Uncertainty, validity and reliability

Covers AS91168

Key ideas

Uncertainty describes the range within which the true value of a measurement is likely to lie. It is not the same as a mistake — even a carefully taken measurement has some uncertainty.
Absolute uncertainty has the same units as the measurement (e.g. $L = 50.0 \pm 0.1$ cm). Percentage uncertainty expresses this as a fraction of the reading:

$\text{\% uncertainty} = \frac{\text{absolute uncertainty}}{\text{measured value}} \times 100\%$

For repeated readings, a simple estimate of uncertainty is half the range:

$\text{uncertainty} \approx \frac{\text{maximum} - \text{minimum}}{2}$

On a graph, uncertainty is shown with error bars — vertical (and sometimes horizontal) lines through each point showing its uncertainty range.
The gradient uncertainty can be estimated by drawing the steepest and shallowest acceptable lines that still pass through all the error bars, finding the gradient of each, and taking half the difference between them.

Validity vs reliability

Reliability is about consistency — would repeating the investigation give similar results? It's improved by taking repeated readings and averaging, and by using more precise equipment.
Validity is about whether the investigation actually tests what it claims to test — did the method properly isolate the relationship between the intended independent and dependent variable, with other variables genuinely controlled? A very reliable (repeatable) result can still be invalid if a variable wasn't properly controlled.
Common validity issues: a controlled variable that wasn't actually kept constant, a systematic error in the measuring equipment (e.g. an un-zeroed instrument), or a range too narrow to reveal the true shape of the relationship.

Worked ExampleEstimating uncertainty from repeated readings

A student times 10 oscillations of a pendulum three times, getting $18.2$ s, $18.4$ s, $18.1$ s. Find the mean period per oscillation and its uncertainty.

Step 1 — Find the mean time for 10 oscillations

$\bar{t} = \frac{18.2 + 18.4 + 18.1}{3} = 18.23\overline{3}\text{ s} \approx 18.2 \text{ s}$

Step 2 — Find the uncertainty using half the range

$\text{uncertainty} = \frac{18.4 - 18.1}{2} = 0.15 \text{ s} \approx 0.2 \text{ s (1 s.f., matching the data precision)}$

So the time for 10 oscillations is $18.2 \pm 0.2$ s.

Step 3 — Convert to the period of one oscillation

Dividing both the value and the uncertainty by 10 (since dividing by an exact number divides the uncertainty by the same factor):

$T = \frac{18.2}{10} = 1.82 \text{ s}, \qquad \text{uncertainty} = \frac{0.2}{10} = 0.02 \text{ s}$

$\textbf{T = 1.82 ± 0.02 s}$

Practice question

A student measures the diameter of a ball three times: $4.0$ cm, $4.2$ cm, $4.1$ cm. State the mean diameter with its uncertainty from half the range.

Worked solution: Mean $= \dfrac{4.0+4.2+4.1}{3} = 4.10$ cm. Half the range $= \dfrac{4.2-4.0}{2} = 0.1$ cm. Diameter $= 4.1 \pm 0.1$ cm.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Three repeated length measurements are $20.1$ cm, $20.3$ cm, $19.9$ cm. Find the mean and the uncertainty using half the range.

Merit

Explain the difference between the reliability and the validity of an investigation, using an example of an investigation that could be reliable but not valid.

Excellence

A student's best-fit line through error-bar data has gradient $2.1$ . The steepest acceptable line (through the error bars) has gradient $2.4$ , and the shallowest has gradient $1.8$ . State the gradient with its uncertainty, and explain what this uncertainty means for the student's conclusion about the power $n$ in $y = kx^n$ .

Uncertainty, validity and reliability

Covers AS91168

Key ideas

Uncertainty describes the range within which the true value of a measurement is likely to lie. It is not the same as a mistake — even a carefully taken measurement has some uncertainty.
Absolute uncertainty has the same units as the measurement (e.g. $L = 50.0 \pm 0.1$ cm). Percentage uncertainty expresses this as a fraction of the reading:

$\text{\% uncertainty} = \frac{\text{absolute uncertainty}}{\text{measured value}} \times 100\%$

For repeated readings, a simple estimate of uncertainty is half the range:

$\text{uncertainty} \approx \frac{\text{maximum} - \text{minimum}}{2}$

On a graph, uncertainty is shown with error bars — vertical (and sometimes horizontal) lines through each point showing its uncertainty range.
The gradient uncertainty can be estimated by drawing the steepest and shallowest acceptable lines that still pass through all the error bars, finding the gradient of each, and taking half the difference between them.

Validity vs reliability

Reliability is about consistency — would repeating the investigation give similar results? It's improved by taking repeated readings and averaging, and by using more precise equipment.
Validity is about whether the investigation actually tests what it claims to test — did the method properly isolate the relationship between the intended independent and dependent variable, with other variables genuinely controlled? A very reliable (repeatable) result can still be invalid if a variable wasn't properly controlled.
Common validity issues: a controlled variable that wasn't actually kept constant, a systematic error in the measuring equipment (e.g. an un-zeroed instrument), or a range too narrow to reveal the true shape of the relationship.

Worked ExampleEstimating uncertainty from repeated readings

A student times 10 oscillations of a pendulum three times, getting $18.2$ s, $18.4$ s, $18.1$ s. Find the mean period per oscillation and its uncertainty.

Step 1 — Find the mean time for 10 oscillations

$\bar{t} = \frac{18.2 + 18.4 + 18.1}{3} = 18.23\overline{3}\text{ s} \approx 18.2 \text{ s}$

Step 2 — Find the uncertainty using half the range

$\text{uncertainty} = \frac{18.4 - 18.1}{2} = 0.15 \text{ s} \approx 0.2 \text{ s (1 s.f., matching the data precision)}$

So the time for 10 oscillations is $18.2 \pm 0.2$ s.

Step 3 — Convert to the period of one oscillation

Dividing both the value and the uncertainty by 10 (since dividing by an exact number divides the uncertainty by the same factor):

$T = \frac{18.2}{10} = 1.82 \text{ s}, \qquad \text{uncertainty} = \frac{0.2}{10} = 0.02 \text{ s}$

$\textbf{T = 1.82 ± 0.02 s}$

Practice question

A student measures the diameter of a ball three times: $4.0$ cm, $4.2$ cm, $4.1$ cm. State the mean diameter with its uncertainty from half the range.

Worked solution: Mean $= \dfrac{4.0+4.2+4.1}{3} = 4.10$ cm. Half the range $= \dfrac{4.2-4.0}{2} = 0.1$ cm. Diameter $= 4.1 \pm 0.1$ cm.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Three repeated length measurements are $20.1$ cm, $20.3$ cm, $19.9$ cm. Find the mean and the uncertainty using half the range.

Merit

Explain the difference between the reliability and the validity of an investigation, using an example of an investigation that could be reliable but not valid.

Excellence