Choose your reading route
eduKateSG · Why Mathematics? · Bland–Altman Limits, Paired Differences and Practical Agreement
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 4–7
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 8–13
Check assumptions
See where design and calibration matter.Route 4 · Chapters 14–18
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 19–24
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why is mathematics important when two measurement methods produce similar numbers but may differ enough to change a decision? Bland–Altman analysis studies the paired differences directly. It estimates average bias, displays whether differences vary across the measurement range and gives limits of agreement for individual differences under stated assumptions.
This chapter uses eight original paired readings. The differences have mean 1.5 units and sample standard deviation sqrt(4/7), about 0.756. Approximate 95% limits are 0.018 to 2.982 using mean difference plus or minus 1.96 standard deviations. The calculation is simple; deciding whether those limits are acceptable requires units, context and a tolerance chosen independently of the data.
Full chapter index
Chapters 1–3 · Understand the question
Chapters 4–7 · Reproduce the example
Chapters 8–13 · Check assumptions
Chapters 14–18 · Explore applications
Chapters 19–24 · Study and verify
For each item, subtract method X from method Y. Our X readings are 100, 102, 104, 106, 108, 110, 112 and 114. The Y readings are 101, 103, 105, 108, 109, 112, 113 and 117.
The differences Y − X are 1, 1, 1, 2, 1, 2, 1 and 3. Positive values mean Y reads higher.
The sign convention must be declared. Reversing subtraction changes the sign of bias and both limits, while the width remains the same.
| Item | X | Y | Pair mean | Y − X |
|---|---|---|---|---|
| 1 | 100 | 101 | 100.5 | 1 |
| 2 | 102 | 103 | 102.5 | 1 |
| 3 | 104 | 105 | 104.5 | 1 |
| 4 | 106 | 108 | 107 | 2 |
| 5 | 108 | 109 | 108.5 | 1 |
| 6 | 110 | 112 | 111 | 2 |
| 7 | 112 | 113 | 112.5 | 1 |
| 8 | 114 | 117 | 115.5 | 3 |
A Bland–Altman plot places each difference on the vertical axis and the mean of the two readings on the horizontal axis. The first point is mean 100.5 with difference 1.
Plotting against pair means lets us inspect whether disagreement grows with magnitude, curves across the range or changes near a threshold.
Do not plot difference against only X without understanding the induced relationship. The standard mean axis treats both paired methods symmetrically for the graphical comparison.
The eight differences sum to 12, so their mean is 12/8 = 1.5 units. On average, method Y reads 1.5 units higher.
This is an estimate of constant bias over the sampled range. It does not say every Y reading is exactly 1.5 higher. Individual differences range from 1 to 3.
A confidence interval for mean bias addresses uncertainty in the average. It is different from limits intended to describe individual differences.
Subtract 1.5 from every difference. Five values of 1 contribute five squared deviations of 0.25. Two values of 2 contribute two more 0.25 terms. The value 3 contributes 2.25.
The total squared deviation is 4. Divide by n − 1 = 7 for sample variance 4/7. The sample standard deviation is sqrt(4/7), approximately 0.7559.
This spread describes observed variation in paired differences. With only eight pairs, its uncertainty is substantial.
Under a roughly stable, approximately normal difference distribution, conventional 95% limits are mean difference ± 1.96 times the standard deviation.
Here 1.96 × 0.7559 is about 1.4817. The lower limit is about 0.0183 and the upper limit about 2.9817 units.
These are estimated limits, not fixed population boundaries. Confidence intervals around each limit may be important, particularly with a small sample.
| Quantity | Value |
|---|---|
| Mean difference | 1.5 |
| Sample variance | 4/7 |
| Sample SD | 0.7559 |
| Lower limit | 0.0183 |
| Upper limit | 2.9817 |
Suppose the application can tolerate differences only between −1 and +1. Our estimated range would be too wide and shifted upward. If differences up to three units are harmless, the conclusion may differ.
The data do not choose the tolerance. Engineers, educators, scientists or decision owners must set it from consequences and measurement purpose.
Reporting “good agreement” without units and an acceptable difference is therefore incomplete.
Did you know that adding the same wide range to both methods can create a very high correlation even when every pair differs? Correlation is driven strongly by between-item variation.
Bland and Altman’s 1986 paper warned that correlation is misleading for method agreement. The difference plot asks a more direct question.
Correlation can still describe association, but it does not replace bias and limit analysis.
If differences increase as pair means increase, a single constant set of limits may be inadequate. The plot may show a funnel or trend.
A regression of differences on pair means can explore trend, but inference requires care because variables share measurements. A logarithmic transformation may be useful for multiplicative error.
Do not decide on transformation solely because it improves appearance. Connect it to the measurement scale and report results in interpretable units or ratios.
When variability grows with magnitude and relative error matters, analyse log-transformed measurements. Differences of logs correspond to log ratios.
After analysis, exponentiating bias and limits gives ratio-scale summaries. A ratio of 1 means equality.
Logs require positive values and change the interpretation. State the base or use natural logs consistently, and explain the back-transformation.
Our example has one pair per item. Repeated readings create within-item dependence and separate sources of variation.
Naively treating every replicate pair as independent can make limits too narrow. Methods for repeated-measures agreement partition within-item and between-item components.
Record operator, device, day and replicate identifiers so the design can be reconstructed.
Heteroscedasticity means difference variability changes across the range. A fan-shaped plot is a warning.
Possible responses include transformation, modelling variance as a function of magnitude or reporting range-specific limits. The choice depends on how the measurement is used.
One global standard deviation should not be forced onto data when the plot shows clearly different error bands.
The difference of 3 in our sample lies just above the approximate upper limit because rounding gives about 2.982. With eight pairs, one point also influences the estimated mean and standard deviation.
Do not delete it merely for crossing a line computed from the same data. Investigate recording, conditions and instrument behaviour.
Report a justified correction and sensitivity analysis if an error is confirmed. Genuine extremes may be exactly what a safety decision needs to know.
A confidence interval for mean bias quantifies uncertainty about the average difference. Limits of agreement aim to describe where individual differences may fall.
Confidence intervals can also be calculated around the estimated lower and upper limits. These address how precisely the limits are known.
Mixing these concepts can produce a report that appears precise while answering the wrong question.
The 1.96 rule relies on a model for differences. Inspect a histogram, quantile plot and the difference plot, while remembering that small samples give limited diagnostic power.
For skewed or heavy-tailed differences, transformation or robust/nonparametric limits may be considered. Their coverage and interpretation should be stated.
No method removes the need for an adequate, representative sample.
Collect pairs across the range where interchangeability matters. A method may agree well in the middle but poorly near low or high values.
Oversample critical thresholds if necessary, then describe the design. Check whether bias or spread changes there.
Limits from a narrow convenience sample should not be extended automatically to unobserved ranges.
If one method is always measured first, warming, fatigue or drift can be confused with method bias. Randomise or balance order when feasible.
Keep timing comparable and document environmental conditions. If the quantity changes between readings, the pair no longer measures the same state.
Good experimental design makes the mathematics meaningful.
Define paired units and acceptable differences before collection. Gather data across the operating range under comparable conditions.
Calculate differences and pair means, draw the plot, estimate bias and limits, and add uncertainty. Look for trend, unequal spread, outliers and clusters.
Conclude in units: whether the estimated differences are narrow enough for the intended decision. Avoid universal labels.
Add the eight differences to verify 12 and mean 1.5. Compute the seven small squared deviations and one large one to obtain total 4.
Divide by seven, take the square root and multiply by 1.96. Reproduce both limits.
Then reverse subtraction. Explain why bias and limits change sign but practical disagreement does not disappear.
- High correlation does not prove agreement.
- Limits of agreement are not confidence intervals for the mean.
- About 95% is a model-based expectation, not a guarantee for every future reading.
- A point outside a limit is not automatically a data error.
- Acceptability comes from context, not from 1.96.
- Repeated readings are not automatically independent.
The best safeguard is to show the full plot with units and the predeclared tolerance.
Using the eight pairs:
- list the pair means and differences;
- verify mean difference 1.5;
- show sample variance 4/7;
- calculate standard deviation about 0.7559;
- reproduce limits about 0.018 and 2.982; and
- identify the point just beyond the upper line.
Now add 10 to every X and Y reading. Explain why pair means shift but differences, bias and limits remain unchanged.
Parents can illustrate the method with two thermometers or two timing apps. The key question is not only whether they rise together, but how far apart their readings are.
Students should write units beside every difference and choose a tolerable error before looking at results. This prevents convenient conclusions after the fact.
The activity builds numeracy, graph reading, statistics and scientific reasoning without requiring advanced calculus.
Why use 1.96?
It is the conventional standard-normal multiplier for approximate central 95% coverage under the model. Small-sample and uncertainty adjustments may be needed.
Can the bias be zero but agreement poor?
Yes. Positive and negative differences can average to zero while individual differences are wide.
Are the methods interchangeable if all points are inside the limits?
Only if the limits themselves are acceptably narrow and the design and assumptions are suitable.
What if errors grow with magnitude?
Consider transformation or a model with magnitude-dependent variance rather than forcing constant limits.
The primary paper is Bland and Altman, Statistical Methods for Assessing Agreement Between Two Methods of Clinical Measurement, published 8 February 1986. The NIST Bland–Altman documentation documents paired plotting and analytic or bootstrap interval options.
Use the full chapter index above for the connected measurement-agreement sequence and continue to the Mathematics Learning Hub.
Verify pairing, subtraction direction, units, range, sample standard deviation, model assumptions, repeated-measure structure, uncertainty and predeclared tolerance. Show the plot rather than only the limits.
The importance of mathematics is practical: it converts “these methods seem close” into a visible distribution of differences that can be compared with a real decision threshold.
Extended practice laboratory
Practice case 1: Reverse subtraction
Setup: Use X minus Y. Task: Recalculate signs.
Check: Bias and limits reverse, while disagreement width is unchanged. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 2: Add constant
Setup: Add ten to both methods. Task: Track the plot.
Check: Pair means shift by ten; differences and limits remain unchanged. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 3: Multiply units
Setup: Convert both methods from metres to centimetres. Task: Transform results.
Check: Bias, standard deviation and limits all multiply by one hundred. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 4: Zero bias
Setup: Balance positive and negative differences. Task: Assess agreement.
Check: Mean bias can be zero while limits remain unacceptably wide. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 5: Constant bias
Setup: Every difference equals two. Task: Calculate limits.
Check: Standard deviation is zero and both limits equal two for the sample. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 6: One outlier
Setup: Change one difference to ten. Task: Audit influence.
Check: Mean and standard deviation widen; investigate provenance before removal. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 7: Funnel shape
Setup: Differences spread as magnitude grows. Task: Choose a response.
Check: Consider logs or a magnitude-dependent variance model. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 8: Ratio scale
Setup: Errors are proportional. Task: Transform carefully.
Check: Log differences can be back-transformed to ratios when readings are positive. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 9: Repeated readings
Setup: Three pairs belong to each item. Task: Avoid false precision.
Check: Use a repeated-measures method that accounts for within-item dependence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 10: Small sample
Setup: Only eight pairs are available. Task: Discuss limits.
Check: Estimated limits have substantial uncertainty and need confidence intervals. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 11: Narrow range
Setup: Study only middle values. Task: Restrict claims.
Check: Do not extend limits to low or high ranges not observed. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 12: Order effect
Setup: X is always measured first. Task: Improve design.
Check: Randomise or balance order when the quantity may drift. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 13: Time gap
Setup: The underlying value changes between readings. Task: Diagnose the problem.
Check: The pair no longer represents the same state, confounding method and time. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 14: Decision threshold
Setup: A three-unit error changes an action. Task: Use tolerance.
Check: Compare the upper and lower limits with consequences, not an adjective scale. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 15: Correlation trap
Setup: Both methods rise across a wide range. Task: Choose evidence.
Check: High r can coexist with systematic and wide differences. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 16: Confidence interval
Setup: A report shows only limits. Task: Ask for precision.
Check: Confidence intervals around bias and limits show estimation uncertainty. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 17: Non-normal tail
Setup: Several extreme differences appear. Task: Check the 1.96 model.
Check: Transformation or robust alternatives may be needed and must be explained. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 18: Clusters
Setup: Measurements come from several schools. Task: Plan resampling.
Check: Use the independent cluster when school-level dependence matters. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 19: Missing extremes
Setup: Large values fail on method Y. Task: Assess bias.
Check: Complete-case limits can understate disagreement at the high end. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 20: Rounding
Setup: Both instruments round to whole units. Task: Inspect ties.
Check: Discrete differences may not resemble a smooth normal distribution. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 21: Predeclared tolerance
Setup: Analysts see results before choosing limits. Task: Protect inference.
Check: Set acceptable difference from consequences before analysis. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 22: New device lot
Setup: A replacement instrument is tested. Task: Define generalisation.
Check: One device pair may not represent future lots or operators. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 23: Coverage claim
Setup: All sample points fall inside limits. Task: Correct the statement.
Check: Limits are estimated from the same sample and do not guarantee future coverage. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 24: Audit file
Setup: Recreate the plot. Task: Preserve essentials.
Check: Save pairs, units, subtraction direction, mean, standard deviation and interval settings. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.