Approximations and Errors

Syllabus tag: KCSE | Mathematics | Form 3 | Topic 2 Approximations and Errors

Lesson objectives

By the end of this topic, you should be able to:

• Make reasonable approximations and estimations in computations and measurements.
• Distinguish rounding off from truncating and find the resulting errors.
• Calculate absolute, relative and percentage errors.
• Determine how errors propagate through addition, subtraction, multiplication and division.

Approximations and Errors

Every measurement is approximate. This topic quantifies how approximate, and tracks what happens when approximate values are combined.

a) Rounding and truncating

Rounding adjusts the last kept digit according to what follows. 3.678 to 2 decimal places is 3.68.

Truncating simply cuts off the extra digits. 3.678 truncated to 2 decimal places is 3.67.

Truncating always gives a value at or below the true one. Rounding may go either way, and is usually closer.

b) Bounds of a measurement

A length given as 7 cm to the nearest centimetre lies between 6.5 and 7.5 cm.

The lower bound is 6.5 and the upper bound is 7.5.

The bounds are always half a unit either side of the stated value.

For a value to 1 decimal place, such as 4.3, the bounds are 4.25 and 4.35.

c) Absolute error

The absolute error is the largest possible difference between the measurement and the true value.

For a value rounded to the nearest unit, it is half that unit.

So 7 cm to the nearest cm has an absolute error of 0.5 cm.

Absolute error carries the same units as the measurement.

d) Relative and percentage error

Relative error = absolute error ÷ measurement. It has no units.

Percentage error = relative error × 100.

Relative error is the more useful comparison. An error of 1 cm matters greatly on a 5 cm measurement. On 5 km it is nothing.

e) Propagation in addition and subtraction

When measurements are added or subtracted, the absolute errors add.

For 7 ± 0.5 plus 12 ± 0.5, the sum is 19 ± 1.

Note that errors add even for subtraction. They never cancel, because each measurement could be wrong in either direction, and we take the worst case.

Subtraction of near-equal values is dangerous. Take 13 ± 0.5 minus 12 ± 0.5. That gives 1 ± 1, so the percentage error jumps to 100%.

f) Propagation in multiplication and division

When measurements are multiplied or divided, the relative errors add.

So a rectangle 7 ± 0.5 by 4 ± 0.5 has relative errors 0.0714 and 0.125, totalling 0.1964.

The area is 28. So the absolute error in the area is 28 × 0.1964, about 5.5.

You may instead work out the maximum and minimum products directly. Those are 7.5 × 4.5 = 33.75 and 6.5 × 3.5 = 22.75.

g) Practical advice

Give answers to a sensible accuracy. An answer quoted to six figures from data measured to two is misleading.

The final answer should be no more accurate than the least accurate measurement it came from.

h) Where this is used

Engineering tolerances. Laboratory measurement. Surveying. Any calculation built from physical measurements.

Words to know

• Rounding off -- adjusting the last retained digit according to what follows.
• Truncating -- cutting digits off without adjusting.
• Absolute error -- the difference between the stated and true values.
• Relative error -- the absolute error divided by the stated value.
• Bounds -- the smallest and largest values a measurement could represent.

:::checkpoint Check yourself

1. Give the bounds of a mass stated as 250 g to the nearest 10 g.
2. Find the percentage error in a length of 8 cm measured to the nearest cm.
3. What happens to absolute errors when two measurements are added?
4. Why is subtracting two nearly equal measurements risky? :::

Bridge to practice

The exercises begin with rounding, truncating and bounds, move through the three errors, and finish with propagation and estimation. For every measurement, write down the last unit first — halving it gives the absolute error and everything else follows.