U5 Estimation and Error: Essential Formulas

This topic covers the concepts of approximation, significant figures, rounding, and the calculation of absolute, relative, and percentage errors. Understanding these is crucial for handling measurements and data in scientific and practical contexts.

1 Approximation and Rounding

Rounding to a Given Place Value

To round a number to $n$ decimal places, look at the $(n+1)$th digit. If it is $5$ or more, round up the $n$th digit; otherwise, leave it unchanged.

$$ \text{Round}(x, n) $$

Significant Figures

The first non-zero digit from the left is the first significant figure. To round to $k$ significant figures, apply the rounding rule to the $k$th significant figure.

2 Types of Error

Absolute Error

The absolute difference between the measured value $M$ and the true value $T$.

$$ \text{Absolute Error} = |M - T| $$

Relative Error

The ratio of the absolute error to the true value. It is a dimensionless measure of accuracy.

$$ \text{Relative Error} = \frac{|M - T|}{|T|} \quad (T \neq 0) $$

Percentage Error

The relative error expressed as a percentage.

$$ \text{Percentage Error} = \frac{|M - T|}{|T|} \times 100\% $$

3 Maximum Absolute Error and Error Intervals

Maximum Absolute Error

If a measurement is given as $a$ (to the nearest unit), the maximum absolute error is half of that unit. For example, if measured to the nearest $0.1$, the maximum absolute error is $0.05$.

$$ \text{Max. Abs. Error} = \frac{1}{2} \times \text{Unit of Measurement} $$

Error Interval

The range of possible true values given a measured value $M$ and the maximum absolute error $E$.

$$ M - E \leq T < M + E $$

Note: The upper bound is usually taken as $M+E$ for continuous quantities, but the inequality may be strict ($<$) depending on the rounding convention.

4 Propagation of Errors in Calculations

Addition and Subtraction

When adding or subtracting approximate values, the maximum absolute errors add up.

$$ \text{Max. Abs. Error}(A \pm B) = E_A + E_B $$

Multiplication and Division

When multiplying or dividing approximate values, the maximum relative (or percentage) errors add up approximately.

$$ \text{Max. Rel. Error}(A \times B \text{ or } A \div B) \approx \frac{E_A}{|A|} + \frac{E_B}{|B|} $$

The maximum percentage error is approximately the sum of the individual percentage errors.

Struggling with complex problems?

Learner App features AI step-by-step analysis technology. Snap a photo and it will guide you through the solution!

Download Learner Now