Every measurement carries some uncertainty
Definition
Measurement uncertainty
The estimated interval around a measured value within which the result is expected to lie.
- No measurement is ever perfectly exact, so every result has some uncertainty attached to it.
- The resolution of the equipment and small differences in how it is read both add to this uncertainty.
Key Idea
Repeating a measurement and comparing the results is how you judge how confident you can be in the value.
Repeat the reading, then take a mean
Definition
Repeated measurement
A measurement of the same quantity made again using the same method.
Definition
Mean
The value found by adding all the measurements and dividing by the number of measurements.
- Make several repeated measurements of the same quantity using the same method.
- Spot any anomalous result that is much larger or smaller than the rest and leave it out.
- Add the remaining results and divide by how many there are to find the mean.
- The mean is more reliable than any single reading because random errors partly cancel out.
Example
- Four readings give 24.224.224.2, 24.324.324.3, 24.424.424.4 and 25.9 cm325.9\ \text{cm}^325.9 cm3.
- The 25.9 cm325.9\ \text{cm}^325.9 cm3 value is anomalous, so it is left out.
- The mean is 24.2+24.3+24.43=24.3 cm3\frac{24.2 + 24.3 + 24.4}{3} = 24.3\ \text{cm}^3324.2+24.3+24.4=24.3 cm3.
The range shows how spread out the results are
Definition
Range
The difference between the highest and lowest values in a set of measurements.
- The range is the difference between the highest and lowest of the valid results.
- A smaller range means the results are more consistent, so you can trust the mean more.
Example
For the three valid readings above, the range is 24.4−24.2=0.2 cm324.4 - 24.2 = 0.2\ \text{cm}^324.4−24.2=0.2 cm3.
Estimate the uncertainty from the range about the mean
- You can estimate the uncertainty as plus or minus half of the range, measured about the mean.
- For the readings above this gives 24.3±0.1 cm324.3 \pm 0.1\ \text{cm}^324.3±0.1 cm3.
- Taking more careful repeats narrows the range and so reduces the uncertainty.
Exam technique
- Always remove anomalous results before you calculate either the mean or the range.
- Quote a measured value with its unit and its uncertainty, for example 24.3±0.1 cm324.3 \pm 0.1\ \text{cm}^324.3±0.1 cm3.
A distribution shows every value and how often it occurs
Definition
Distribution of results
The pattern showing the values obtained and how often each value occurs.
- Recording the distribution of results in a table or chart shows which values came up and how often.
- The spread of that distribution makes both the anomalies and the size of the uncertainty easy to see.
Self review
- Why does every measurement have some uncertainty?
- How do you calculate a mean from a set of repeated readings?
- How do you work out the range of a set of results?
- Readings of 18.418.418.4, 18.618.618.6 and 18.5 cm318.5\ \text{cm}^318.5 cm3 are collected; estimate the value and its uncertainty.
