# Calibration and Systematic Errors

Physics I · Principles of Laboratory Practice · https://tryals.app/learn/physics-i/calibration-and-systematic-errors

## Calibration Fundamentals

**Calibration** links an instrument's reading to a known standard. Without it, all measurements remain **systematically shifted**.

| Error Type | Description | Example |
| --- | --- | --- |
| **Zero offset** | Doesn't read zero at zero | Scale reads $0.3\text{ g}$ empty |
| **Scale factor** | Consistently high or low | Ruler expanded by heat |
| **Non-linearity** | Error varies across range | Spring past Hooke's law |
| **Hysteresis** | Depends on approach direction | Mechanical backlash |

**Calibration procedure** measures **known standards** across your range, plots measured versus true values, and fits a **calibration curve** ($y = mx + b$). Apply the inverse to correct future readings.

## Errors & Accuracy

Detect systematic errors by comparing with **accepted values**, using **different methods**, or checking **residuals** (measured minus fitted values).

| Term | Definition |
| --- | --- |
| **Accurate** | Close to true value; low systematic error |
| **Precise** | Measurements cluster tightly; low random error |

**Common pitfall:** Averaging cures **only** random error. A miscalibrated instrument delivers precision without accuracy. **Systematic errors do not decrease with more measurements.** The only cure is identifying and removing the source.

## Practice questions

5 of this lesson's 14 practice questions, with answers. The full set is in the app.

### 1. Sort each correction method by the type of error it addresses.

**Answer:**

- Corrects systematic error: Subtracting a known zero offset, Applying a calibration curve, Using a reference standard before each run, Recalibrating the instrument
- Reduces random error: Averaging multiple measurements, Increasing the number of trials

**Why:** Systematic errors require calibration/correction; random errors are reduced by averaging.

Page: https://tryals.app/practice/physics-i/calibration-and-systematic-errors/sort-each-correction-method-by-the-type-of-error-it-addresses

### 2. Your electronic balance reads **0.3 g with nothing on it**. You then weigh ten samples without fixing this. What kind of error infects the data, and what repairs it?

A. Random error, average enough samples and it disappears
B. No error, since all samples share the shift
C. A blunder, all ten measurements must be discarded
D. A systematic offset: subtract 0.3 g from every reading

**Answer:** D. A systematic offset: subtract 0.3 g from every reading

**Why:** A zero offset biases every reading identically, invisible to averaging, invisible to error bars, deadly to accuracy. The cures are calibration habits: tare before use, check against known masses. Precision guards against scatter; **calibration** guards against being consistently wrong.

Page: https://tryals.app/practice/physics-i/calibration-and-systematic-errors/your-electronic-balance-reads-0-3-g-with-nothing-on-it-you-then

### 3. Averaging many measurements reduces random error but not systematic error. What makes the two kinds of error behave differently?

A. Random errors are eliminated entirely by a single repeat measurement
B. Random errors cancel on average; a systematic one does not
C. Systematic errors are larger and so dominate the average
D. Systematic errors are unpredictable and so cannot be averaged

**Answer:** B. Random errors cancel on average; a systematic one does not

**Why:** A bias survives averaging untouched. This is exactly why calibration against a known standard is a separate procedure from repeating the measurement.

Page: https://tryals.app/practice/physics-i/calibration-and-systematic-errors/averaging-many-measurements-reduces-random-error-but-not-systematic

### 4. Order the steps for calibrating a measurement instrument.

**Answer:**

1. Obtain a set of known reference standards
2. Measure each standard with the instrument
3. Plot instrument readings vs true values
4. Fit a calibration curve (e.g., linear)
5. Use the calibration curve to correct future readings

**Why:** Calibration maps instrument readings to true values using known standards.

Page: https://tryals.app/practice/physics-i/calibration-and-systematic-errors/order-the-steps-for-calibrating-a-measurement-instrument

### 5. Precision vs. accuracy: classify each archery-style measurement outcome.

**Answer:**

- Precise AND accurate: Readings tightly clustered on the true value
- Precise, NOT accurate: Readings tightly clustered, all 5% too high, A miscalibrated but very stable instrument
- Accurate on average, NOT precise: Readings scattered widely around the true value

**Why:** Tight grouping and centred grouping are different virtues: repetition improves neither if the sights are bent. Real measurements are judged on both axes at once, and each axis has its own remedy, statistics for precision, calibration for accuracy.

Page: https://tryals.app/practice/physics-i/calibration-and-systematic-errors/precision-vs-accuracy-classify-each-archery-style-measurement