How to Calculate Absolute Error

Absolute error is the difference between the measured value and the true value. ^{[1] X Research Source} This is a way to check the accuracy of the values when measuring. If you know the true and measured values, you can calculate the absolute error by performing a simple subtraction. However, if sometimes the true value is not available, you can take the maximum possible error as the absolute error. ^{[2] X Research Source} If you know the true value and the relative error, you can also work backwards to find the absolute error.

Table of Contents

Use measured value and actual value

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Set the absolute error formula. Recipe:

Δ

x

=

x

0

-

x

{displaystyle Delta x=x_{0}-x}

{displaystyle Delta x=x_{0}-x}

, in there

Δ

x

{displaystyle Delta x}

{displaystyle Delta x}

is the absolute error (or the deviation, the change between the measured value and the true value),

x

0

{displaystyle x_{0}}

{displaystyle x_{0}}

represents the measured value and

x

{displaystyle x}

x

represent real value. ^{[3] X Research Sources}

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Substitute the actual value in the formula. The actual value will be given. Otherwise, use the accepted reference value. Substitute this value in

x

{displaystyle x}

x

in the formula.

Example: You are measuring the length of a football field. In Vietnam, the true value or accepted length of a professional football field is about 110 meters. Substituting the actual value of 110 into the formula, we have: $Δ$ $x$ $=$ $x$ $0$ $-$ $110$ ${displaystyle Delta x=x_{0}-110}$ ${displaystyle Delta x=x_{0}-110}$ .

Use true value and relative error

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Set up the formula for calculating the relative error. Recipe:

Δ

x

=

x

0

-

x

x

{displaystyle delta x={frac {x_{0}-x}{x}}}

{displaystyle delta x={frac {x_{0}-x}{x}}}

, in there

Δ

x

{displaystyle delta x}

{displaystyle delta x}

is the relative error (the ratio of the absolute error to the true value),

x

0

{displaystyle x_{0}}

{displaystyle x_{0}}

represents the measured value and

x

{displaystyle x}

x

represent real value. ^{[4] X Research Sources}

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Enter the relative error in the formula. Usually, the relative error will be in the form of a decimal or a percentage. You need to replace this value in

Δ

x

{displaystyle delta x}

{displaystyle delta x}

.

For example, if the relative error is 0.025, the formula becomes: $0$ $,$ $025$ $=$ $x$ $0$ $-$ $x$ $x$ ${displaystyle 0.025={frac {x_{0}-x}{x}}}$ ${displaystyle 0.025={frac {x_{0}-x}{x}}}$ .

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Enter the actual value in the formula. This parameter is usually given first. Proceed to replace this value in

x

{displaystyle x}

x

.

For example, if the given true value is 360 m, the formula becomes: $0$ $,$ $025$ $=$ $x$ $0$ $-$ $360$ $360$ ${displaystyle 0.025={frac {x_{0}-360}{360}}}$ ${displaystyle 0.025={frac {x_{0}-360}{360}}}$ .

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Multiply both sides of the equation by the real value to cancel out the fraction.

For example:
$0$ $,$ $025$ $=$ $x$ $0$ $-$ $360$ $360$ ${displaystyle 0.025={frac {x_{0}-360}{360}}}$ ${displaystyle 0.025={frac {x_{0}-360}{360}}}$
$0$ $,$ $025$ $\times$ $360$ $=$ $x$ $0$ $-$ $360$ $360$ $\times$ $360$ ${displaystyle 0.025times 360={frac {x_{0}-360}{360}}times 360}$ ${displaystyle 0.025times 360={frac {x_{0}-360}{360}}times 360}$
$9$ $=$ $x$ $0$ $-$ $360$ ${displaystyle 9=x_{0}-360}$ ${displaystyle 9=x_{0}-360}$

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Add the real value to both sides. You will find measurable value

x

0

{displaystyle x_{0}}

{displaystyle x_{0}}

.

For example:
$9$ $=$ $x$ $0$ $-$ $360$ ${displaystyle 9=x_{0}-360}$ ${displaystyle 9=x_{0}-360}$
$9$ $+$ $360$ $=$ $x$ $0$ $-$ $360$ $+$ $360$ ${displaystyle 9+360=x_{0}-360+360}$ ${displaystyle 9+360=x_{0}-360+360}$
$369$ $=$ $x$ $0$ ${displaystyle 369=x_{0}}$ ${displaystyle 369=x_{0}}$

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Subtract the actual value from the measured value. Since the absolute error is always positive, you need to get the absolute value of the deviation by removing the minus sign. This would be the absolute error.

For example, if the measured value is 369 m and the actual value is 360 m, we have: $369$ $-$ $360$ $=$ $9$ ${displaystyle 369-360=9}$ ${displaystyle 369-360=9}$ . So, the absolute error is 9 m.

Use the maximum possible error

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Determine the unit of measurement. This is the “correct to” value mentioned. This quantity may be stated (e.g., “Building is measured to the nearest meter”), but this is not always the case. To determine the unit of measure, you simply need to look at the number of decimal places to which the measurement is rounded.

For example, if the measured length of a road is 357 m, you will know the road is measured to the nearest meter. So, the measurement value is 1 m.

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Determine the maximum possible error. The maximum possible error is

first

2

{displaystyle {frac {1}{2}}}

{frac {1}{2}}

measurement value. ^{[5] X Research Source} This value is usually written in the form

\pm

{displaystyle pm }

pm

number.

Continuing the example above, if the unit of measurement is meters, the maximum possible error is 0.5 m. Therefore, the measure of the path length will be $357$ $\pm$ $0$ $,$ $5$ $m$ ${displaystyle 357pm 0.5m}$ ${displaystyle 357pm 0.5m}$ . This means that the actual value of the path length may be 0.5 m shorter or longer than the measured value. Where this length is shorter/longer, the measurement value can be 356 or 358 m.

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Take the maximum possible error as the absolute error. ^{[6] X Research Source} Since the absolute error is always positive, you need to get the absolute value of the deviation by removing the minus sign. This would be the absolute error.

In the example, if you find the measure of the road length is $357$ $\pm$ $0$ $,$ $5$ $m$ ${displaystyle 357pm 0.5m}$ ${displaystyle 357pm 0.5m}$ then the absolute error will be 0.5 m.

Advice

If the actual value is not given, you can take the accepted value or the theoretical value.

Steps

Use measured value and actual value

Use true value and relative error

Use the maximum possible error

Advice

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