Types of Errors

Every measurement has an error. Identifying its type tells you how to reduce or correct for it.

Concept

By Source

• Systematic errors — Consistent, reproducible bias. Same direction every time.
  • Instrumental (zero error, miscalibrated scale)
  • Imperfect technique (parallax, mounting tilt)
  • Environmental (constant temperature drift)
• Random errors — Unpredictable fluctuations in either direction. Caused by inherent fluctuations, judgement noise, vibrations.
• Gross errors — Outright mistakes: misreading, mis-recording, wrong scale.

Reducing Each Type

Quantifying Error

Suppose true value is $a$, measured value is $a_m$.

• Absolute error $|\Delta a| = |a_m - a|$. Same units as the quantity.
• Mean absolute error for $n$ readings: $\overline{|\Delta a|} = \frac{1}{n}\sum|a_i - \bar{a}|$.
• Relative error $= \frac{|\Delta a|}{a}$ (dimensionless).
• Percent error $= \frac{|\Delta a|}{a} \times 100\%$.

Best Estimate

When no "true" value is known, the mean of repeated readings is the best estimate. Uncertainty is the mean absolute error or the standard deviation.

Worked Example

Q: Five measurements of a resistor: $99.2, 100.5, 99.8, 100.3, 100.2\,\Omega$. Find the best estimate, mean absolute error, relative error, and percent error.

Solution:

Mean: $\bar{R} = (99.2 + 100.5 + 99.8 + 100.3 + 100.2)/5 = 500.0/5 = 100.0\,\Omega$.

Deviations: $|99.2 - 100| = 0.8$, $0.5$, $0.2$, $0.3$, $0.2$.

Mean absolute error: $(0.8 + 0.5 + 0.2 + 0.3 + 0.2)/5 = 2.0/5 = 0.4\,\Omega$.

Relative error: $0.4/100.0 = 0.004$.

Percent error: $0.4\%$.

So we report $R = 100.0 \pm 0.4\,\Omega$, or $R = 100.0\,\Omega \pm 0.4\%$.

Common Confusions

• "Random errors can be eliminated by careful work." — They can be reduced by averaging but never eliminated; they are statistical in nature.
• "A zero error is a random error." — No, it's systematic. The needle reads the same offset each time.
• Absolute error has units; relative and percent errors do not.

Key Takeaways

• Systematic errors bias accuracy; random errors degrade precision; gross errors are mistakes.
• Best estimate = mean of repeated readings.
• Uncertainty can be reported as mean absolute error, relative error, or percent error.
• Always report measurements as best estimate ± uncertainty.