer 1

Your formula is a real method — but it is not the only one, and which you should use depends on the operation and on what your syllabus expects.

The rule depends on the arithmetic

Adding or subtracting quantities → combine ABSOLUTE uncertainties.
(mass by difference, a temperature change, a volume delivered from a burette)

Multiplying or dividing → combine RELATIVE (percentage) uncertainties.
(density = m/V, concentration = n/V, rate = amount/time)

Mixing these up is the single most common error in this topic. You cannot add an uncertainty in grams to one in seconds — but you can add 2% to 1%.

Linear vs quadrature — the part you are asking about

Once you know which uncertainties to combine, there are two conventions for combining them:

Linear (simple) rule: e_total = e₁ + e₂ + e₃ + ...
Quadrature (statistical) rule: e_total = √(e₁² + e₂² + ...)

Your formula is the quadrature rule, and it is correct — it is the statistically proper way to combine independent, random uncertainties, and it is what university and research work uses. It always gives a smaller total, because independent errors partly cancel rather than all conspiring in the same direction.

The linear rule is a deliberately conservative worst-case estimate. IB and most high-school syllabuses require the linear rule, so check which your course wants before switching.

Your table is already using the linear rule

Look at your electronic scale: reading uncertainty ±0.1 g, total ±0.6 g. That is 6 × 0.1 — six measurements combined linearly.

Had you used quadrature, the same six would give √(6 × 0.1²) = ±0.24 g. So your table and your formula currently disagree with each other. Pick one convention and apply it throughout — a report that mixes them is inconsistent even if each individual number is defensible.

Your instrument uncertainties are right, and for the right reasons

• Digital instruments: ± the smallest displayed increment. Scale ±0.1 g, thermometer ±0.1 °C — both correct.
• Analogue instruments: ± half the smallest division. Your 1.0 mL cylinder giving ±0.5 mL is exactly right.
• Stopwatch: good instinct writing ±0.25 s rather than ±0.01 s. The display resolves to 0.01 s, but human reaction time (roughly 0.2–0.3 s) dominates completely. When a human limitation exceeds the instrument's, use the larger value — that is a point examiners specifically look for.

Two more things worth doing

1. Identify the dominant term. In quadrature especially, squaring makes small contributions vanish: if one uncertainty is more than about three times another, the smaller one barely registers. Saying which measurement limits your precision is often the most valuable sentence in an evaluation.
2. Count every reading. Measuring mass by difference uses the balance twice, so it carries two uncertainties, not one. The same applies to initial and final burette readings.