1.1.4

Limitation of Physical Measurements

Test yourself

Physical Measurement Errors

All measurement has an associated uncertainty. We call this uncertainty "error".

Measurement model

• We can model a measurement you make as two parts.
• The measurement you make is called the "observed value".
• What value the measurement should be is called the "true value".
• The uncertainty we have about our measurement is called the "error". So the model is:
• Observed value = true value + error

Error

• The error in our measurement model is made up of two parts:
• Random error.
• Systematic error.

Measurement model revisited

• We can now rewrite our measurement model as:
• Observed value = true value + random error + systematic error

The model in practice

• Normally, the model always has a positive error.
• In practice, we know that the error could make our observed value look larger or smaller than the true value.
• One example of this is measuring the temperature of a cup of tea.
• To show our uncertainty, we might say that the tea has a temperature of 95 ± 3oC.
• This means our true value is likely to be between 92oC and 98oC.

Systematic and Random Errors

A systematic error is an error that follows a set pattern. A random error follows no set pattern.

Systematic error

• A systematic error is an error that follows a set pattern.
• E.g. If you were taking the mass of some flour and forgot to zero your mass balance, all your measurements would be off by a set amount. This type of systematic error is called zero error.
• It is hard to avoid systematic error.
• To avoid systematic error, you should use the measuring equipment to measure a known value.
• This process is called calibration.

Systematic error cont.

• Let's say you knew a mass of flour was 500g, and the mass balance read differently, you would then know you have a systematic error.
• You would also have an estimate of the size of the systematic error.
• You can then either subtract this systematic error from all of your incorrect readings or you could redo the experiment with the equipment correctly calibrated.

Random error

• Random error is error that follows no set pattern.
• Random error could be due to reading the measuring equipment in different ways.
• The error could be because the measuring equipment is changing slightly in different ways.

Random error cont.

• If an experiment was carried out a large number of times, we would expect to see these random errors mostly cancel each other out.
• Doing lots of repeats is the one way to reduce random error.
• If there was no systematic error, we assume that the observed value we get after lots of repeats is the true value.

Describing Measurements

You will need to know the following terms for describing measurements:

Accuracy

• Accuracy is how close a measurement is to the correct value for that measurement.
• This image shows accurate shots at the target.

Precision

• The precision of a measurement system refers to how close to each other the repeated measurements are.
• It is independent of the "true" value of the measurement.
• This image shows precise (but not accurate) shots at the target.

Repeatability

• A measurement is repeatable if the same person performing the same experiment with the same apparatus gives the same results (within random errors).

Reproducibility

• A measurement is reproducible if a different person can perform the same experiment with the same apparatus and get the same results (within errors).

Resolution

• The resolution of a measuring instrument describes its maximum precision.
• A higher resolution TV screen will have a sharper (more detailed) image.
• A higher resolution telescope will be able to separate points of light more easily than a lower resolution one.