# Calculating Uncertainties ## Combining Uncertainties ### Absolute & Percentage Uncertainties - There is always a degree of uncertainty when measurements are taken; the uncertainty can be thought of as the difference between the **actual** reading taken (caused by the equipment or techniques used) and the **true value** - Uncertainties are **not** the same as errors - Errors can be thought of as issues with equipment or methodology that cause a reading to be different from the true value - The uncertainty is a range of values around a measurement within which the true value is expected to lie, and is an **estimate** - For example, if the true value of the mass of a box is 950 g, but a systematic error with a balance gives an actual reading of 952 g, the uncertainty is ±2 g

**Absolute Uncertainty:** uncertainty given as a fixed quantity

**Fractional Uncertainty:** uncertainty given as a fraction of the measurement

**Percentage Uncertainty:** uncertainty given as a percentage of the measurement

- These uncertainties can be represented in a number of ways using the three forms above - To convert between these forms: $$\text{percentage uncertainty} = \frac{\text{uncertainty}}{\text{measured value}} \times 100\%$$ ### Finding Uncertainties in Different Situations | **Situation** | **Uncertainty** | |---------------|-----------------| | The uncertainty in a **reading** | ± half the smallest division | | The uncertainty in a **measurement** | at least ±1 smallest division | | The uncertainty in **repeated data** | half the range i.e. ± ½ (largest - smallest value) | | The uncertainty in **digital readings** | ± the last significant digit unless otherwise quoted | | The uncertainty in the **natural log of a value** | absolute uncertainty in $\ln(x) = \frac{\text{uncertainty in } x}{x}$ |

Always make sure your absolute or percentage uncertainty is to the same number of significant figures as the reading

### Example: Different Forms of Uncertainty - Consider an ammeter with smallest division = 0.2 mA and a reading of I = 4.6 mA **Absolute Uncertainty:** - $\Delta I = \frac{1}{2} \times 0.2 \text{ mA} = 0.1 \text{ mA}$ - $I = 4.6 \pm 0.1 \text{ mA}$ **Fractional Uncertainty:** - Fractional uncertainty = $\frac{\text{uncertainty}}{\text{value}} = \frac{0.1}{4.6} = \frac{1}{46}$ - $I = 4.6 \pm \frac{1}{46} \text{ mA}$ **Percentage Uncertainty:** - Percentage uncertainty = $\frac{\text{uncertainty}}{\text{value}} \times 100 = \frac{0.1}{4.6} \times 100 = 6.2\%$ - $I = 4.6 \pm 6.2\% \text{ mA}$ ## Combining Uncertainties When combining uncertainties, the rules are as follows: ### Adding / Subtracting Data

**Add** together the absolute uncertainties

**Example: Measuring Tyre Thickness** - Diameter of tyre $(d_1) = 55.0 \pm 0.5$ cm - Diameter of inner tyre $(d_2) = 21.0 \pm 0.7$ cm Calculate the difference in diameters: 1. Difference = $d_1 - d_2 = 55.0 - 21.0 = 34.0$ cm 2. Uncertainty in difference = $\pm(0.5 + 0.7) = \pm 1.2$ cm 3. Result: $d_1 - d_2 = 34.0 \pm 1.2$ cm ### Multiplying / Dividing Data

**Add** the percentage or fractional uncertainties

**Example: Calculating Speed** - Distance = $50.0 \pm 0.1$ m - Time = $5.00 \pm 0.05$ s Calculate speed $v = \frac{\text{distance}}{\text{time}}$: 1. Speed = $v = \frac{50.0}{5.0} = 10.0$ m s$^{-1}$ 2. Fractional uncertainty in speed: $$\frac{\Delta v}{v} = \frac{\Delta s}{s} + \frac{\Delta t}{t} = \frac{0.1}{50.0} + \frac{0.05}{5.00} = 0.002 + 0.01 = 0.012$$ 3. Absolute uncertainty: $\Delta v = 10.0 \times 0.012 = \pm 0.12$ m s$^{-1}$ 4. Result: $v = 10.0 \pm 0.12$ m s$^{-1}$ ### Raising to a Power

**Multiply** the percentage uncertainty by the power

**Example: Volume of a Sphere** - Radius of sphere: $r = 2.50 \pm 0.02$ cm - Volume formula: $V = \frac{4}{3}\pi r^3$ Calculate volume: 1. $V = \frac{4}{3}\pi(2.50)^3 = 65.5$ cm$^3$ 2. Fractional uncertainty in radius: $$\frac{\Delta r}{r} = 3 \times \frac{0.02}{2.50} = 0.024$$ 3. Absolute uncertainty: $\Delta V = 65.5 \times 0.024 = 4.57$ cm$^3$ 4. Percentage uncertainty: $\%\Delta V = 100 \times 0.024 = 2.4\%$ 5. Result: $V = 65.5 \pm 4.6$ cm$^3$ or $V = 65.5 \pm 2.4\%$ cm$^3$ ## Summary of Rules | **Operation** | **Uncertainty Rule** | |---------------|---------------------| | Addition or Subtraction | Add the **absolute** uncertainties | | Multiplication or Division | Add the **percentage** or **fractional** uncertainties | | Raising to a power $n$ | Multiply the **percentage** uncertainty by $n$ |

Absolute uncertainties (denoted by $\Delta$) have the same units as the quantity; percentage uncertainties have no units

The uncertainty in constants, such as $\pi$, is taken to be zero

Uncertainties in trigonometric and logarithmic functions will not be tested in the exam

## Worked Scenarios ### Scenario 1: Temperature Change - Initial temperature: $T_1 = 20.5 \pm 0.5$ °C - Final temperature: $T_2 = 85.2 \pm 0.5$ °C - Temperature change: $\Delta T = T_2 - T_1 = 85.2 - 20.5 = 64.7$ °C - Uncertainty: $\pm(0.5 + 0.5) = \pm 1.0$ °C - Result: $\Delta T = 64.7 \pm 1.0$ °C ### Scenario 2: Electrical Power - Voltage: $V = 12.0 \pm 0.3$ V (percentage uncertainty = 2.5%) - Current: $I = 0.50 \pm 0.02$ A (percentage uncertainty = 4.0%) - Power: $P = VI = 12.0 \times 0.50 = 6.0$ W - Percentage uncertainty in power: $2.5\% + 4.0\% = 6.5\%$ - Absolute uncertainty: $6.0 \times 0.065 = 0.39$ W - Result: $P = 6.0 \pm 0.4$ W ### Scenario 3: Area of a Square - Side length: $L = 8.0 \pm 0.2$ cm (percentage uncertainty = 2.5%) - Area: $A = L^2 = (8.0)^2 = 64$ cm$^2$ - Percentage uncertainty in area: $2 \times 2.5\% = 5.0\%$ - Absolute uncertainty: $64 \times 0.05 = 3.2$ cm$^2$ - Result: $A = 64 \pm 3$ cm$^2$