# Uncertainty

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## Error vs Uncertainty

In the realm of physics, it’s important to distinguish between ‘error’ and ‘uncertainty.’

• Error pertains to the disparity between the measured quantity of a physical parameter and its ‘true value.’ In experimental endeavors, efforts are made to address known errors, such as rectifying zero-point discrepancies. Any error whose magnitude remains unknown becomes a source of uncertainty.
• Uncertainty denotes the extent of variation in values, likely encompassing the ‘true value.’ We can quantify the degree of uncertainty associated with a particular measurement.

## Uncertainties in measurements

For instance, consider a scaler timer capable of measuring with a resolution of ±0.1%. In a ‘g by free fall’ experiment, the timer gauges the time taken for a ball bearing to descend 2 meters.

• If the recorded time is 638.5 milliseconds, the result is expressed as 638.5 ms ± 0.6 ms.
• In this case, the uncertainty value is the absolute uncertainty, determined by the instrument’s specified resolution of 0.1%.
• The absolute uncertainty shares the same units as the measurement and signifies the potential range of measurement values.

When conducting a series of repeated measurements, the absolute uncertainty is calculated as half the range between the highest and lowest obtained values.

In general, uncertainty of a reading is determined to the nearest half of the smallest graduation.

• Always quoted to one significant figure.

Numerical value of uncertainty is the absolute uncertainty or actual uncertainty.

Uncertainty of an instrument determined the number of decimal places that should be quoted for the readings taken from it. The number of decimal places in a reading is the same as that in the uncertainty.

A lower percentage uncertainty will mean the instrument used to measure it is more acceptable.

## Combining uncertainties

During physics experiments, it’s common to obtain measurements from multiple instruments and use them to compute a different quantity. When adding or subtracting these measurements, the uncertainty in the overall result is determined by adding the absolute uncertainties. In mathematical terms:

$$(a \pm \Delta a) + (b \pm \Delta b) = (a + b) \pm (\Delta a + \Delta b)$$

$$(a \pm \Delta a) – (b \pm \Delta b) = (a – b) \pm (\Delta a + \Delta b)$$

If $Y = a + b$ OR $Y = a – b$, uncertainty of Y is:

$$\Delta Y = \Delta a + \Delta b$$

Because we are always maximizing the possible uncertainty that can occur.

Important: Make X the subject to find the uncertainty of X.

### Multiplication and Division

However, when the calculated quantity is the result of multiplying or dividing the measured quantities, the combined percentage error of the calculated quantity is obtained by summing the percentage errors of the individual measurements.

$$\text{Fractional Uncertainty} = \frac{\Delta R}{R}$$

$$\text{Percentage Uncertainty} = \frac{\Delta R}{R} \times 100 \%$$

If $Y = a \times b$ OR $Y = \frac{a}{b}$, uncertainty of Y, $\Delta Y$ is given by:

$$\frac{\Delta Y}{Y} = \frac{\Delta a}{a} + \frac{\Delta b}{b}$$

OR

$$\text{Fractional Uncertainty of Y} = \text{Fractional Uncertainty of a} + \text{Fractional Uncertainty of b}$$

$$\text{Percentage Uncertainty of Y} = \text{Percentage Uncertainty of a} + \text{Percentage Uncertainty of b}$$

### Other

If $Y = a^{n} \: = \: a \: \times \: a \: \times \: a \: \times \: a \:….$, then uncertainty of Y, $\Delta Y$ is given by:

$$\frac{\Delta Y}{Y} = n \frac{\Delta a}{a}$$

### Example 1

The measured potential difference over a resistor is 10.0 V with a range of ± 0.3 V. Simultaneously, the measured current flowing through the resistor is 1.3 A with a range of ± 0.2 A. What are the percentage and absolute uncertainties in the resistor’s resistance?

\begin{aligned} \text{Resistance} &= \frac{\text{Potential Difference}}{\text{Current}} \\ &= \frac{10.0 \, \text{V}}{1.3 \, \text{A}} \\ &= 7.7 \, \Omega \end{aligned}

\begin{aligned} \% \, \text{uncertainty in p.d.} &= \frac{0.3 \, \text{V}}{10.0 \, \text{V}} \times 100\% \\ &= 3\% \end{aligned}

\begin{aligned} \% \, \text{uncertainty in current} &= \frac{0.2 \, \text{A}}{1.3 \, \text{A}} \times 100\% \\ &= 15\% \end{aligned}

\begin{aligned} \% \, \text{uncertainty in resistance} &= \% \, \text{uncertainty in p.d.} + \% \, \text{uncertainty in current} \\ &= 3\% + 15\% \\ &= 18\% \end{aligned}

\begin{aligned} \text{Absolute uncertainty} &= \frac{\left( \text{value} \times \% \, \text{uncertainty} \right)}{100} \\ \text{Resistance} &= 7.7 \, \Omega \pm 1.4 \, \Omega \end{aligned}

## Example 2

The duration for an athlete to cover a distance of 100.00 m is recorded as 9.63 seconds. The uncertainty in the distance measurement is 0.01 m, and in the time measurement, it is 0.01 seconds. Determine the absolute uncertainty in the athlete’s speed.

\begin{aligned} v &= \frac{d}{t} \\ &= \frac{100 \text{ m}}{9.63 \text{ s}} \\ &= 10.384 \text{ m s}^{-1} \end{aligned}

\begin{aligned} \% \, \text{uncertainty in d} &= \frac{0.01 \text{ m}}{100 \text{ m}} \times 100\% \\ &= 0.01 \% \end{aligned}

\begin{aligned} \% \, \text{uncertainty in t} &= \frac{0.01 \, \text{s}}{9.63 \, \text{s}} \times 100\% \\ &= 0.10384\% \end{aligned}

\begin{aligned} \% \text{ Uncertainty of v} &= \% \text{ Uncertainty of d} + \% \text{ Uncertainty of t} \\ &= 0.01 \% + 0.10384 \% \\ &= 0.11384 \% \end{aligned}

\begin{aligned} \text{Absolute uncertainty} &= \frac{\left( \text{value} \times \% \, \text{uncertainty} \right)}{100} \\ \text{speed} &= 10.38 \text{ m s}^{-1} \pm 0.01 \text{ m s}^{-1} \end{aligned}

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