Sixteen paired measurements with two entered wrongly — one far too high, one negative. Passing-Bablok gives a slope of 1.15313 with an interval of 1.0884 to 1.3038. Ordinary least squares on the same points gives 0.50893, less than half. Twenty-two of the 120 pairwise slopes fall below −1 here, and the shift that corrects for them is what keeps the fit symmetric.
16 pairs · 120 pairwise slopes · shift 22
y = 1.15313x + 4.8632
The slope's 95% interval is 1.0884 to 1.3038 and the intercept's is -2.0206 to 7.3749. Least squares on the same points gives 0.50893 — 55.9% away, which means a small number of points are pulling it. The slope interval excludes 1: a proportional difference between the methods.
Slope
1.15313
CI 1.088 to 1.304
Intercept
4.8632
CI -2.021 to 7.375
Least squares slope
0.50893
on the same points
Negative-slope shift
22
of 120 pairwise slopes
Of the 120 pairwise slopes, 22 fall below −1. The estimator shifts the median by that count, which is what makes the fit symmetric: swap the two methods and the slope inverts exactly. Without the shift it would not.
The slopes being taken a median of
How the pairwise slopes are distributed around the fitted slope
Pairwise slope
Count
Share
below −1
22
18.3%
−1 to 0
0
0.0%
0 to half the fit
3
2.5%
half to the fit
57
47.5%
the fit to double
31
25.8%
above double
7
5.8%
A tight cluster around the fit gives a narrow interval; slopes scattered across these bands give a wide one. The interval comes from the ranks of this distribution, so it reports the spread of the data rather than the spread a normal model would predict.
Both fits on the same points
Symmetric in X and Y No distribution assumed Rank-based interval Needs a linear relationship
What this tool shows
Sixteen paired measurements with two readings entered wrongly: Passing-Bablok returns a slope of 1.15313 and ordinary least squares returns 0.50893. Less than half, from two bad values in sixteen. The fit is the median of every pairwise slope, shifted so that swapping the two methods inverts the answer exactly — a symmetry ordinary regression does not have.
Passing-Bablok slope and intercept with rank-based confidence intervals
The negative-slope shift that makes the fit symmetric in X and Y, reported rather than hidden
The distribution of the pairwise slopes the median is taken from
Ordinary least squares on the same points, plotted as a second line
A preset where the two agree, showing what robustness costs on clean data
Slope and intercept intervals read together, for proportional and constant differences
Symmetric in X and Y No distribution assumed Rank-based intervals Slope distribution shown
It assumes the relationship is linear. It cannot check that for you.
Updated 13 September 2026 · Works in any browser, no installation
Passing-Bablok fits the line between two measurement methods without assuming anything about how their errors are distributed. It takes the slope between every pair of points and uses the median, which is why a few wrong values cannot move it. Its one addition to a plain median of slopes is a shift that counts the pairwise slopes below −1 — and that shift is what makes the answer the same whichever method you put on which axis.
At a glance
Formula shown
Compute Sᵢⱼ = (yⱼ − yᵢ)/(xⱼ − xᵢ) for every pair with distinct x, discard any equal to −1, and sort. Let K be the number below −1. The slope is the (N/2 + K)-th ranked value rather than the plain median, and the intercept is the median of yᵢ − b·xᵢ. The interval uses Sen’s rank formula: C = z·√(n(n−1)(2n+5)/18) with the endpoints at ranks (N − C)/2 + K and N − (N − C)/2 + 1 + K.
Scenario support
Comparing a new assay against an established one in clinical chemistry, validating an instrument where a handful of samples are known to be mishandled, any method comparison where normality has not been checked, and regulatory method-comparison work where a distribution-free procedure is expected.
Educational estimate
Planning support from the values you enter — not professional advice.
Two wrong values in sixteen
The first two presets differ only in two numbers, and that is enough to separate the two fitting methods completely.
With the errors present, Passing-Bablok gives 1.15313 and least squares gives 0.50893. Two readings in sixteen, and the least-squares slope has lost more than half its value.
With the errors corrected, the two agree closely. Robustness costs almost nothing when nothing is wrong.
Which is the argument for using it as the default. A method chosen after seeing the answer is not a method; a method that is safe on clean data and survives dirty data can simply be the one you always use.
The median of pairwise slopes has a breakdown point near 29%. Roughly three in ten points would have to be corrupted before it fails. Least squares fails at one.
The shift, and why it exists
Passing-Bablok is a plain median of pairwise slopes plus one correction, and the correction is the reason to prefer it over the plain version in a method comparison.
Count the pairwise slopes below −1 and shift the median by that count. On the first preset, 22 of the 120 slopes qualify.
Without it, swapping the two methods does not invert the slope. You would get one answer with method A on the x axis and a different, incompatible answer with method B there.
With it, the fit is exactly symmetric — which is the property a method comparison needs, because neither instrument is the truth and the choice of axis is arbitrary.
On clean positive data the shift is zero and the fit coincides with Theil-Sen exactly. The verification suite asserts that equality whenever the shift is zero.
Two intervals, two different findings
A method comparison asks two questions, and the slope alone answers only one of them.
A slope interval excluding 1 is a proportional difference: one method reads a fixed percentage higher, and the gap grows with the measurement.
An intercept interval excluding 0 is a constant offset: one method reads a fixed amount higher across the whole range.
They call for different corrections and can occur together. The proportional-difference preset here has both, which is the common case for an assay with a calibration problem.
The intervals come from the ranks of the pairwise slopes, so they widen when the slopes scatter and narrow when they cluster — the distribution table on the page shows which is happening.
Against Deming, and against least squares
Three fits, three sets of assumptions, and the data usually decides rather than preference.
Least squares assumes x is exact. In a method comparison it never is, and the result is a slope biased toward zero even with perfectly clean data.
Deming assumes normal errors with a known variance ratio. Given those, it is more efficient than this fit — narrower intervals from the same data.
Passing-Bablok assumes only that the relationship is linear. It pays for that with wider intervals, and it needs no λ at all.
The practical rule: use this one unless you have measured λ from duplicates and have checked that the scatter is well behaved. Both of those are more work than most method comparisons do.
The assumption it cannot drop
Passing-Bablok is distribution-free, which is often read as assumption-free. It is not.
It assumes the true relationship is a straight line. If the two methods diverge at high concentrations, no amount of robustness recovers that.
A curved relationship produces a slope that is an average of two different slopes, and a confidence interval that says nothing about the curvature.
So plot the points first. The scatter and the two fitted lines are on this page for that reason, and a visible bend is a reason to stop rather than to read the numbers.
The original paper pairs the fit with a cusum linearity test for the same reason. In its absence, the residual against the fitted line does the same job by eye.
Reporting the fit
Four items, and the third is the one a reader cannot reconstruct.
Give slope and intercept, each with its interval. Both, always — they describe different kinds of difference between the methods.
Give n and the range of measurements. A fit over a narrow range extrapolates badly, and method comparisons often cover only part of the working range.
Say that you checked linearity, and how. It is the one assumption this method still makes.
And say whether any points were excluded. The whole appeal of a robust fit is that nothing has to be, so an exclusion needs a reason outside the statistics.
Method. Every pairwise slope is computed explicitly rather than approximated, slopes of exactly −1 are discarded as the original procedure requires, and the count of slopes below −1 is used to shift the median — that shift is reported on the page rather than folded away, because it is what makes the fit symmetric in X and Y. The interval uses Sen’s rank formula on the sorted slopes with the same shift applied to both endpoints, and the intercept interval is derived from the slope endpoints rather than estimated separately. The distribution of the pairwise slopes is shown, since it is what decides whether the interval is narrow or wide. The suite asserts that on an exactly linear dataset the slope and intercept come back as exact equalities rather than approximations, that replacing one y value with 400 leaves both untouched, that the fit coincides with Theil-Sen whenever the shift is zero, and that swapping the two methods inverts the slope. That engine is verified on every change against 103 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Deming RegressionRegression when both variables carry error, with a settable error-variance ratio, jackknife intervals and least squares shown as the limiting case.
Theil-SenThe median-of-slopes estimator with a distribution-free interval, least squares beside it, and a leave-one-out table naming what each fit depends on.
Bland-AltmanLimits of agreement with confidence intervals on the limits themselves, a proportional-bias test and the correlation printed beside them for contrast.
Concordance CorrelationLin's concordance correlation with its exact decomposition into precision and accuracy, the scale and location shifts separated, and a bounded interval.
Linear RegressionThe least-squares line with r and r² — and the regression of x on y beside it, because those are two different lines rather than one line rearranged.
OutlierRuns four outlier rules together and says when each is failing — the 3-sigma rule cannot flag anything below n=11, two outliers hide each other, and the quartile convention alone can decide the answer.
An educational tool. Passing-Bablok is distribution-free but not assumption-free: it requires the true relationship between the two methods to be linear, and a curved relationship produces a slope that averages two different ones with an interval that says nothing about the curvature. It is also less efficient than Deming regression when Deming’s assumptions genuinely hold.