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Percent error and uncertainty propagation calculator

Percent error from measured and accepted values, and uncertainty propagation through sums, products, quotients and powers, in quadrature or worst case.

更新日 検証済みの例：7

試す
Percent error
%
Percent error: 0.9174 %
小数点以下の最大桁数：4；最も近い値へ、等距離ならゼロから遠い値へ
Signed percent error
−0.9174%
Absolute error
0.09

The measurement is 0.9174% below the accepted value.

Signed percent error (0 = the accepted value)

over 5% low1–5% low1–5% highover 5% high-10-5-11510accepted−0.9174%
計算方法 S

  1. Difference from the accepted value

    xm−xa=9.72−9.81=−0.09x_m - x_a = 9.72 - 9.81 = -0.09

  2. Percent error

    ∣−0.09∣∣9.81∣×100=0.917431%\frac{|-0.09|}{|9.81|}\times100 = 0.917431\%

Percent error and uncertainty propagation calculatorについて

Percent error compares a measurement with an accepted value: |measured − accepted| / |accepted| × 100. Uncertainty propagation works out how the ± uncertainties of inputs carry into a calculated result. For sums and differences the absolute uncertainties combine; for products and quotients the relative uncertainties combine; a power aⁿ multiplies the relative uncertainty by |n|. Independent random errors combine in quadrature (the square root of the sum of squares), and the worst-case option adds them linearly.

The default, a measured g of 9.72 m/s² against the accepted 9.81 m/s², is 0.917% low. Multiplying (2.0 ± 0.1) by (3.0 ± 0.2) gives relative uncertainties of 5% and 6.7%, which combine in quadrature to 8.3%, so the product is reported as 6.0 ± 0.5.

Results are rounded as Taylor's An Introduction to Error Analysis recommends: the uncertainty to one significant figure, or two when it starts with 1, and the value to the same decimal place. The formulas are first-order, so they assume each uncertainty is small compared with its value.

計算例

Measured g = 9.72 against 9.81

Calculate
Percent error
Measured value
9.72
Accepted (true) value
9.81
Percent error
0.9174%
Signed percent error
-0.9174%
Absolute error
0.09

照合元：Python 3.8 decimal: |9.72 − 9.81| / 9.81 × 100 = 0.9174312

Sum in quadrature: (10.0 ± 0.3) + (5.0 ± 0.4)

Calculate
Sum or difference a ± b
Operation
a + b or a × b
a
10.0
Uncertainty in a (±)
0.3
b
5.0
Uncertainty in b (±)
0.4
Combine uncertainties
In quadrature
Result q
15
Uncertainty δq
0.5
Report as
15.0 ± 0.5

照合元：Taylor §3.5: √(0.3² + 0.4²) = 0.5

Same sum, worst case

Calculate
Sum or difference a ± b
Operation
a + b or a × b
a
10.0
Uncertainty in a (±)
0.3
b
5.0
Uncertainty in b (±)
0.4
Combine uncertainties
Worst case
Uncertainty δq
0.7
Report as
15.0 ± 0.7

照合元：Taylor §3.3: δa + δb = 0.7

Product (2.0 ± 0.1) × (3.0 ± 0.2)

Calculate
Product or quotient a × b, a ÷ b
Operation
a + b or a × b
a
2.0
Uncertainty in a (±)
0.1
b
3.0
Uncertainty in b (±)
0.2
Combine uncertainties
In quadrature
Result q
6
Uncertainty δq
0.5
Report as
6.0 ± 0.5

照合元：Python 3.8 decimal: 6 × √(0.05² + 0.0667²) = 6 × 0.083333 = 0.5

よくある質問

How do you calculate percent error?

Percent error = |measured − accepted| / |accepted| × 100. Measuring g as 9.72 m/s² against an accepted 9.81 m/s² gives 0.09/9.81 × 100 = 0.917%. Keep the sign when the direction matters: −0.917% says the measurement came out low. The formula fails when the accepted value is 0; report the absolute error instead.

How do you propagate uncertainty when multiplying or dividing?

Combine the relative uncertainties in quadrature: δq/|q| = √((δa/a)² + (δb/b)²). For (2.0 ± 0.1) × (3.0 ± 0.2) the relative uncertainties are 5% and 6.7%, giving 8.3% of 6.0, so q = 6.0 ± 0.5. Division follows the same rule: (10 ± 0.5) ÷ (4 ± 0.2) = 2.50 ± 0.18.

How do you add or subtract uncertainties?

Combine the absolute uncertainties: δq = √(δa² + δb²) when the errors are independent and random, or δa + δb for a worst-case bound. (10.0 ± 0.3) + (5.0 ± 0.4) = 15.0 ± 0.5 in quadrature, or 15.0 ± 0.7 worst case. Subtraction uses the same rule, so the difference of two close values can carry a large relative uncertainty.

When should uncertainties be added in quadrature?

When the errors in the inputs are independent and random, the assumption behind the combined standard uncertainty in NIST Technical Note 1297 and the GUM. Quadrature never exceeds the linear sum: 0.3 and 0.4 combine to 0.5 rather than 0.7. Use the linear worst case when errors may be correlated, such as two readings taken with the same miscalibrated instrument.

How many significant figures should an uncertainty have?

One, or two when its leading digit is 1, following Taylor's An Introduction to Error Analysis (§2.2); the value is then rounded to the same decimal place. So 0.1768 becomes 0.18 and the result is written 2.50 ± 0.18, while 0.52 becomes 0.5 and the result 6.0 ± 0.5. Writing 6.0 ± 0.5234 claims more precision than the measurement has.

「Percent error and uncertainty propagation calculator」の精度はどのくらいですか？

精度は入力値と計算方法の前提に依存します。十進演算には有効数字50桁を使いますが、推定、数値計算手法、元データの精度はそれより低い場合があります。表示の丸め処理でこれらの制約がなくなるわけではありません。 独立した出典の解答と照合した計算例：7。 例えば、「Measured g = 9.72 against 9.81」はPython 3.8 decimal: |9.72 − 9.81| / 9.81 × 100 = 0.9174312と照合しています。

この計算方法の出典は何ですか？

Taylor, An Introduction to Error Analysis (2nd ed., 1997), ch. 2–3 (rules for reporting and propagating uncertainties); NIST Technical Note 1297 — Guidelines for evaluating and expressing the uncertainty of NIST measurement results.

この計算機について

% error=∣xm−xa∣∣xa∣×100;δ(a±b)=δa2+δb2;δ(ab)∣ab∣=(δaa)2+(δbb)2;δ(an)∣an∣=∣n∣δa∣a∣\%\,\text{error} = \frac{|x_m - x_a|}{|x_a|}\times100;\quad \delta(a\pm b) = \sqrt{\delta a^2 + \delta b^2};\quad \frac{\delta(ab)}{|ab|} = \sqrt{\Big(\frac{\delta a}{a}\Big)^2 + \Big(\frac{\delta b}{b}\Big)^2};\quad \frac{\delta(a^n)}{|a^n|} = |n|\frac{\delta a}{|a|}

出典

  1. Taylor, An Introduction to Error Analysis (2nd ed., 1997), ch. 2–3 (rules for reporting and propagating uncertainties)
  2. NIST Technical Note 1297 — Guidelines for evaluating and expressing the uncertainty of NIST measurement results

出典と照合済み

この計算機には、独立した出典の解答を使った計算例が 7 件あります。テストに組み込まれており、ここでも実行できます。

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