Percent error from measured and accepted values, and uncertainty propagation through sums, products, quotients and powers, in quadrature or worst case.
Aggiornato Esempi verificati: 7
The measurement is 0.9174% below the accepted value.
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.
Fonte di verifica: Python 3.8 decimal: |9.72 − 9.81| / 9.81 × 100 = 0.9174312
Fonte di verifica: Taylor §3.5: √(0.3² + 0.4²) = 0.5
Fonte di verifica: Taylor §3.3: δa + δb = 0.7
Fonte di verifica: Python 3.8 decimal: 6 × √(0.05² + 0.0667²) = 6 × 0.083333 = 0.5
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.
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.
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 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.
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.
La precisione dipende dai dati inseriti e dalle ipotesi del metodo. Il calcolo decimale usa 50 cifre significative, ma stime, metodi numerici e dati di origine possono essere meno precisi; l’arrotondamento visualizzato non elimina questi limiti. Esempi svolti verificati con fonti indipendenti: 7. Per esempio, «Measured g = 9.72 against 9.81» viene verificato con Python 3.8 decimal: |9.72 − 9.81| / 9.81 × 100 = 0.9174312.
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.
Questa calcolatrice include 7 esempi svolti con risposte da fonti indipendenti. Fanno parte della suite di test e puoi eseguirli anche qui.
Kinematics calculator for the SUVAT equations: enter three of displacement, initial and final velocity, acceleration and time to get the other two.
Ohm's law calculator: enter any two of voltage, current, resistance and power to get the other two, shown in the units you pick (mA, kΩ…).