Accuracy & Precision

You measure your textbook with a ruler and write down 21.3 cm. But is the book exactly 21.300000… cm long? Impossible. Every measurement is only an approach to the true value — the point is not to "eliminate error" but to know how big the error is and report it honestly.

Measurement has limits

The smallest division of a millimetre ruler is 1 mm. When you read 21.3 cm, the true length actually lies somewhere between 21.25 cm and 21.35 cm — the error is at most half a division, that is $\pm 0.5\ \mathrm{mm}$ . This is the precision of the measurement: set by the tool's smallest division, not by how hard you squint. A millimetre ruler can never measure 21.32 cm — an extra digit you write in is fiction.

Significant figures: numbers speak for themselves

How a number is written carries a precision promise: a height of 1.62 m has three significant figures and claims accuracy to 1 cm (error $\pm 0.5$ cm); reporting only 1.6 m is two significant figures, accurate to 10 cm (error $\pm 5$ cm). The count of significant figures is the declaration "this is how far I am exact". The number 0.0012 has just two significant figures — the leading zeros only hold places.

Errors compound

Estimate first, then compute exactly

To work out $8.4 \times 2.1$ : first round to whole numbers for an estimate, $8 \times 2 = 16$ — the true answer should live near 16. The exact product is $8.4 \times 2.1 = 17.64$ . If your exact calculation produced 176.4, the estimate would cry foul at once. The estimating you practised in rounding is the smoke alarm for errors.

Rounding snowballs

A rectangle's length and width are rounded to whole centimetres and recorded as 7 cm and 4 cm. Using $7 \times 4 = 28$ for the area is convenient, but the true lengths could be anywhere in 6.5–7.5 and 3.5–4.5, so the area could be as small as $6.5 \times 3.5 = 22.75$ or as large as $7.5 \times 4.5 = 33.75$ . Each number is only a little off, yet multiplied together the errors swell. So: avoid rounding in the middle of a calculation — estimate first, round once at the end.

Report it honestly

An engineering drawing reads "diameter $20 \pm 0.1$ mm" — the allowed deviation is called the tolerance, and it is the bedrock of modern manufacturing. In daily life, honest reporting takes just two little marks: write "about" when you are unsure, and "±" when you know the precision. A temperature of 36.8 °C or a long jump of 3.4 m both declare "this is the digit I measured to". When you restate a value in different units, the promise must travel with the number — metric units and conversion helps you keep it.

Check yourself

Quick quiz

1. A millimetre ruler reads 21.3 cm. What is the largest possible error?
2. How many significant figures does 0.0012 have?
3. Why avoid rounding in the middle of a calculation?