### Limitations of Measurement

#### Types of Error

- Measurements of quantities are made with the aim of finding the true value of that quantity
- In reality, it is impossible to obtain the true value of any quantity as there will always be a degree of
**uncertainty**- This can be seen when you repeat a measurement and you get different results

- An error is the difference between the measurement result and the true value if a true value is thought to exist
- This is not a mistake in the measurement
- The error can be due to both systematic and random effects and an error of unknown size is a source of uncertainty.

- Random and systematic errors are two types of measurement errors that lead to uncertainty

**Random error**

- Random errors cause
**unpredictable fluctuations**in an instrument’s readings as a result of uncontrollable factors, such as environmental conditions - This affects the
**precision**of the measurements taken**,**causing a wider spread of results about the mean value - To
**reduce**random error:**Repeat**measurements several times and calculate an average from them

**Systematic error**

- Systematic errors arise from the use of
**faulty instruments**or from**flaws**in the**experimental method** - This type of error is repeated consistently every time the instrument or method are used, which affects the
**accuracy**of all readings obtained - To
**reduce**systematic errors:- Instruments should be
**recalibrated**, or different instruments should be used - Corrections or adjustments should be made to the technique

- Instruments should be

**Systematic errors on graphs are shown by the offset of the line from the origin**

*Representing precision and accuracy on a graph*

**Zero error**

- This is a type of
**systematic error**that occurs when an instrument gives a reading when infact the**true reading is zero** - This introduces a
**fixed error**into the readings which must be accounted for when the results are recorded - Zero error is a type of systematic error since all the values will be displaced by the same amount

#### Precision vs Accuracy

**Precision**

- Precise measurements denote the
**closeness**of agreement (consistency) between values obtained by repeated measurement- This is influenced only by random effects and can be expressed numerically by measures such as standard deviation.
- A measurement is precise if the values ‘cluster’ closely together.

- Precise measurements have very little spread about the mean value, in other words, how close the measured values are to each other
- If a measurement is repeated several times, it can be described as precise when the values are very similar to, or the same as, each other
- The precision of a measurement is reflected in the values recorded - measurements to a greater number of decimal places are said to be more
**precise**than those to a whole number

**Accuracy**

- A measurement is considered accurate if it is close to the true value
- It is a quality denoting the
**closeness**of agreement between measurement and true value- It cannot be quantified and is influenced by random and systematic errors

- The accuracy can be increased by repeating measurements and finding a mean of the results
- Repeating measurements also helps to identify anomalies that can be omitted from the final results

*The difference between precise and accurate results*

#### Resolution

- Resolution is the
**smallest change**in the quantity being measured- It gives a perceptible change in the reading
- It is also the source of uncertainty in a single reading

- For example, the resolution of a wristwatch is 1 s, whereas the resolution of a digital stop-clock is typically 10 ms (0.01 s)
- In imaging, resolution can also be described as the ability to see two structures as two separate structures rather than as one fuzzy entity

**Good resolution and poor resolution in an ultrasound scanner. The good image manages to resolve the two objects into two distinct structures, whereas the poor image shows one fuzzy entity.**

#### Uncertainties

- Uncertainty is an estimate of the difference between a measurement reading and the true value
- In other words, it is the interval within which the true value can be considered to lie with a given level of
**confidence**or**probability** - Any measurement will have some uncertainty about the result, this will come from variation in the data obtained and be subject to systematic or random effects

- In other words, it is the interval within which the true value can be considered to lie with a given level of
- 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
- These uncertainties can be represented in a number of ways:
**Absolute Uncertainty:**where uncertainty is given as a fixed quantity**Fractional Uncertainty:**where uncertainty is given as a fraction of the measurement**Percentage Uncertainty:**where uncertainty is given as a percentage of the measurement

- Percentage uncertainty is defined by the equation:

**Percentage uncertainty = × 100 %**

- To find uncertainties in different situations:
**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

*How to calculate absolute, fractional and percentage uncertainty*

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

**Combining Uncertainties**

- When combining uncertainties, the rules are as follows:

**Adding / Subtracting Data**

**Add**together the absolute uncertainties

**Multiplying / Dividing Data**

**Add**the percentage or fractional uncertainties

**Raising to a Power**

**Multiply**the percentage uncertainty by the power

#### Worked Example

A student achieves the following results in their experiment for the angular frequency, *ω* of a rotating ball bearing.

0.154, 0.153, 0.159, 0.147, 0.152

Calculate the percentage uncertainty in the mean value of *ω.*

**Step 1: Calculate the mean** **value **

mean *ω = = *0.153 rad s^{–1}

**Step 2: Calculate half the range (this is the uncertainty for multiple readings)**

× (0.159 – 0.147) = 0.006 rad s^{–1}

**Step 3: Calculate percentage uncertainty**

× 100 % = × 100 %

× 100 % = 3.92 %

#### Exam Tip

It is a very common mistake to confuse precision with accuracy - measurements can be precise but **not** accurate if each measurement reading has the same error. Make sure you learn that **precision** refers to the ability to take multiple readings with an instrument that are close to each other, whereas **accuracy** is the closeness of those measurements to the true value.

Remember:

- Absolute uncertainties have the same units as the quantity
- Percentage uncertainties have
**no**units - The uncertainty in numbers and constants, such as π, is taken to be zero