Combining Uncertainties and Reading a Result off a Graph · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Combining uncertainty bounds

Science · Scientific Inquiry · ages 17-18
Name ______________________   Date ____________

  1. Using absolute upper bounds, add 5.0 ± 0.1 cm and 3.0 ± 0.2 cm. What is the total's absolute uncertainty bound?

    • 0.1 cm
    • 0.2 cm
    • 0.3 cm

  2. A report gives a measured length with a stated ± bound. What does the bound describe?

    • Limited knowledge of the length
    • The known signed measurement error
    • The exact difference from truth

  3. Reporting a measured result with its unit and uncertainty bound, and explaining how the bound was estimated, helps a reader judge its stated measurement limits.

    Circle one:   True   False

  4. Under the stated error-bar model, the shallowest acceptable line joins (0 s, 1.2 m) to (4 s, 8.8 m). The steepest joins (0 s, 0.8 m) to (4 s, 9.2 m). Using the midpoint and half-difference convention, report the gradient.

    • 2.0 ± 0.2 m/s
    • 2.0 ± 0.1 m/s
    • 4.0 ± 0.1 m/s

  5. For a product of two first-power factors, use the first-order percentage-bound rule. One factor has 1% uncertainty and the other 8%. Which contributes more?

    • The factor with 1% uncertainty
    • Both factors contribute equally
    • The factor with 8% uncertainty

  6. L is the sum of 6.0 ± 0.1 cm and 4.0 ± 0.1 cm. Area is L times 5.0 ± 0.4 cm. Add absolute bounds for L, then use the first-order product rule. What is the area's percentage uncertainty estimate?

    Answer: ______________

  7. Measured lengths 3.2 cm and 1.4 cm are multiplied. Using the basic product significant-figure rule, how should 4.48 cm² be reported?

    • 4.48 cm²
    • 4.5 cm²
    • 4 cm²

  8. Lengths are reported as 10.0 ± 0.2 cm and 10.3 ± 0.2 cm, using closed interval bounds. Call them compatible if their intervals overlap. Which conclusion follows?

    • Their intervals do not overlap
    • They are compatible, not proven equal
    • Their central values prove equality
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Answer key

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Combining uncertainty bounds W1-mt_kn8RhTDEOK-s1

  1. 0.3 cm · For this sum-bound rule, add 0.1 + 0.2 = 0.3 cm.
  2. Limited knowledge of the length · The bound describes uncertainty. The actual signed error requires knowledge of the true value.
  3. True · The unit identifies the measurement scale. The bound states its reported limits, and its basis explains how those limits were estimated.
  4. 2.0 ± 0.1 m/s · The stated extreme gradients are 1.9 and 2.1 m/s. Their midpoint is 2.0 m/s and half their difference is 0.1 m/s.
  5. The factor with 8% uncertainty · For this product and convention, the percentage contributions are 1% and 8%. The 8% contribution is larger.
  6. 10 · L = 10.0 ± 0.2 cm, giving 2%. The other length has 8%. The first-order product estimate is 10%.
  7. 4.5 cm² · Both measured inputs have two significant figures. Round 4.48 to two figures: 4.5 cm².
  8. They are compatible, not proven equal · The intervals are 9.8 to 10.2 cm and 10.1 to 10.5 cm. They overlap from 10.1 to 10.2 cm.
Worksheet · LightMySky