Accuracy and Precision

Introduction

In everyday speech "accurate" and "precise" mean almost the same thing. In science they are two quite different ideas, and confusing them is one of the most common mistakes students make. Accuracy is about hitting the right answer. Precision is about getting the same answer each time. A good measurement is both, but a set of results can easily be one without the other.

Core explanation

Accuracy. A measurement is accurate if it is close to the true value of the quantity. In practice we rarely know the true value exactly, so we compare with an accepted value — for example, a data-book value such as the boiling point of pure water at standard pressure, 100 °C. If a student measures 99.8 °C, the result is accurate; if they measure 94 °C, it is not.

Precision. Measurements are precise if repeated readings are close to one another — that is, the spread or scatter is small. Precision says nothing about whether the readings are right. It is judged from the data alone, without needing to know the true value.

Four combinations. Suppose the true mass of a sample is 10.00 g and four students each weigh it three times:

Student Readings (g) Precise? Accurate? --- --- --- --- A 10.01, 9.99, 10.00 Yes Yes B 10.52, 10.50, 10.51 Yes No C 9.6, 10.4, 10.1 No Mean fairly close, readings not D 10.9, 11.6, 11.2 No No

Student B's readings agree closely but are all about 0.5 g too high — a sign of a consistent bias such as an unzeroed balance. Student C's readings scatter widely; the mean (10.03 g) happens to be close, but no single reading can be trusted.

Links to errors. Poor precision is usually caused by random errors , which scatter readings on both sides of the mean. Poor accuracy with good precision is usually caused by systematic errors , which push every reading the same way. Repeating readings and averaging improves the effect of random errors but does nothing to remove a systematic error.

Precision and resolution. An instrument with finer resolution can give more precise readings — a balance reading to 0.001 g can reveal smaller differences than one reading to 0.1 g. But a high-resolution instrument that is badly calibrated will still give inaccurate results.

Step-by-step reasoning

To judge a set of repeated results:

1. Look at the spread of the readings (for example, the range). A small spread means high precision. 2. Calculate the mean of the readings. 3. Compare the mean with the accepted value. A small difference means high accuracy. 4. Describe both qualities separately, giving evidence for each.

Visual explanation

The classic picture is a dartboard. Darts clustered tightly in the bullseye are accurate and precise. Darts clustered tightly but off to one side are precise but not accurate. Darts spread all over the board are imprecise, and if they surround the centre only their average is near the bullseye.

Real-world analogy

A kitchen scale that always reads 50 g too heavy gives the same reading every time you weigh a bag of sugar — it is precise. But every reading is wrong by the same amount, so it is not accurate. Resetting its zero fixes the accuracy without changing its precision.

Real-world example

Clinical laboratories check blood-glucose analysers every day using control samples of known concentration. The staff look for two things: whether repeat results agree with each other (precision) and whether their average matches the known value (accuracy). A drift in either can lead to a wrong diagnosis.

Why?

Why must accuracy and precision be judged separately? They have different causes and different cures. Scatter comes from unpredictable variations and is reduced by repeating and averaging. Bias comes from a fault in the method or instrument and is reduced only by finding and correcting the fault. Mixing the ideas up would lead you to apply the wrong cure.

Common misconception

"If my repeat results all agree, my answer must be correct." Close agreement shows high precision only. Every reading could share the same systematic error, so precise results can still be far from the true value.

Worked example

Question: The accepted density of aluminium is 2.70 g/cm³. Group X obtains 2.69, 2.71 and 2.70 g/cm³. Group Y obtains 2.84, 2.85 and 2.83 g/cm³. Comment on the accuracy and precision of each group.

Reasoning: Both groups have a range of 0.02 g/cm³, so both are equally precise. Group X's mean is 2.70 g/cm³, equal to the accepted value. Group Y's mean is 2.84 g/cm³, 0.14 g/cm³ too high.

Answer: Both are precise; X is accurate and Y is not, suggesting a systematic error in Y's method.

Quick check

1. Readings of 5.1, 5.1 and 5.2 cm are taken for a length whose true value is 6.0 cm. Are they precise, accurate, both or neither? Answer: Precise but not accurate — they agree closely but are all about 0.9 cm too small.

Exam focus

Use the words carefully: "accurate" means close to the true value; "precise" means close together. Always back up a judgement with numbers, for example "the range is only 0.1 cm³, so the results are precise". Never write "more accurate" when you mean "more precise".

Advanced insight

Metrologists (measurement scientists) formally avoid assigning a single number to "accuracy". Instead, they state a measured value with its uncertainty and describe the separate contributions of random effects and systematic effects. Precision is often quantified by the standard deviation of repeated readings, a more refined measure of spread than the range.

Summary

Accuracy is how close a result is to the true or accepted value; precision is how close repeated results are to each other. Results can be precise without being accurate, usually because of a systematic error. Random errors reduce precision, and systematic errors reduce accuracy. Always judge and report the two qualities separately.

Practice questions

1. Define accuracy and precision. Answer: Accuracy is how close a measurement is to the true value; precision is how close repeated measurements are to one another. 2. A thermometer consistently reads 2 °C too high. Does this affect accuracy, precision or both? Answer: Accuracy; the readings can still agree closely with each other, so precision is not affected. 3. Titres of 22.10, 22.40 and 21.80 cm³ are obtained. The true titre is 22.10 cm³. Comment on precision and accuracy. Answer: The range is 0.60 cm³, so precision is poor; the mean is 22.10 cm³, so the mean is accurate even though individual readings are not reliable. 4. Explain why repeating readings can improve precision of the mean but not remove a systematic error. Answer: Random errors scatter above and below the true value and partly cancel when averaged, but a systematic error shifts every reading the same way, so it is still present in the mean.