# Measurement, Uncertainty, and Error Analysis

Physics I · Principles of Laboratory Practice · https://tryals.app/learn/physics-i/measurement-uncertainty-and-error-analysis

## Measurement & Error

Every measurement has **uncertainty**. Reporting a result without it is incomplete; an error bar is your claim's precision, not a failure.

$$x = x_{\text{best}} \pm \delta x$$

| Type | Cause | How to Reduce |
| --- | --- | --- |
| **Random** | Fluctuations | Repeat and average |
| **Systematic** | Calibration, offset | Recalibrate |
| **Resolution** | Instrument limit | Finer instrument |

**Absolute vs Relative Uncertainty**:

$$\text{Absolute: } \delta x, \quad \text{Relative: } \frac{\delta x}{x} \times 100\%$$

**Significant figures**: The last digit reflects the uncertainty ($L = 3.45$ m implies $\pm 0.005$ m).

## Sig Figs & Statistics

**Rules**: Add/subtract matches the fewest **decimal places**. Multiply/divide matches the fewest **significant figures**.

**Standard deviation** measures spread in repeated trials:

$$\sigma = \sqrt{\frac{1}{N-1}\sum_{i=1}^N (x_i - \bar{x})^2}$$

The **standard error of the mean** shrinks with more trials:

$$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{N}}$$

> **Tip**: Report uncertainty to **one** sig fig (two at most) and match the measurement: $g = 9.81 \pm 0.03$ m/s$^2$.

## Practice questions

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

### 1. Sort each description into accuracy or precision.

**Answer:**

- Accuracy: Results are close to the accepted value, Low systematic error, Improved by calibration
- Precision: Results cluster tightly together, Low random error, Improved by averaging many trials

**Why:** Accuracy relates to systematic error; precision relates to random error.

Page: https://tryals.app/practice/physics-i/measurement-uncertainty-and-error-analysis/sort-each-description-into-accuracy-or-precision

### 2. Five stopwatch timings of one pendulum swing: 2.01, 1.98, 2.03, 1.99, 2.02 s. A classmate reports the result as **1.98 s** "because it was measured too". What is the right report?

A. The mean 2.01 s, with uncertainty from the spread
B. Whichever value appeared twice
C. The smallest value, to be conservative
D. The most recent measurement, practice makes it best

**Answer:** A. The mean 2.01 s, with uncertainty from the spread

**Why:** Averaging beats picking: random errors partially cancel in the mean, and the scatter (standard deviation ≈ 0.02 s) becomes the honest uncertainty: $T = 2.01 \pm 0.01$ s. Reporting spread is what separates a measurement from a guess.

Page: https://tryals.app/practice/physics-i/measurement-uncertainty-and-error-analysis/five-stopwatch-timings-of-one-pendulum-swing-2-01-1-98-2-03-1-99

### 3. A measurement of 2.50 cm claims more than one of 2.5 cm, though the numbers are equal. What does that extra digit assert?

A. Nothing: trailing zeros are cosmetic
B. That the value is exactly 2.50 with no uncertainty
C. That the hundredths place was actually resolved
D. That the measurement was repeated at least twice

**Answer:** C. That the hundredths place was actually resolved

**Why:** Notation carries a claim about precision. Reporting more digits than the apparatus resolves overstates what was learned, which is why significant figures propagate through a calculation.

Page: https://tryals.app/practice/physics-i/measurement-uncertainty-and-error-analysis/a-measurement-of-2-50-cm-claims-more-than-one-of-2-5-cm-though-the

### 4. Order the steps for analyzing measurement uncertainty.

**Answer:**

1. Identify all sources of uncertainty
2. Estimate the magnitude of each uncertainty
3. Classify uncertainties as systematic or random
4. Combine uncertainties using propagation rules
5. Report the final result with its total uncertainty

**Why:** Uncertainty analysis proceeds from identification through quantification to combination.

Page: https://tryals.app/practice/physics-i/measurement-uncertainty-and-error-analysis/order-the-steps-for-analyzing-measurement-uncertainty