Measurement uncertainty and error Transitional Reasoning Download PDF

Questions

1. 1.

   A student uses a standard meter rule graduated in millimeters to measure the length of a wooden block. The student claims that aligning one end of the block precisely with the 0 mm mark eliminates any uncertainty at that end, leaving only the reading uncertainty of $\pm 0.5\text{ mm}$ at the opposite end. Which of the following statements correctly evaluates the student's claim and explains the actual absolute uncertainty associated with this measurement?

   • A) The student's claim is valid because the zero graduation represents a fixed datum point, which eliminates any observational error at the starting position of the object.
   • B) The student's claim is valid only if the object's length exceeds 10 cm, where the relative uncertainty at the starting mark becomes negligible compared to the total length.
   • C) The student's claim is invalid because aligning an end with the zero mark is an observational judgment subject to reading uncertainty, meaning the measurement inherently combines the uncertainties from both ends.
   • D) The student's claim is invalid because placing an object at the zero graduation creates an unavoidable systematic offset that consistently overestimates the measured length.
2. 2.

   A student determines the volume of an irregularly shaped rock by water displacement using a graduated cylinder. When recording both the initial water volume and the final volume after fully submerging the rock, the student consistently reads the scale from the upper outer edge of the concave water meniscus instead of the bottom of the curve. Which of the following correctly classifies this experimental error and explains its effect on the calculated volume of the rock?

   • A) It is a random error that introduces unpredictable fluctuations into both measurements, which significantly broadens the experimental uncertainty of the rock's volume.
   • B) It is a systematic error that overestimates both volume readings equally, so the constant positive offset cancels out when subtracting the two values to determine the rock's volume.
   • C) It is a systematic error that causes both individual volume readings to be too high, resulting in an accumulated overestimation of the rock's calculated volume.
   • D) It is an instrumental calibration error arising from the graduated markings, which permanently introduces a proportional scaling error into the calculated volume.
3. 3.

   To determine the electrical power dissipated by an ohmic resistor, a student considers two possible methods: measuring current and potential difference to calculate $P = VI$, or measuring current and resistance to calculate $P = I^2 R$. All available laboratory meters (voltmeter, ammeter, and ohmmeter) measure their respective quantities with a relative percentage uncertainty of $\pm 2\%$. According to the standard rules of experimental uncertainty propagation, which method yields the lower relative uncertainty in the calculated power, and why?

   • A) Calculating $P = I^2 R$ yields an overall percentage uncertainty of 4%, which is preferred because squaring the current diminishes its fractional contribution to the final result.
   • B) Both methods result in an identical percentage uncertainty of 4%, because the power dissipated by the circuit component is physically identical regardless of the measuring instruments used.
   • C) Calculating $P = VI$ yields an overall percentage uncertainty of 2%, which is preferred because multiplying two quantities divides their relative uncertainties.
   • D) Calculating $P = VI$ yields an overall percentage uncertainty of 4%, which is preferred over the 6% resulting from $P = I^2 R$ because the squared term doubles the uncertainty contribution of the current measurement.
4. 4.

   To determine the cross-sectional area of a copper wire, a student is instructed to measure its diameter using a micrometer screw gauge. Rather than repeating measurements at a single marked location, the experimental protocol requires measuring the diameter at several positions along the wire and at two mutually perpendicular orientations at each position. Which of the following best explains why this experimental procedure is necessary?

   • A) It improves the micrometer's inherent resolution by subdividing the smallest graduation of the instrument scale across multiple trials.
   • B) It accounts for geometrical defects such as ovality and tapering, preventing localized cross-sectional variations from systematically biasing the calculated area.
   • C) It eliminates any mechanical zero error present in the micrometer screw gauge without the need to record a closed-jaw zero reading.
   • D) It transforms random fluctuations in the wire's physical dimensions into a uniform systematic offset that cancels during calculations.
5. 5.

   A student investigates the period of a simple pendulum using a stopwatch. Rather than starting and stopping the timer when the pendulum bob reaches its maximum displacement (amplitude), the student places a vertical reference mark at the lowest point of the swing (equilibrium position) and records the timing as the bob passes this mark moving in a specified direction. Which of the following best explains why this experimental technique reduces the timing uncertainty?

   • A) The bob has maximum speed at the equilibrium position, minimizing the time interval over which the crossing occurs and reducing ambiguity in identifying the exact start and stop instants.
   • B) The bob comes to a momentary rest at the amplitude, which eliminates human reaction time and allows for a perfectly synchronized start.
   • C) Air resistance acts only at the extreme positions of the swing, so timing at the equilibrium position eliminates systematic dampening errors from the measured period.
   • D) The period of a pendulum is defined only between points of zero gravitational potential energy, so measuring from the amplitude would yield an incorrect theoretical frequency.
6. 6.

   A student uses a handheld digital stopwatch with a display resolution of 0.01 s to measure the transit time of a cart rolling down an inclined track. From a single trial, the student records a time of 3.48 s and reports the result as $t = (3.48 \pm 0.01)\text{ s}$, asserting that the measurement uncertainty is defined solely by the instrument's display. Which of the following statements correctly evaluates the student's methodology and identifies the primary factor determining the experimental uncertainty in this measurement?

   • A) The methodology is invalid because human reaction time introduces an operational uncertainty of roughly 0.1–0.3 s, which far exceeds the instrument resolution and dominates the measurement error.
   • B) The methodology is valid because digital timing devices automatically compensate for user latency by registering high-frequency electronic quartz oscillations.
   • C) The methodology is invalid because digital timers inherently suffer from continuous calibration drift that imposes a systematic additive zero offset of at least 1.0 s.
   • D) The methodology is valid because taking repeated trials is only required when using an analogue stopwatch with visual scale division interpolation.
7. 7.

   Two laboratory groups measure the boiling temperature of pure water at standard atmospheric pressure, where the accepted value is 100.0 °C. Group X records values of 96.8 °C, 97.0 °C, 96.9 °C, and 96.9 °C, while Group Y records 98.5 °C, 101.4 °C, 99.1 °C, and 101.0 °C. Which statement correctly evaluates the accuracy, precision, and predominant types of experimental error in each group's data set?

   • A) Group X demonstrates high precision but low accuracy due to a systematic error, whereas Group Y demonstrates high accuracy but low precision due to significant random error.
   • B) Group X demonstrates high accuracy because its measurements are tightly clustered, whereas Group Y demonstrates high precision because its mean value matches the accepted temperature.
   • C) Group X is both accurate and precise because the spread is only 0.2 °C, whereas Group Y is neither accurate nor precise because its individual readings fluctuate widely around the boiling point.
   • D) Group X is affected primarily by random error causing a constant offset, whereas Group Y is affected primarily by systematic error causing large scatter between trials.
8. 8.

   A student uses a digital multimeter to measure a steady current in a circuit, expecting a value around 18 mA. To avoid blowing the meter's internal fuse, the student selects the 10 A range (resolution 0.01 A) and records a reading of 0.02 A, instead of using the 200 mA range (resolution 0.1 mA) or the 20 mA range (resolution 0.01 mA). Which of the following evaluations correctly explains why this choice is poor experimental practice and identifies the optimal range to minimize measurement uncertainty?

   • A) Selecting the 10 A range yields a reading with only one significant digit and an unacceptably large relative uncertainty; switching to the 20 mA range provides the finest resolution without exceeding the range limit, thereby minimizing relative uncertainty.
   • B) Selecting the 10 A range introduces a severe systematic zero error; switching to the 200 mA range is required because digital instruments must always operate below 50% of their maximum limit to avoid non-linear distortion.
   • C) Selecting the 10 A range is acceptable because the absolute uncertainty is identical across all ranges, but the 20 mA range must be avoided as its lower internal resistance would draw excessive current from the circuit.
   • D) Selecting the 10 A range produces excessive random thermal fluctuations in the meter; the student should choose the 200 mA range to achieve an optimal trade-off between heat dissipation and display stability.
9. 9.

   To determine the acceleration of a cart rolling down an inclined ramp, a student plots velocity on the vertical axis against time on the horizontal axis, placing experimental uncertainty bars on each data point. Taking into account the limits imposed by the uncertainty bars, the student constructs the steepest acceptable line of best fit with a gradient of $3.40\text{ m/s}^2$ and the shallowest acceptable line of best fit with a gradient of $3.00\text{ m/s}^2$. Which of the following correctly states the experimental acceleration with its absolute uncertainty and explains how the uncertainty is determined from these lines?

   • A) $a = (3.20 \pm 0.40)\text{ m/s}^2$; the best estimate is the average of the two gradients, and the uncertainty is the total range between the steepest and shallowest gradients.
   • B) $a = (3.40 \pm 0.20)\text{ m/s}^2$; the best estimate is given by the steepest gradient, and the uncertainty is half the difference between the steepest and shallowest gradients.
   • C) $a = (3.20 \pm 0.20)\text{ m/s}^2$; the best estimate is the average of the two gradients, and the uncertainty is half the difference between the steepest and shallowest gradients.
   • D) $a = (3.20 \pm 0.10)\text{ m/s}^2$; the best estimate is the average of the two gradients, and the uncertainty is one-quarter of the range between the steepest and shallowest gradients.
10. 10.

    An experimenter checks the calibration of a digital balance using three certified reference masses of 10.0 g, 20.0 g, and 50.0 g, recording readings of 10.5 g, 21.0 g, and 52.5 g, respectively. The experimenter considers subtracting a fixed value of 0.5 g from all future readings to correct for the discrepancy. Which of the following correctly identifies the type of error present and explains why this proposed correction is invalid?

    • A) It is a proportional systematic error, because the measured value exceeds the true mass by a constant factor of 1.05 rather than a constant additive offset; correcting it requires dividing readings by 1.05.
    • B) It is a constant zero-offset systematic error, because every reading deviates above the certified mass; subtracting 0.5 g will accurately calibrate readings across the entire range.
    • C) It is a random error caused by ambient air currents, because the deviations change between the different masses; it should instead be minimized by averaging repeated measurements of each mass.
    • D) It is a precision limit error resulting from the display resolution, because digital balances cannot accurately register masses above 10.0 g without recalibrating the zero tare point.
11. 11.

    A high-precision analogue ammeter contains a flat mirror strip positioned directly beneath the scale markings and the pointer needle. When taking a reading of a steady direct current, a student shifts their viewing position until the needle completely overlaps and conceals its own reflection. Which of the following correctly explains the scientific rationale for this technique and identifies the error it eliminates?

    • A) Aligning the needle with its reflection doubles the effective resolution of the graduated scale, which minimizes the instrumental uncertainty associated with the smallest scale division.
    • B) Viewing the needle directly over its reflection allows the observer to average the two visual positions, which cancels out random errors caused by momentary current fluctuations.
    • C) The mirror compensates for mechanical wear and thermal expansion in the needle's pivot spring, thereby eliminating zero-setting error before the circuit is closed.
    • D) Obscuring the reflection ensures that the observer's line of sight is strictly perpendicular to the dial, eliminating parallax error that would otherwise produce a systematic reading bias.
12. 12.

    A student determines the resistance of an ohmic component by recording multiple pairs of current and voltage measurements and plotting voltage on the vertical axis versus current on the horizontal axis. Unknown to the student, the voltmeter has a constant zero offset, displaying +0.4 V when disconnected from the circuit. Which of the following correctly describes how this systematic error affects the resulting graph and the resistance determined from its gradient?

    • A) The slope of the line increases because every voltage reading is larger, resulting in an overestimated resistance.
    • B) The vertical intercept shifts upward by 0.4 V, while the slope and the calculated resistance remain unaffected.
    • C) Both the vertical intercept and the slope increase proportionally, leading to an overestimation of the resistance.
    • D) The vertical intercept remains at the origin, but the random scatter of data points around the line increases significantly.
13. 13.

    Two laboratory groups measure the speed of sound in the same room under identical environmental conditions. Group A reports a value of $342 \pm 3\text{ m/s}$, while Group B reports $351 \pm 4\text{ m/s}$. A student claims that the two results are in agreement because the percentage uncertainty of each measurement is under 2%. Which of the following statements correctly evaluates the student's claim?

    • A) The claim is valid because both groups achieved relative uncertainties below 2%, which guarantees mutual agreement regardless of the measured central values.
    • B) The claim is valid because the midpoint between the two values, $346.5\text{ m/s}$, falls within the experimental uncertainty range of both groups.
    • C) The claim is invalid because their uncertainty intervals, $339\text{ to }345\text{ m/s}$ and $347\text{ to }355\text{ m/s}$, do not overlap, indicating a significant discrepancy likely due to systematic error.
    • D) The claim is invalid because two experimental results can only be considered in agreement if their central values and absolute uncertainties are completely identical.
14. 14.

    A student determines the density of a solid metal cube by measuring its mass $m$ and edge length $L$ using the relationship $\rho = \frac{m}{L^3}$. The relative uncertainty in the mass measurement is $1.0\%$, while the relative uncertainty in the edge length is $2.0\%$. Which of the following correctly states the total percentage uncertainty in the calculated density and explains the relative impact of the edge length measurement?

    • A) 3.0%, because relative uncertainties in experimental quantities always add directly regardless of powers.
    • B) 7.0%, because raising the edge length to the third power multiplies its relative uncertainty by three during error propagation.
    • C) 5.0%, because the edge length contributes proportionally to the area of the cube's faces rather than its full volume.
    • D) 9.0%, because the percentage uncertainty of the edge length must be cubed when calculating the volume.
15. 15.

    To determine the diameter of a precision ball bearing, a student takes four repeated measurements using a digital caliper with a resolution of 0.01 mm: 18.24 mm, 18.32 mm, 18.26 mm, and 18.30 mm. To report the final result, the student calculates the mean and estimates the experimental uncertainty from the spread of the data. Which of the following correctly presents the result with its uncertainty and explains why this uncertainty exceeds the instrument's resolution?

    • A) $(18.28 \pm 0.08)\text{ mm}$, because the uncertainty must equal the full difference between the extreme values to cover the entire data range.
    • B) $(18.28 \pm 0.04)\text{ mm}$, because random variations between trials produce a spread whose half-range is larger than the instrument resolution.
    • C) $(18.28 \pm 0.01)\text{ mm}$, because the uncertainty of any digital tool is strictly defined by its smallest display division regardless of scatter.
    • D) $(18.32 \pm 0.04)\text{ mm}$, because the reported central value must be the maximum recorded measurement to avoid underestimating the diameter.
16. 16.

    A student investigates the thermal expansion of a metal rod by measuring its length before and after heating. The initial length is recorded as $L_1 = 50.0 \pm 0.1$ cm and the heated length as $L_2 = 50.5 \pm 0.1$ cm. The student calculates the change in length as $\Delta L = L_2 - L_1$ and observes that the percentage uncertainty in $\Delta L$ is substantially greater than that of either original measurement. Which of the following correctly states the value of $\Delta L$ with its absolute uncertainty and explains why its relative uncertainty increased so significantly?

    • A) $\Delta L = 0.5 \pm 0.0$ cm; subtracting the two measured values cancels out their absolute uncertainties.
    • B) $\Delta L = 0.5 \pm 0.1$ cm; the absolute uncertainty of a difference cannot exceed the largest individual uncertainty.
    • C) $\Delta L = 0.5 \pm 0.2$ cm; the relative uncertainty increases because repeated readings compound random errors multiplicatively.
    • D) $\Delta L = 0.5 \pm 0.2$ cm; absolute uncertainties add together while the calculated difference is much smaller than the original lengths.
17. 17.

    A student investigates the electrical resistance of a component using the relationship $R = \frac{V}{I}$, recording a potential difference of $V = 10.0 \pm 0.2\text{ V}$ and a current of $I = 2.00 \pm 0.08\text{ A}$. To minimize the overall relative uncertainty in the calculated resistance, the student evaluates the individual contribution of each instrument. Which of the following correctly identifies the total percentage uncertainty in the calculated resistance and the measurement that should be prioritized for improvement?

    • A) 2.0%, and the voltage measurement should be prioritized because its absolute uncertainty ($\pm 0.2\text{ V}$) is larger than that of the current.
    • B) 4.0%, and the current measurement should be prioritized because current appears in the denominator of the resistance formula.
    • C) 6.0%, and the current measurement should be prioritized because its fractional uncertainty is twice that of the voltage measurement.
    • D) 10.0%, and both measurements should be prioritized equally because absolute uncertainties add directly regardless of the operation.
18. 18.

    To determine the local acceleration due to gravity, a student conducts five trials using a timing gate and a falling sphere, obtaining values of 9.38 m/s², 9.40 m/s², 9.39 m/s², 9.41 m/s², and 9.38 m/s² against an accepted value of 9.81 m/s². The student claims that conducting ten additional trials using the identical procedure and equipment will shift the experimental mean to the accepted value. Which statement correctly evaluates this claim based on the principles of measurement error?

    • A) The claim is valid, because taking more measurements systematically removes calibration offsets from the experimental setup.
    • B) The claim is valid, because increasing the sample size always improves both the precision and the accuracy of experimental data.
    • C) The claim is invalid, because the large spread among the measurements indicates that random errors cannot be reduced by averaging.
    • D) The claim is invalid, because repeating identical trials reduces random uncertainty but cannot eliminate the prevailing systematic error.
19. 19.

    Before measuring the diameter of a cylindrical brass rod, a student notices that a digital caliper reads +0.06 mm when its measuring jaws are fully closed. The student then takes five repeated measurements along the rod, obtaining values that cluster closely around an average of 8.42 mm. Which statement correctly identifies the type of error caused by the closed-jaw reading and the appropriate step to find the rod's actual diameter?

    • A) It is a random error that increases the scatter of the data, and it is automatically eliminated by calculating the average of the five measurements.
    • B) It is a random error caused by fluctuating grip pressure, so the zero reading should be added to each measurement before reporting the mean.
    • C) It is a systematic error that shifts all measurements unpredictably, meaning the five measurements must be discarded and repeated with a ruler.
    • D) It is a systematic error that consistently shifts all readings upward, and the actual diameter is found by subtracting 0.06 mm from the average measurement.
20. 20.

    A student uses a vernier caliper with an instrumental uncertainty of $\pm 0.05\text{ mm}$ to measure the thickness of a single sheet of paper, obtaining a value of $0.10\text{ mm}$. This single measurement carries an unacceptably high relative uncertainty of 50%, which undermines the reliability of the experimental findings. Which of the following procedures will most effectively minimize the percentage uncertainty in determining the thickness of one sheet using the same instrument?

    • A) Calibrate the zero point repeatedly before re-measuring the same single sheet.
    • B) Measure the thickness of the single sheet at several positions and record the maximum spread.
    • C) Clamp the caliper jaws firmly onto the single sheet to eliminate any microscopic air pockets.
    • D) Measure the total thickness of a stack of 100 identical sheets and divide the result by 100.