understand the distinction between precision and accuracy

Errors and Uncertainties

In experimental physics, every measurement carries some degree of error. Understanding the nature of these errors is essential for interpreting results correctly and for improving experimental technique.

Key Concepts

• True value: The exact value of a quantity, which is usually unknown.
• Measured value: The value obtained from an experiment.
• Error: The difference between the measured value and the true value.
• Uncertainty: An estimate of the range within which the true value is expected to lie.

Precision vs Accuracy

Two distinct aspects of measurement quality are often confused: precision and accuracy. They are illustrated in the figure below.

Definitions

• Accuracy – How close a measurement is to the true value.
• Precision – How closely repeated measurements agree with each other, regardless of the true value.

Mathematical Expressions

For a set of \$n\$ measurements \$x_i\$ with mean \$\bar{x}\$ and true value \$X\$, we can write:

\$\text{Accuracy error} = \bar{x} - X\$

\$\text{Precision (standard deviation)} = \sigma = \sqrt{\frac{1}{n-1}\sum{i=1}^{n}(xi-\bar{x})^2}\$

Comparison Table

Sources of Error

1. Systematic errors – affect accuracy; they shift all measurements in the same direction.
2. Random errors – affect precision; they cause scatter around the true value.
3. Instrumental limitations – finite resolution, zero‑error.
4. Environmental factors – temperature, humidity, vibrations.

Improving Measurements

To achieve both high accuracy and high precision, consider the following strategies:

• Calibrate instruments before use (improves accuracy).
• Take multiple readings and use the mean (improves precision).
• Identify and eliminate sources of systematic bias (e.g., parallax error).
• Use equipment with finer resolution where possible.
• Control environmental conditions during the experiment.

Common Misconceptions

• “A precise measurement is automatically accurate.” – Not true; a set of consistently wrong readings is precise but inaccurate.
• “A single measurement can tell you about precision.” – Precision is a statistical property; it requires multiple readings.
• “Reducing random error will fix systematic error.” – Different error types require different corrective actions.

Summary

Understanding the distinction between precision and accuracy allows students to diagnose experimental problems correctly and to apply appropriate remedies. Accurate measurements reflect correct calibration and bias correction, while precise measurements reflect careful technique and stable instrumentation.