Do Overlapping Error Bars Mean Two Results Agree?

Two groups measure the period of the same pendulum. The first reports \(1.42\pm0.04\ \mathrm{s}\), while the second reports \(1.47\pm0.02\ \mathrm{s}\). Their error bars overlap slightly, between \(1.45\) and \(1.46\ \mathrm{s}\).

Does this prove that both groups measured the same period?

No. Overlapping error bars show that the results may be compatible within their stated uncertainties. They do not prove that the results are identical. More importantly, the strength of that conclusion depends on what the error bars actually represent.

This is where students often make a mistake. They see two overlapping intervals and immediately conclude that the experiments agree. To make a proper comparison, you need to understand the measurement intervals, the uncertainty convention, and how uncertainty behaves when two results are compared.

01The experiment behind the two numbers

Consider an experiment investigating the period of a simple pendulum.

The pendulum consists of a bob attached to a string, suspended from a fixed clamp. It swings through a central equilibrium position, completing one oscillation each time it returns to its starting position and direction.

Two groups want to measure its period using different methods.

Group A: Stopwatch method

The first group uses a stopwatch to measure the time taken for 10 complete oscillations.

They start and stop the stopwatch as the bob passes through the equilibrium position in the same direction. Timing 10 oscillations rather than one reduces the relative effect of human reaction time.

Their measured time is:

\[
t_A=(14.2\pm0.4)\ \mathrm{s}
\]

The period is the time divided by the number of oscillations:

\[
T_A=\frac{14.2}{10}=1.42\ \mathrm{s}
\]

The uncertainty is also divided by 10:

\[
U_A=\frac{0.4}{10}=0.04\ \mathrm{s}
\]

Therefore:

\[
T_A=(1.42\pm0.04)\ \mathrm{s}
\]

Group B: Video analysis

The second group records the pendulum using a smartphone camera operating at 120 frames per second.

They count the number of frames between two successive crossings of the equilibrium position in the same direction. Using the same direction ensures that they measure a complete oscillation rather than half an oscillation.

They count \(176\pm2\) frames.

Since the camera records 120 frames each second:

\[
T_B=\frac{176}{120}
=1.467\ \mathrm{s}
\]

The uncertainty in the period is:

\[
U_B=\frac{2}{120}
=0.0167\ \mathrm{s}
\]

Therefore, to appropriate significant figures:

\[
T_B=(1.47\pm0.02)\ \mathrm{s}
\]

The results are summarised below.

All data are illustrative. For this initial comparison, the uncertainties are treated as maximum estimated bounds.

What should the experiment show?

If both groups measure the same stable pendulum under the same small-angle conditions, they should be estimating the same underlying period.

However, their central estimates do not need to be identical. Different measurement methods introduce different uncertainties.

The stopwatch method is affected by reaction time. Video analysis is affected by frame rate and the choice of frames marking the start and end of an oscillation.

A first attempt to time just one oscillation with a stopwatch would make reaction time a substantial fraction of the measurement. Timing 10 oscillations and dividing by 10 improves the method.

The two groups now have results that can be compared. The important question is how to interpret their uncertainty intervals.

02What does an uncertainty bar actually mean?

An error bar is a graphical representation of uncertainty or another measure of variation associated with a result.

For a measurement written as

\[
x\pm U
\]

the central value is \(x\), and \(U\) is the stated uncertainty.

If \(U\) represents a maximum uncertainty bound, the corresponding measurement interval is:

\[
[x-U,\ x+U]
\]

For example, consider:

\[
x=(12.4\pm0.3)\ \mathrm{cm}
\]

The lower limit is:

\[
12.4-0.3=12.1\ \mathrm{cm}
\]

The upper limit is:

\[
12.4+0.3=12.7\ \mathrm{cm}
\]

The measurement interval is therefore:

\[
[12.1,\ 12.7]\ \mathrm{cm}
\]

This interval tells us which values are allowed by the stated maximum uncertainty.

But there is an important complication: not all uncertainty bars represent maximum bounds.

Three common uncertainty conventions

These conventions are not interchangeable.

1. Maximum uncertainty

This is commonly used in introductory experimental comparisons.

If a measurement is \(x\pm U\), the stated limits are \(x-U\) and \(x+U\).

For this convention, interval overlap means that a common value is possible within both sets of stated bounds.

2. Standard uncertainty

Standard uncertainty is expressed as a standard deviation.

If the distribution is approximately normal, an interval of \(x\pm u\) covers approximately 68% of the distribution of values that could reasonably be attributed to the measurand.

It is not a guaranteed maximum or minimum.

3. Expanded uncertainty

Expanded uncertainty is calculated using:

\[
U=ku
\]

where:

• \(U\) is the expanded uncertainty;
• \(u\) is the standard uncertainty;
• \(k\) is the coverage factor.

For an approximately normal distribution with sufficient data, \(k=2\) corresponds to approximately 95% coverage.

This is useful when comparing measurements that need a specified level of coverage.

The formal distinction between these conventions follows international measurement terminology. See the BIPM International Vocabulary of Metrology and the NIST guide to measurement uncertainty.

The key lesson: always establish what the error bars represent before deciding whether they overlap meaningfully.

03Visualising overlapping and non-overlapping intervals

Let’s return to two measurements.

Suppose:

\[
x_1=10.0\pm0.4
\]

\[
x_2=10.5\pm0.3
\]

Assume both uncertainties are maximum bounds and both measurements refer to the same physical quantity.

The first interval is:

\[
I_1=[9.6,\ 10.4]
\]

The second interval is:

\[
I_2=[10.2,\ 10.8]
\]

The intervals overlap between 10.2 and 10.4.

This means a common value is possible within the two stated uncertainty intervals.

However, it does not tell us that the true value must be somewhere in the overlap, nor does it prove that the two central estimates are equal.

Now consider a contrasting case:

\[
x_1=10.0\pm0.1
\]

\[
x_2=10.5\pm0.1
\]

The intervals are:

\[
I_1=[9.9,\ 10.1]
\]

\[
I_2=[10.4,\ 10.6]
\]

There is no overlap.

Under the maximum-bound convention, the two results are inconsistent with a common value within their stated intervals.

This suggests that something needs investigating. Perhaps one method has an unaccounted systematic effect, the experimental conditions changed, or the uncertainty was underestimated.

It does not automatically mean that one group made a careless mistake.

04Error, uncertainty, and standard deviation are different

These three terms are related, but they describe different things.

Confusing them can lead to completely incorrect conclusions about experimental results.

Measurement error

Measurement error is the difference between a measured value and a reference or true value.

Using the sign convention:

\[
e=x_{\mathrm{measured}}-x_{\mathrm{reference}}
\]

A positive error means the measurement is higher than the reference. A negative error means it is lower.

For example, suppose a calibrated reference gives a pendulum period of \(1.460\ \mathrm{s}\), while the stopwatch method measures \(1.42\ \mathrm{s}\).

The measurement error is:

\[
e=1.42-1.460
\]

\[
e=-0.040\ \mathrm{s}
\]

The stopwatch result underestimates the reference value by \(0.040\ \mathrm{s}\).

Notice that the uncertainty was also \(0.040\ \mathrm{s}\).

These numbers happen to be equal, but they describe different things.

The error is the signed difference from the reference. The uncertainty describes the doubt associated with the measurement.

In a real experiment, the true value is usually unknown. This means the actual measurement error is generally unknown too.

Measurement uncertainty

Measurement uncertainty describes the range or dispersion of values that can reasonably be attributed to a measured quantity, based on the available information.

It can arise from several sources, including:

• Instrument resolution.
• Human reaction time.
• Random variation between measurements.
• Calibration limitations.
• Systematic effects that have not been fully corrected.

An important distinction is that uncertainty is not simply the amount by which a measurement is wrong.

A measurement can have small uncertainty but still be systematically inaccurate. For example, a poorly calibrated instrument might produce almost identical readings every time, all shifted away from the reference value.

Standard deviation

Standard deviation describes how spread out a set of repeated measurements is around its mean.

Consider five measurements of a pendulum’s period:

The mean period is:

\[
\bar T=\frac{1.41+1.43+1.42+1.44+1.40}{5}
\]

\[
\bar T=1.42\ \mathrm{s}
\]

The sample standard deviation is calculated using:

\[
s=\sqrt{\frac{\sum(T_i-\bar T)^2}{n-1}}
\]

where:

• \(s\) is the sample standard deviation;
• \(T_i\) is an individual measurement;
• \(\bar T\) is the mean;
• \(n\) is the number of measurements.

For these five readings:

\[
s=\sqrt{\frac{0.0010}{4}}
\]

\[
s=0.0158\ \mathrm{s}
\]

So the standard deviation is approximately:

\[
s=0.016\ \mathrm{s}
\]

Does this mean the uncertainty in the mean period is automatically \(0.016\ \mathrm{s}\)?

No.

The standard deviation describes the spread of individual readings. If the readings are independent and random variation is the only component being considered, the standard error of the mean is:

\[
u_{\bar T}=\frac{s}{\sqrt n}
\]

For this example:

\[
u_{\bar T}=\frac{0.0158}{\sqrt5}
\]

\[
u_{\bar T}\approx0.0071\ \mathrm{s}
\]

The standard error describes the expected variation in the mean, rather than the spread of individual readings.

Even this is not necessarily the total uncertainty of the mean. Instrument calibration, resolution, and other systematic effects may contribute additional uncertainty.

A small standard deviation does not automatically mean that a result is accurate.

05How should you actually compare two results?

There are two useful approaches, depending on the uncertainty convention.

1. Comparing maximum uncertainty intervals

Suppose two measurements are:

\[
x_1\pm U_1
\]

\[
x_2\pm U_2
\]

Their intervals overlap when:

\[
|x_1-x_2|\leq U_1+U_2
\]

This is simply the mathematical condition for the two intervals to share at least one value.

Use this method when the uncertainties are explicitly given as maximum bounds.

• If the intervals overlap, a common value is possible within the stated bounds.
• If the intervals do not overlap, there is no common value within the stated bounds.

Do not interpret this as proof that the underlying physical quantities are identical or different.

2. Comparing independent standard uncertainties

If the uncertainties are standard uncertainties, there is a more useful way to compare the results.

Instead of comparing the individual error bars, calculate the uncertainty in the difference between the two measurements.

Let:

\[
d=x_1-x_2
\]

For independent measurements, the combined standard uncertainty is:

\[
u_d=\sqrt{u_1^2+u_2^2}
\]

where \(u_1\) and \(u_2\) are the standard uncertainties of the two measurements.

Why do we use a square root of the sum of squares rather than simply adding the uncertainties?

Because independent standard uncertainties combine through their variances. Adding them directly would instead give a conservative maximum-bound estimate, not the combined standard uncertainty.

A useful comparison is the ratio:

\[
R=\frac{|d|}{u_d}
\]

For approximately normal, independent uncertainties:

• \(R\lesssim1\): The difference is within approximately one combined standard uncertainty.
• \(R\lesssim2\): The difference is within approximately two combined standard uncertainties, corresponding to roughly 95% coverage.
• \(R>2\): The difference exceeds approximately two combined standard uncertainties, suggesting a possible disagreement at that level.

These are useful approximate comparisons, not universal statistical significance rules. The actual interpretation depends on the uncertainty distributions and coverage assumptions.

There is also an important limitation: the formula assumes independent uncertainties.

If both measurements share a systematic effect, such as the same calibration offset, that effect may partly or completely cancel when calculating their difference. We will examine this more carefully in the final practice question.

What if the intervals do not overlap?

Consider two acceleration measurements with maximum uncertainty bounds:

\[
a_1=(2.10\pm0.03)\ \mathrm{m\,s^{-2}}
\]

\[
a_2=(2.20\pm0.02)\ \mathrm{m\,s^{-2}}
\]

Their intervals are:

\[
I_1=[2.07,\ 2.13]
\]

\[
I_2=[2.18,\ 2.22]
\]

There is a gap of:

\[
2.18-2.13=0.05\ \mathrm{m\,s^{-2}}
\]

The intervals do not overlap.

Under the stated maximum-bound convention, the results are inconsistent with a common value.

The appropriate response is to investigate the discrepancy. Check whether the experimental conditions were identical, whether the instruments were calibrated, and whether the uncertainty estimates included all important sources of variation.

06A practical decision rule for HSC Physics

When you encounter overlapping error bars in a practical investigation or exam question, follow this sequence.

1. Identify the quantity being measured. Are both experiments measuring the same physical quantity under comparable conditions?
2. Identify what the error bars represent. Are they maximum uncertainty bounds, standard deviations, standard uncertainties, or expanded uncertainties?
3. If they are maximum bounds, calculate the measurement intervals and check for overlap.
4. If they are independent standard uncertainties, calculate the combined uncertainty of the difference.
5. Consider any shared systematic effects that could influence both measurements.
6. State whether the results are compatible within the stated uncertainties. Do not claim that they are exactly equal.

A strong experimental conclusion might read:

“The uncertainty intervals overlap, indicating that the two measurements are consistent with a common value within their stated maximum uncertainties. However, this does not establish that the measured values are identical.”

That is much more scientifically accurate than simply writing, “The error bars overlap, so the results are the same.”

For more practice with constructing error bars and using maximum and minimum gradient lines, see Error Bars and Maximum-Minimum Gradients: HSC Physics Guide.

If you are comparing a measured value with a reference value, rather than comparing two uncertain measurements, see Percentage Error vs Percentage Uncertainty: HSC Physics Guide.

07Questions and solutions

Question 1

Two students measure the constant speed of a trolley using different methods.

Their results are:

\[
v_1=(3.60\pm0.10)\ \mathrm{m\,s^{-1}}
\]

\[
v_2=(3.66\pm0.08)\ \mathrm{m\,s^{-1}}
\]

Both uncertainties represent maximum bounds.

Calculate the measurement intervals and determine whether the results are compatible with a common speed.

Solution 1

The intervals overlap between \(3.58\) and \(3.70\ \mathrm{m\,s^{-1}}\), so the results are compatible with a common speed within their stated maximum uncertainties.

For the first student:

\[
v_{1,\min}=3.60-0.10=3.50
\]

\[
v_{1,\max}=3.60+0.10=3.70
\]

Therefore:

\[
I_1=[3.50,\ 3.70]\ \mathrm{m\,s^{-1}}
\]

For the second student:

\[
v_{2,\min}=3.66-0.08=3.58
\]

\[
v_{2,\max}=3.66+0.08=3.74
\]

Therefore:

\[
I_2=[3.58,\ 3.74]\ \mathrm{m\,s^{-1}}
\]

The common interval is:

\[
[3.58,\ 3.70]\ \mathrm{m\,s^{-1}}
\]

The central-value difference is:

\[
|3.66-3.60|=0.06\ \mathrm{m\,s^{-1}}
\]

The sum of the maximum uncertainty bounds is:

\[
0.10+0.08=0.18\ \mathrm{m\,s^{-1}}
\]

Since \(0.06<0.18\), a common speed is possible within the stated bounds.

The correct conclusion is that the results are consistent, not that the two central measurements are exactly equal.

Question 2

Two independent experiments are conducted to determine gravitational acceleration.

The first experiment gives:

\[
g_1=(9.72\pm0.12)\ \mathrm{m\,s^{-2}}
\]

The second gives:

\[
g_2=(9.90\pm0.09)\ \mathrm{m\,s^{-2}}
\]

Both uncertainties are standard uncertainties, corresponding to one standard deviation.

(a) Determine whether the individual uncertainty intervals overlap.

(b) Calculate the combined standard uncertainty in the difference between the results.

(c) Assess whether the results are consistent at approximately the 95% level, assuming independent, approximately normal uncertainties.

Solution 2

The individual intervals overlap, and the difference is within approximately two combined standard uncertainties. Therefore, the results are consistent at approximately the 95% level under the stated assumptions.

(a) Compare the intervals.

For the first measurement:

\[
I_1=[9.60,\ 9.84]\ \mathrm{m\,s^{-2}}
\]

For the second measurement:

\[
I_2=[9.81,\ 9.99]\ \mathrm{m\,s^{-2}}
\]

The intervals overlap between:

\[
9.81\text{ and }9.84\ \mathrm{m\,s^{-2}}
\]

(b) Calculate the combined standard uncertainty.

First calculate the difference:

\[
d=|9.90-9.72|
\]

\[
d=0.18\ \mathrm{m\,s^{-2}}
\]

For independent standard uncertainties:

\[
u_d=\sqrt{u_1^2+u_2^2}
\]

Substituting:

\[
\begin{aligned}
u_d&=\sqrt{0.12^2+0.09^2}\\
&=\sqrt{0.0144+0.0081}\\
&=\sqrt{0.0225}\\
&=0.15\ \mathrm{m\,s^{-2}}
\end{aligned}
\]

(c) Assess the consistency.

Calculate the ratio:

\[
R=\frac{0.18}{0.15}=1.2
\]

The difference is 1.2 combined standard uncertainties.

For an approximate 95% comparison:

\[
2u_d=2(0.15)=0.30\ \mathrm{m\,s^{-2}}
\]

Since:

\[
0.18<0.30
\]

the difference is within the approximate 95% comparison range.

The measurements are therefore compatible at this level.

Notice that overlapping intervals alone did not provide the complete assessment. Calculating the uncertainty of the difference gave us a more useful comparison.

Question 3

A student performs five independent measurements of gravitational acceleration and records:

The student calculates a mean of \(9.80\ \mathrm{m\,s^{-2}}\) and a sample standard deviation of approximately \(0.038\ \mathrm{m\,s^{-2}}\).

They report:

\[
g=(9.80\pm0.038)\ \mathrm{m\,s^{-2}}
\]

(a) Calculate the standard error of the mean.

(b) Explain why the student should not automatically identify the standard deviation as the uncertainty in the mean.

(c) Explain why even the standard error may underestimate the total measurement uncertainty.

Solution 3

The standard error of the mean is approximately \(0.017\ \mathrm{m\,s^{-2}}\). The standard deviation describes the spread of individual measurements, not directly the uncertainty of their mean.

(a) Calculate the standard error.

The standard error is:

\[
u_{\bar g}=\frac{s}{\sqrt n}
\]

where:

• \(s=0.038\ \mathrm{m\,s^{-2}}\);
• \(n=5\).

Substituting:

\[
u_{\bar g}=\frac{0.038}{\sqrt5}
\]

\[
u_{\bar g}\approx0.017\ \mathrm{m\,s^{-2}}
\]

This estimates the random variation expected in the mean if the measurements are independent.

(b) Explain the difference.

The standard deviation describes how much individual readings vary around their mean.

The standard error describes how much the calculated mean is expected to vary between repeated sets of measurements.

As the number of independent measurements increases, the standard error generally decreases because averaging reduces the influence of random variation.

The standard deviation does not automatically decrease in the same way.

Therefore, reporting the sample standard deviation as the uncertainty of the mean confuses two different statistical quantities.

(c) Consider other uncertainty sources.

The standard error only accounts for the random variation represented by the repeated readings.

It does not automatically include instrument calibration, systematic effects, or limitations in the experimental method.

For example, if the measuring instrument consistently reads too high, repeating the experiment may produce a very small standard deviation while the mean remains systematically biased.

The total uncertainty must account for the relevant sources, not just the scatter of repeated readings.

Question 4

Two independent laboratories measure the spring constant of a material.

Their results are:

\[
k_1=(10.00\pm0.10)\ \mathrm{N\,m^{-1}}
\]

\[
k_2=(10.15\pm0.10)\ \mathrm{N\,m^{-1}}
\]

The uncertainties shown are expanded uncertainties with a coverage factor of \(k=2\), corresponding to approximately 95% coverage.

Assume both underlying standard uncertainties are independent and approximately normally distributed.

(a) Determine whether the individual 95% uncertainty intervals overlap.

(b) Calculate the combined standard uncertainty of the difference.

(c) Determine whether the approximate 95% uncertainty interval for the difference includes zero.

(d) Explain what this reveals about using overlapping error bars as a test of agreement.

Solution 4

The individual 95% uncertainty intervals overlap, but the approximate 95% uncertainty interval for their difference excludes zero. Under the stated assumptions, the results provide evidence of a difference despite the overlapping bars.

(a) Compare the individual intervals.

For the first result:

\[
I_1=[9.90,\ 10.10]\ \mathrm{N\,m^{-1}}
\]

For the second result:

\[
I_2=[10.05,\ 10.25]\ \mathrm{N\,m^{-1}}
\]

The intervals overlap between:

\[
10.05\text{ and }10.10\ \mathrm{N\,m^{-1}}
\]

(b) Calculate the combined standard uncertainty.

The displayed uncertainties are expanded uncertainties with \(k=2\).

Therefore, the individual standard uncertainties are:

\[
u_1=u_2=\frac{0.10}{2}=0.05\ \mathrm{N\,m^{-1}}
\]

For independent measurements:

\[
u_d=\sqrt{u_1^2+u_2^2}
\]

\[
\begin{aligned}
u_d&=\sqrt{0.05^2+0.05^2}\\
&=\sqrt{0.005}\\
&=0.0707\ \mathrm{N\,m^{-1}}
\end{aligned}
\]

(c) Calculate the approximate 95% interval for the difference.

The central-value difference is:

\[
d=10.15-10.00=0.15\ \mathrm{N\,m^{-1}}
\]

The expanded uncertainty in the difference is:

\[
U_d=2u_d
\]

\[
U_d=2(0.0707)=0.1414\ \mathrm{N\,m^{-1}}
\]

The interval for the difference is:

\[
d\pm U_d
\]

\[
[0.0086,\ 0.2914]\ \mathrm{N\,m^{-1}}
\]

This interval does not include zero.

Under the stated approximate normal model, this indicates evidence of a difference at roughly the 95% level.

(d) Explain the implication.

The individual uncertainty intervals overlap, but that does not guarantee that the difference between the results is statistically compatible with zero at the same coverage level.

The reason is that the uncertainty of a difference is not calculated by simply adding the widths of the individual intervals.

Instead, independent standard uncertainties combine using the square root of the sum of their squares.

This is a more careful comparison than the simple overlap test.

The result is relatively close to the approximate 95% boundary, so it should not be overstated. A full statistical analysis would also need to account for the quality of the uncertainty estimates and the underlying distributions.

Question 5

A research team investigates the thermal expansion of a metal rod.

They measure its length at two temperatures using the same digital length sensor.

The sensor remains at a constant room temperature throughout the experiment. Its calibration has an unknown additive zero offset that is unchanged between measurements.

The measurements are:

Each reading has two uncertainty components:

• A common calibration zero-offset standard uncertainty of \(u_c=0.20\ \mathrm{cm}\), shared by both measurements.
• An independent reading and repeatability standard uncertainty of \(u_r=0.10\ \mathrm{cm}\) for each measurement.

Assume the two repeatability components are independent and approximately normally distributed. The common calibration offset is purely additive, and there are no other relevant uncertainty sources.

A student combines the total uncertainty of each length independently and concludes that the change in length is not resolved.

Investigate this conclusion.

(a) Calculate the total standard uncertainty associated with each individual length measurement.

(b) Determine the change in length between the two temperatures.

(c) Calculate the correct standard uncertainty in the change in length, taking account of the shared calibration offset.

(d) Determine whether the expansion is resolved at approximately the 95% level.

(e) Explain why treating the two total uncertainties as independent gives the wrong conclusion.

Solution 5

The rod’s measured increase in length is \(0.30\ \mathrm{cm}\). Because the calibration zero offset is identical in both measurements, it cancels when the difference is calculated. The resulting standard uncertainty is approximately \(0.141\ \mathrm{cm}\), meaning the increase is just resolved at the approximate 95% level under the stated assumptions.

(a) Calculate the total uncertainty of each measurement.

For each individual length, the calibration and repeatability components are independent.

Therefore, they combine in quadrature:

\[
u_L=\sqrt{u_c^2+u_r^2}
\]

Substituting:

\[
\begin{aligned}
u_L&=\sqrt{0.20^2+0.10^2}\\
&=\sqrt{0.04+0.01}\\
&=\sqrt{0.05}\\
&=0.224\ \mathrm{cm}
\end{aligned}
\]

Each individual measurement therefore has a standard uncertainty of approximately:

\[
u_L=0.224\ \mathrm{cm}
\]

(b) Calculate the change in length.

Define the change as:

\[
\Delta L=L_{80}-L_{20}
\]

Substituting:

\[
\Delta L=51.10-50.80
\]

\[
\boxed{\Delta L=0.30\ \mathrm{cm}}
\]

The positive result indicates that the rod has increased in measured length.

(c) Calculate the uncertainty in the change.

This is the important part of the problem.

The two measurements share the same calibration zero offset.

Represent the measured lengths as:

\[
L_{20}=L_{20,\mathrm{true}}+b+\epsilon_1
\]

\[
L_{80}=L_{80,\mathrm{true}}+b+\epsilon_2
\]

where:

• \(b\) is the common additive calibration offset;
• \(\epsilon_1\) is the independent reading fluctuation in the first measurement;
• \(\epsilon_2\) is the independent reading fluctuation in the second measurement.

Subtracting the two measurements gives:

\[
\begin{aligned}
\Delta L
&=L_{80}-L_{20}\\
&=(L_{80,\mathrm{true}}+b+\epsilon_2)\\
&\quad-(L_{20,\mathrm{true}}+b+\epsilon_1)
\end{aligned}
\]

The common offset cancels:

\[
b-b=0
\]

Therefore, the uncertainty in the difference comes from the two independent repeatability components:

\[
u_{\Delta L}=\sqrt{u_r^2+u_r^2}
\]

Substituting:

\[
\begin{aligned}
u_{\Delta L}
&=\sqrt{0.10^2+0.10^2}\\
&=\sqrt{0.02}\\
&=0.141\ \mathrm{cm}
\end{aligned}
\]

(d) Assess whether the expansion is resolved.

For an approximately 95% comparison:

\[
U_{\Delta L}\approx2u_{\Delta L}
\]

Therefore:

\[
U_{\Delta L}=2(0.141)
\]

\[
U_{\Delta L}=0.282\ \mathrm{cm}
\]

The measured change is:

\[
\Delta L=0.30\ \mathrm{cm}
\]

Since:

\[
0.30>0.282
\]

the approximate 95% interval for the change excludes zero.

The increase is therefore just resolved at approximately the 95% level under the assumptions in the question.

This is marginal evidence, not proof. Repeating the measurements and checking the calibration would strengthen the conclusion.

(e) Explain the tempting mistake.

The student’s approach would be to combine the two complete individual uncertainties as if they were independent:

\[
u_{\Delta L,\mathrm{wrong}}
=\sqrt{0.224^2+0.224^2}
\]

\[
u_{\Delta L,\mathrm{wrong}}
\approx0.317\ \mathrm{cm}
\]

The approximate 95% uncertainty would then be:

\[
2(0.317)=0.634\ \mathrm{cm}
\]

Since \(0.30<0.634\), the student would conclude that the change is not resolved.

But this calculation counts the same calibration uncertainty twice as though it were two unrelated errors.

The calibration offset is shared. It moves both measurements in the same direction by the same amount, so subtracting the measurements removes it.

This is the crucial distinction: the uncertainty of an individual measurement is not necessarily the uncertainty of the difference between two measurements.

If the sensor had been recalibrated between measurements, its offset had changed, or its scale factor varied with temperature, the cancellation might not apply.

The conclusion depends on the physical relationship between the uncertainty sources, not just the numerical size of the individual error bars.

The next time you compare experimental results, don’t stop at asking whether the error bars overlap. First identify what they represent, then work out what uncertainty actually applies to the comparison.

That same reasoning becomes important when you calculate gradients, determine physical constants from graphs, or propagate uncertainty through an equation. In particular, understanding the difference between individual measurement uncertainty and uncertainty in a derived quantity will make your analysis of experimental data much more reliable.