Making measurements and analyzing data

1. Measurements and uncertainties

• Measurements and uncertainties are critical aspects of scientific research and engineering applications.
• Measurements:
• – Accuracy: The degree of closeness to the true value.
• – Precision: The degree of consistency in repeated measurements.
• – Resolution: The smallest change in the measured quantity that can be detected.
• – Sensitivity: The ability to detect small changes in the measured quantity.
• Uncertainties:
• – Random errors: Unpredictable fluctuations in measurements.
• – Systematic errors: Consistent biases in measurements.
• – Instrumental errors: Errors due to instrument limitations.
• – Methodological errors: Errors due to measurement methods.
• Quantifying uncertainties:
• – Standard deviation: A measure of the spread of measurements.
• – Confidence intervals: A range of values within which the true value is likely to lie.
• – Uncertainty budgets: A detailed breakdown of individual uncertainty sources.

(a) Systematic errors (including zero errors) and random errors in measurements:

• ⇒Random errors:

• Random errors are unpredictable fluctuations in measurements that occur due to various factors, such as:
• – Instrument noise: Random variations in instrument readings.
• – Environmental factors: Changes in temperature, humidity, or other environmental conditions.
• – Human error: Accidental mistakes made by the person taking the measurement.
• – Sampling variability: Natural fluctuations in the population or sample being measured.
• – Measurement instrument variability: Differences in measurements due to variations in instrument calibration or performance.
• Characteristics of random errors:
• – Unpredictable: Cannot be anticipated or corrected for.
• – Variable: Change from measurement to measurement.
• – Zero mean: Average value of random errors is zero.
• – Normally distributed: Follow a normal distribution (bell-shaped curve).
• Effects of random errors:
• – Increase measurement uncertainty
• – Reduce precision
• – Make it difficult to identify systematic errors
• Minimizing random errors:
• – Use high-quality instruments
• – Take multiple measurements
• – Use averaging techniques
• – Control environmental factors
• – Implement quality control procedures

• ⇒ Systemic error:

• Systemic error, also known as systematic error, is a type of error that occurs due to a flaw or bias in the measurement process, instrumentation, or method. Unlike random errors, systemic errors are consistent and repeatable, and cannot be eliminated by simply repeating the measurement.
• Examples of systemic errors:
• – Instrument calibration error: Incorrect calibration of an instrument leads to consistent errors.
• – Measurement bias: Consistent differences between measured and true values due to faulty equipment or methods.
• – Sampling bias: Selecting samples that are not representative of the population.
• – Data processing errors: Systematic mistakes during data analysis or processing.
• – Human bias: Consistent errors due to observer expectations or biases.
• – Environmental factors: Uncontrolled environmental conditions affecting measurements.
• – Methodological errors: Flaws in the measurement method or procedure.
• – Standards or reference material errors: Inaccurate standards or reference materials.
• – Modeling errors: Inaccurate assumptions or simplifications in models.
• – Algorithmic errors: Systematic mistakes in data analysis algorithms.
• – The scale printed on a meter rule is incorrect and the ruler scale is only 99.0 cm long.
• – The needle on an ammeter-points to 0.1 A when no current is flowing. Each value recorded will, therefore, be bigger than the true value by 0.1 A. This is an example of a zero error – the apparatus shows a non-zero value when it should be registering a value of exactly zero.
• – A thermometer has been incorrectly calibrated, so it constantly gives temperature readings that are 2°C lower than the true temperature.
• – A parallax error is caused by reading a scale at the wrong angle, for example when your eye is not parallel with the meniscus when using a measuring cylinder
• Systemic errors can be reduced or eliminated by:
• – Calibration and maintenance
• – Improving measurement methods
• – Using multiple measurement techniques
• – Data validation and verification
• – Training and education
• – Quality control procedures
• – Regular equipment checks
• – Using standards and reference materials
• – Model validation and refinement
• – Algorithm testing and validation
• 
• Figure 1 Systematic and random error

(b) Precision and accuracy:

• ⇒ Precision:

• Precision refers to the consistency or repeatability of a measurement or value. It indicates how close individual measurements are to each other, even if they are not close to the true value.
• Types of Precision:
• – Repeatability: The precision of repeated measurements made under the same conditions.
• – Reproducibility: The precision of measurements made under different conditions, such as different instruments or observers.
• Factors Affecting Precision:
• – Instrumentation: Quality and calibration of instruments.
• – Methodology: Experimental design and procedures.
• – Observer bias: Systematic errors introduced by the observer.
• – Random errors: Unpredictable fluctuations in measurements.
• High precision does not necessarily mean high accuracy. However, high precision is essential in physics to:
• – Detect small changes: Precise measurements allow detection of subtle effects.
• – Test hypotheses: Precise measurements enable rigorous testing of theoretical predictions.
• – Establish trends: Precise measurements help identify patterns and relationships.
• Precision is critical in physics to ensure reliable and consistent measurements, which are essential for advancing our understanding of the physical world.

• ⇒ Accuracy:

• Accuracy refers to how close a measurement or value is to the true or accepted value. It indicates the degree of closeness to the target value.
• Types of Accuracy:
• – Absolute accuracy: The difference between the measured value and the true value.
• – Relative accuracy: The ratio of the absolute accuracy to the true value.
• Factors Affecting Accuracy:
• – Instrumentation: Quality and calibration of instruments.
• – Methodology: Experimental design and procedures.
• – Observer bias: Systematic errors introduced by the observer.
• – Random errors: Unpredictable fluctuations in measurements.
• – Standards and references: Quality of reference materials and standards.
• Importance of Accuracy:
• – Trustworthiness: Accurate measurements are essential for reliable scientific conclusions.
• – Comparability: Accurate measurements enable comparison across different experiments and researchers.
• – Decision-making: Accurate measurements inform decisions in fields like engineering, medicine, and policy-making.
• – Understanding physical phenomena: Accurate measurements help physicists understand and model complex phenomena.
• Accuracy is critical in physics to ensure that measurements reflect the true nature of the physical world, allowing for reliable scientific conclusions, comparisons, and decision-making.
• 
• Figure 2 The closer to the center of the target an arrow lands, the more accurate it is. The closer the arrows cluster together, the more precise the shooting is. The aim is to be accurate and precise.
• When the shots are accurate and precise, they all land in the middle of the target.
• When the shots are precise, they do not all land in the center of the target, but they land very close to one another.
• Low accuracy and low precision can be thought of as arrows landing far away from the center of the target and far away from each other.

(c) Absolute and percentage uncertainties:

• Absolute uncertainty refers to the maximum possible error or uncertainty in a measurement or value. It is a quantitative measure of the uncertainty and is usually expressed in the same units as the measurement.
• Types of Absolute Uncertainty:
• – Instrumental uncertainty: Limitations of the measuring instrument.
• – Methodological uncertainty: Errors in the measurement method.
• – Random uncertainty: Fluctuations in the measurement.
• – Systematic uncertainty: Biases in the measurement.
• Importance of Absolute Uncertainty:
• – Quantifies error: Provides a numerical value for the uncertainty.
• – Allows error propagation: Enables calculation of uncertainties in derived quantities.
• – Facilitates comparison: Enables comparison of measurements with different uncertainties.
• – Informs decision-making: Helps assess the reliability of measurements in practical applications.
• Imagine that you measure the diameter of a ball using a meter ruler that has centimeter and millimeter graduations.
• You find that the diameter of the ball is 63 mm, so you could state that the diameter of the ball is 63±0.5mm. When taking single readings, the absolute uncertainty is usually given as the smallest division on the measuring instrument used.

• ⇒ Percentage uncertainty:

• Percentage uncertainty is a way to express the absolute uncertainty as a percentage of the measured value. It is calculated by dividing the absolute uncertainty by the measured value and multiplying by 100.
• [math]\text{Percentage Uncertainty} = \left( \frac{\text{Uncertainty}}{\text{Measured Value}} \right) \times 100\%[/math]
• Advantages:
• – Easy to understand and compare.
• – Allows for quick assessment of precision.
• – Useful for reporting uncertainties in measurements.

1. For addition and subtraction:

• Absolute uncertainties are added:
• For example, the distance x determines between two separate position measurements
• [math]\begin{gather} X_1 = 12.5 \pm 0.1 \, \text{cm} \qquad X_2 = 32.6 \pm 0.1 \, \text{cm} \\
  X = X_2 – X_1 = (32.6 – 12.5) \pm (0.1 + 0.1) \\
  X = 20.1 \pm 0.2 \, \text{cm} \end{gather}[/math]

2. For multiplication and division:

• Percentage uncertainties are added.
• For example, the maximum possible uncertainty in the value of resistance R of a conductor determined from the measurements of potential difference V and resulting current flow I by using
• [math] \begin{gather} R = \frac{V}{I} \\
  V = 6.8 \pm 0.1 \, \text{V} \\
  I = 0.95 \pm 0.06 \, \text{A} \end{gather}[/math]
• The %age uncertainty for V is [math]\frac{0.1}{6.8} \times 100\% = \text{ about } 1\%[/math]
• The %age uncertainty for I is [math]\frac{0.06}{0.95} \times 100\% = \text{ about } 1\%[/math]
• Hance total uncertainty in the value of resistance R when V is divided by 7%. The result is
• [math] \begin{gather} R = \frac{6.8}{0.95} = 7.16 \, \text{V} \, \text{A}^{-1} = 7.16 \, \Omega \, \text{with a percentage uncertainty of } 7\% \\ = 7.16 \times \frac{7}{100} = 0.5 \\
  R = 7.16 \pm 0.5 \, \Omega \end{gather}[/math]
• The result is round off to two significant digits because both V and R have two significant figures and uncertainty, being an estimate only, is recorded by one significant figure.

3. For power:

• Multiply the percentage uncertainty by that power.
• For example, in the calculation of the volume of a sphere using
• [math]V = \frac{4}{3} πr^3 [/math]
• %age uncertainty in V [math] V = 3 × \% \text{ age uncertainty in radius r}[/math].
• As uncertainty is multiplied by power factor, it increases the precision demand of measurement. If the radius of a small sphere is measured as 1.25 cm by a vernier callipers with least count 0.01 cm, then
• The radius r is recorded as
• [math]r=1.25±0.01cm[/math]
• Absolute uncertainty = least count = [/math]±0.01cm[/math]
• [math]\% \text{ age uncertainty in r} = \frac{0.01}{1.25} \times 100\% = 0.8\% \\
  \text{The percentage uncertainty in } V = 3 \times 0.8\% = 2.4\%[/math]
• Volume is
• [math] \begin{gather}V = \frac{4}{3} \pi r^3 \\
  V = \frac{4}{3} \times 3.14 \times (1.25)^3 \\
  V = 8.17 \, \text{cm}^3 \, \text{with a 2.4\% uncertainty} \\
  = 8.17 \times \frac{2.4}{100} = 0.2  \\ \text{Thus the resut should be recorded as} \\ V = 8.17 \pm 0.2 \, \text{cm}^3 \end{gather}[/math]

4. For uncertainty in the average value of many measurements:
5. i)  Find the average value of measured values.
6. ii) Find deviation of each measured value from the average value.
7. iii)  The mean deviation is the uncertainty in the average value.
8. For example, the six readings of the micrometer screw gauge to measure the diameter of a wire in mm are

9. 2.56, 2.59, 2.14, 1.96, 2.16, 2.87

10. Then
11. [math]\begin{gather} \text{Average} = \frac{2.56 + 2.59 + 2.14 + 1.96 + 2.16 + 2.87}{6} \\
    \text{Average} = 2.38 \, \text{mm} \end{gather}[/math]
12. The deviation of the readings, which are the difference without regards to the sign, between each reading and average value are 0.18, 0.21, 0.24, 0.42, 0.22, 0.49
13. [math] \begin{gather} \text{Mean Deviation} = \frac{0.18 + 0.21 + 0.24 + 0.42 + 0.22 + 0.49}{6} \\
    \text{Mean Deviation} = 0.29 \, \text{mm} \end{gather}[/math]
14. Thus, likely uncertainty in the mean diameter 2.38 mm is 0.29 mm recorded as
15. [math]= 2.38±0.29mm [/math]

(d) Graphical treatment of errors and uncertainties:

• Graphs are a useful, highly visual way of demonstrating the relationship between two variables, showing patterns and trends and allowing us to determine values from measurements of the gradient and the y -intercept.
• Graphs are most effective when:
• – The scale of the graph has been chosen so that the plotted points cover as much of the graph paper as possible in both directions
• – The points are plotted clearly
• – The lines of best fit and worst fit are drawn clearly
• – The gradient can be calculated easily using two points on the line that are as far apart as possible, but within the measured range.
• – The y -intercept can be read clearly and accurately using the scale on the y -axis.

• ⇒Determining the uncertainty in the gradient from the maximum and minimum gradients

• It is possible to determine the uncertainty in a gradient by drawing lines of maximum and minimum gradient through the appropriate points on the graph.
• If there is only a small amount of scatter then error bars can be incorporated into the graph to help this to happen.
• The uncertainty in a gradient can be determined as follows:
• – Add error bars to each point. The size of the error bars is usually the same for each measurement.
• – Draw a line of best fit through the scattered points and within the error bars. The line of best fit should go through as many points as possible, with equal numbers of points above and below the line. Discard any major outliers.
• – Calculate the gradient of the line of best fit.
• – Do the same for the worst fit line, which may be more steep or less steep than the line of best fit.
• – To find the uncertainty from the graph, work out the difference between the gradients of the line of best fit and the line of worst fit. This should be expressed as a positive value (the modulus). The equation you use is
• [math] \text{uncertainty = (gradient of best fit line) – (gradient of worst fit line)}[/math]
• For example:
• If the gradient of the line of best fit has a value of 3.7 and the gradient of the worst acceptable line was 4.3, the percentage uncertainty would be:
• [math]\text{The percentage uncertainty} = (\frac{4.3-3.7}{3.7}) × 100 \% = 16 \%[/math]
• In this case, the uncertainty is half the difference between the maximum and minimum gradients, as shown in graph
• 
• Figure 3 Lines of best fit