Accuracy vs Precision

Accuracy and precision are NOT synonyms. A measurement can be precise but inaccurate, accurate but imprecise, both, or neither. Distinguishing them is crucial for analysing experimental data.

Concept

• Accuracy is how close a measurement is to the true value.
• Precision is how close repeated measurements are to each other (i.e., low scatter).

A good analogy is a dartboard:

Mathematically, for measurements $x_1, x_2, \ldots, x_n$ with true value $X$:

• Accuracy ↔ how small $|\bar{x} - X|$ is.
• Precision ↔ how small the standard deviation $\sigma$ is.

Where $\bar{x} = \frac{1}{n}\sum x_i$ and $\sigma = \sqrt{\frac{1}{n-1}\sum(x_i - \bar{x})^2}$.

Sources of Inaccuracy vs Imprecision

• Inaccuracy (poor accuracy) usually arises from systematic errors: miscalibrated instrument, biased reading method.
• Imprecision (poor precision) usually arises from random errors: fluctuations, parallax, judgement variability.

Worked Example

Q: A student measures gravitational acceleration five times and gets: $9.78, 9.79, 9.80, 9.81, 9.82\,\text{m/s}^2$. The accepted value is $9.81\,\text{m/s}^2$.

(a) Find the mean and standard deviation. (b) Comment on accuracy and precision.

Solution:

Mean: $\bar{x} = (9.78 + 9.79 + 9.80 + 9.81 + 9.82)/5 = 9.80\,\text{m/s}^2.$

Deviations from mean: $-0.02, -0.01, 0, +0.01, +0.02$. Variance: $\sigma^2 = \frac{0.0004 + 0.0001 + 0 + 0.0001 + 0.0004}{4} = \frac{0.0010}{4} = 0.00025.$ $\sigma \approx 0.016\,\text{m/s}^2$.

Accuracy: $|\bar{x} - X| = |9.80 - 9.81| = 0.01$, so about $0.1\%$ off — quite accurate.

Precision: $\sigma/\bar{x} \approx 0.16\%$ — very precise.

The measurement is both accurate and precise.

Compare: If readings were $9.50, 9.51, 9.50, 9.49, 9.50$, the precision is excellent (low $\sigma$) but accuracy is poor (off by $0.31$, about $3\%$ low) — pointing to a systematic error.

Common Confusions

• "More decimal places = more accurate." — More decimals can mean more precise, but not necessarily more accurate.
• High precision with low accuracy almost always indicates a systematic error (instrument calibration).
• Random errors affect precision; systematic errors affect accuracy.

Key Takeaways

• Accuracy = closeness to true value (bias).
• Precision = reproducibility (scatter).
• Systematic errors hurt accuracy; random errors hurt precision.
• Always report both the mean (for accuracy) and an uncertainty estimate like $\sigma$ (for precision).