1.1.5

# Uncertainty

Test yourself

## Uncertainty

Uncertainty is a measure of how confident you can be in a measurement. All measurements contain some amount of uncertainty. Let's say we measure a piece of string to be 10cm (to the nearest cm):

### Absolute uncertainty

• The absolute uncertainty is the range of possible real values for the length of the string.
• Length of string = 10cm ± 0.5cm
• Our absolute uncertainty is ± 0.5cm.

### Fractional uncertainty

• The fractional uncertainty is the absolute uncertainty divided by the measured value.
• Fractional uncertainty = $\frac{0.5cm}{10cm} = \frac{1}{20}$

### Percentage uncertainty

• The percentage uncertainty is the fractional uncertainty expressed as a percentage (i.e. multiplied by 100%).
• Percentage uncertainty = $\frac{1}{20} \times 100 = 5\%$

## Uncertainty on Graphs

The range of possible values for a measurement can be shown as a line on an uncertainty graph. We call this line an error bar. Some points on a graph might not have error bars.

### Error bars

• Small uncertainties are shown by small error bars.
• Large uncertainties are shown by large error bars.

### Finding the uncertainty from a graph

• A systematic uncertainty can be spotted if the theory and the results do not match for when the independent variable is zero.
• For example, when measuring the current in a circuit we would expect the current to be zero when the supply potential difference is zero.

### Finding the uncertainty from a graph 2

• If there is a systematic uncertainty, this expectation would not come true.
• We would see that the intercept of the line wasn't zero.
• The intercept would indicate the size of the systematic error.
• Once the systematic error has been found, we can use the graph to find an estimate for the random error.

### Using error bars

• Error bars can be used to find the steepest and shallowest possible lines of best fit.
• The steepest possible line of best fit will have a gradient g1.
• The shallowest possible line of best fit will have a gradient g2.
• The uncertainty in the measurement should then be quoted as $\frac{g_1 + g_2}{2} \pm \frac{g_1 - g_2}{2}$.