【近似與誤差：絕對誤差、相對誤差及百分誤差】Approximation and Errors: Absolute, Relative and Percentage Error
SketchMath 數學團隊 SketchMath Maths Team ｜ 更新 Updated 2026-10-20
量度結果準確至某單位，最大絕對誤差是該單位的一半。相對誤差 = 絕對誤差 ÷ 量度值，百分誤差 = 相對誤差 × 100%。A measurement correct to a given unit has a maximum absolute error of half that unit. Relative error = absolute error ÷ measured value, and percentage error = relative error × 100%.
改變準確程度 d：量度值 12.5 位於可能真實值區間的中央，區間半闊為 d/2。 Drag d, the degree of accuracy: the measurement 12.5 sits at the centre of the interval of possible true values, of half-width d/2. 自己試試 → Try it yourself →
誤差公式 Error Formulas
最大絕對誤差Maximum absolute error½ × degree of accuracy
量度值範圍Limits of a measurementx − e ≤ true value < x + e
相對誤差Relative errorabsolute error / measured value
百分誤差Percentage errorrelative error × 100%
乘積的最大／最小值Largest / smallest of a product把各個上限相乘／各個下限相乘Multiply the upper limits / the lower limits
某長度量得 12.5 cm，準確至最接近 0.1 cm。求最大絕對誤差、量度範圍及百分誤差。A length is measured as 12.5 cm, correct to the nearest 0.1 cm. Find the maximum absolute error, the limits, and the percentage error.
一個長方形長 5.0 m、闊 3.0 m，兩者均準確至最接近 0.1 m。求面積的最大及最小可能值。A rectangle is 5.0 m by 3.0 m, each correct to the nearest 0.1 m. Find the greatest and least possible area.
長度介乎 4.95 至 5.05；闊度介乎 2.95 至 3.05Length between 4.95 and 5.05; width between 2.95 and 3.05
Greatest = 5.05 × 3.05 = 15.4025
Least = 4.95 × 2.95 = 14.6025
答案： Answer: 最大 15.4025 m²，最小 14.6025 m²Greatest 15.4025 m², least 14.6025 m²
⚠️ 是單位的一半，不是整個單位 Half the unit, not the whole unit準確至最接近 0.1 cm，誤差最多是任何方向 0.05 cm，並非 0.1 cm。Correct to the nearest 0.1 cm means the error can be up to 0.05 cm in either direction, not 0.1 cm.
⚠️ 上限不包括在內 Upper limit is not included範圍寫成 12.45 ≤ x < 12.55。12.55 會進位成 12.6。Write the range as 12.45 ≤ x < 12.55. A value of 12.55 would round up to 12.6.
絕對誤差用與量度相同的單位表示誤差大小；相對誤差則與量度值比較，可用來比較不同量度的準確程度。
Absolute error is the size of the error in the same unit as the measurement. Relative error compares it with the measured value, so it lets you compare accuracy of different measurements.
把最大絕對誤差分別減去及加上：下限包括在內，上限不包括。
Subtract and add the maximum absolute error: the lower limit is included, the upper limit is not.
80 cm：絕對誤差相同（0.5 cm），但所除的量度值較大。
80 cm: the absolute error is the same (0.5 cm) but it is divided by a larger value.