### 誤差

 ${\left({a}\right)}$ The difference in value between an approximation and the actual value is called the absolute error. It is always positive.

If the length of a bridge is ${19.4}\text{km}$, but when Ken tells the reporters he rounds it up and says it is ${20}\text{km}$ long. In this case, the absolute error is ${0.6}\text{km}$.

 ${\left({b}\right)}$ In measurement, the actual value and the absolute error cannot be found. However, the largest possible error of the measured value, which is called maximum absolute error, can be determined. Maximum absolute error $=\dfrac{{1}}{{2}}\times$ scale interval of the measuring tool Lower limit of the actual value $=$ measured value $-$ maximum absolute error Upper limit of the actual value $=$ measured value $+$ maximum absolute error

Peter measures his height with a ruler in which the smallest interval is ${1}\text{cm}$. In this case, the maximum absolute error will be ${0.5}\text{cm}$.

If the result of measurement is ${172}\text{cm}$, then the lower limit of his actual height is ${171.5}\text{cm}$ and the upper limit of his actual height is ${172.5}\text{cm}$ .

 ${\left({c}\right)}$ Relative error $=\dfrac{\text{absolute}}{\text{actual value}}{(}$or $\dfrac{\text{maximum absolute error}}{\ \text{ measured value}}$)

As in above example, if the actual height of Peter is ${172.4}\text{cm}$ , then the "relative error from actual value" is $\dfrac{{{172.4}-{172}}}{{172.4}}\approx{0.00232}$ .

On the other hand, the "relative error of his measurement method" is $\dfrac{{0.5}}{{172}}\approx{0.00291}$ .

 ${\left({d}\right)}$ Percentage error $=$ relative error$\times$100%

As in above example, the "percentage error from actual value" is $\dfrac{{{172.4}-{172}}}{{172.4}}\times{100}\%\approx{0.232}\%$.

On the other hand, the "percentage error of his measurement method" is $\dfrac{{0.5}}{{172}}\times{100}\%\approx{0.291}\%$.

When measurements like length and width are substituted into a formula to find another quantity like perimeter and area, the errors in measurements will lead to an error in the result. We describe errors that arise in this way as accumulated errors.

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