Wikijunior:The Book of Estimation/Absolute error

Absolute error is the simplest type of error. It is the difference between the actual value and the estimated or approximate value.

Absolute error in computation

In computational estimation, the error is the difference between the actual value and the measured value.

Absolute error in measurement

Although it is impossible to find the actual value of a measurement, the actual value does exist. The difference between the actual value and the measured value is called the absolute error.

Why 'absolute'?

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When a number is absolute, it is not negative. Absolute is expressed by vertical bars. For example, 1-2=-1, while |1-2|=1. When we say that the absolute error is the absolute value of the actual value minus the measured value, that means we should subtract the measured value from the actual value, then remove the negative sign (if any). This works because if a-b=c, b-a=-c.

Absolute error, as stated above, is the difference between the actual and measured values. In other words,

${\text{Absolute error}}={\text{Actual value}}-{\text{Measured value}}\,$
${\text{(where Actual value }}\geqslant {\text{Measured value)}}\,$
${\text{Absolute error}}={\text{Measured value}}-{\text{Actual value}}\,$
${\text{(where Measured value }}\geqslant {\text{Actual value)}}\,$

Or, if you want to be exact (see the box on the side):

${\text{Absolute error}}=\vert {\text{Actual value}}-{\text{Measured value}}\vert \,$

Let's look at absolute error in action then!

Example 1
Question	(a)The measured length of a pencil is 6cm. The actual length of the pencil is 6.4cm. Find the absolute error. (b)The actual population of City B is 123406. The government estimate at the beginning of the year was 123000. Find the absolute error.
Solution	(a)${\begin{aligned}{\text{The absolute error of the measurement}}&=(6.4-6){\text{cm}}\\&=0.4{\text{cm}}\end{aligned}}$ (b)${\begin{aligned}{\text{The absolute error of the estimation}}&=(123,456-123000)\\&=406\end{aligned}}$

What about Joe Bloggs?

Example 1	Continued from Example I
Question	Find the absolute error of the measurements if the actual height of the first storey is 3.25m, the actual width of the building is 3.85m, and the actual length of the building is 6.05m.
Solution	Recall: It is ten storeys high and the first storey is around three metres tall. The length the building is around six metres, and the width four metres. ${\begin{aligned}{\text{The absolute error of the measured height of the first storey}}&=(3-3.25){\text{cm}}\\&=0.25{\text{cm}}\end{aligned}}$

Vocabulary list

• Absolute error

Exercises

1. Using the benchmark strategy, I estimated that a library card four one-dollar coins wide. Given that one-dollar coins have a diametre of 1.4 cm and that the card is 6 cm wide, find the absolute error of my estimation.
2. 346 603 - 153 345 ≈ 200 000. What is the absolute error?

Answers:

1. The absolute error = 6 cm - 1.4 cm × 4 = 5.6 cm.
2. The absolute error = |(346 603 - 153 345) - 200 000| = 193 258 - 200 000 = -6 742