Twenty-four observations with one point far out along x. Its Cook’s distance is 1.759588 — a single number saying the fit moves a lot. The DFBETAS say what Cook’s distance cannot: −1.67223 on the intercept and +1.84258 on the slope. The point pushes the intercept down by 1.67 standard errors and the slope up by 1.84, in opposite directions, and a summary measure over all the fitted values has no way to express that. If only the slope matters to the conclusion, that is the number to quote.
24 rows · 2 coefficients · 2 parameters
2 rows move a coefficient past 0.40825 standard errors
1 of them is below the 4/n Cook's distance cutoff of 0.16667. Cook's distance summarises the movement in every fitted value at once, so a row that swings one coefficient while leaving the overall fit nearly intact can pass it and still change the answer you care about.
Largest DFFITS
1.90031
cutoff 0.5774
Largest DFBETA
1.84258
cutoff 0.4082
Coefficient only
1
past DFBETAS, quiet on Cook's
Precision changed
2
covariance ratio outside 1 ± 0.250
Ranked by DFFITS
The twelve rows that move the fit most, with per-coefficient movement and the effect on precision
Row
DFFITS
Cook’s D
Largest DFBETA
On
Covariance ratio
Effect on precision
24
1.90031
1.75959
1.84258
predictor 1
3.12682
keeping it tight
16
-0.58091
0.14411
-0.45594
intercept
0.78115
little change
1
-0.46122
0.09398
-0.20352
predictor 1
0.82332
little change
9
0.44501
0.08924
0.21851
predictor 1
0.85956
little change
23
-0.33203
0.05247
-0.13315
predictor 1
0.95358
little change
18
0.29618
0.04270
0.19299
intercept
0.99960
little change
19
-0.28266
0.03877
-0.08414
predictor 1
0.98709
little change
12
0.27180
0.03798
0.26025
intercept
1.25492
keeping it tight
14
-0.26539
0.03441
-0.06626
predictor 1
0.99922
little change
3
0.23965
0.02836
0.09773
intercept
1.01871
little change
13
-0.22885
0.02601
-0.04957
predictor 1
1.03202
little change
4
0.15780
0.01278
0.10779
intercept
1.11413
little change
A covariance ratio above 1 means deleting that row would widen the standard errors — it is buying precision, usually by sitting far out along a predictor. Below 1 means it is inflating them. Cook’s distance is non-negative and cannot distinguish the two.
Movement per coefficient
DFBETAS for every coefficient, on the rows that move at least one of them
Row
intercept
predictor 1
24
-1.67223
1.84258
16
-0.45594
0.35370
1
0.09576
-0.20352
9
-0.11673
0.21851
23
0.05446
-0.13315
18
0.19299
-0.13197
19
0.01537
-0.08414
12
0.26025
-0.23308
Each entry is how many standard errors that coefficient moves when the row is deleted. Opposite signs across a row are common and informative: a point far out along a predictor typically pushes the slope one way and the intercept the other.
Every value here is computed from a closed form rather than by refitting the model 24times. The verification suite does perform those refits — for every row and every coefficient — and requires agreement to better than 1e-9.
All three measures delete one row at a time, so all three are defeated by the same thing: two similar influential points, each of which looks harmless to remove while the other remains. More measures do not fix that. Deleting small groups and refitting does.
Per coefficient Signed, unlike Cook’s D Effect on precision Still leave-one-out
What this tool shows
Two rows on the second preset have almost the same Cook’s distance — 0.18398 and 0.24561 — and opposite effects on precision. One has a covariance ratio of 3.58480: deleting it takes the slope’s standard error from 0.11751 to 0.21428. The other has 0.34384: deleting it takes the same standard error down to 0.08922. Cook’s distance is one non-negative number about the fitted values and has no way to say that one of these points is buying precision and the other is destroying it.
DFFITS for every row — how far its own fitted value moves when it is deleted, in standard errors
DFBETAS per coefficient, signed, so the direction and the target of the movement are both visible
The covariance ratio, which says whether a row is tightening the estimates or inflating them
Cook’s distance printed alongside, since the whole point is where the measures disagree
All of it from closed forms, checked against actual refits for every row and every coefficient
Why all three measures fail on the same thing, and what to do instead
Per coefficient Signed, unlike Cook’s D Effect on precision Cook’s D alongside
Updated 13 September 2026 · Works in any browser, no installation
These three measures answer the questions Cook’s distance compresses away: how far does this observation move its own prediction, how far does it move each individual coefficient, and does it tighten or loosen the estimates? All are in standard-error units, all are signed except the last, and all come from deleting one row. Cook’s distance is the summary; these are what it is a summary of.
At a glance
Formula shown
DFFITSᵢ = tᵢ·√(hᵢ/(1 − hᵢ)) where tᵢ is the externally studentised residual — the change in fitted value i, in units of its own standard error. DFBETASᵢⱼ is the change in coefficient j when row i is dropped, divided by that coefficient’s standard error from the reduced fit. COVRATIOᵢ = (s²₍ᵢ₎/s²)ᵖ/(1 − hᵢ): above 1 means deleting the row would widen the estimates, below 1 means it would tighten them. Conventional cutoffs are 2√(p/n), 2/√n and 3p/n respectively.
Scenario support
Deciding whether an influential row matters to the coefficient you actually care about, checking whether a result rests on a point that is also supplying most of its precision, auditing a regression before publication, and separating influence on the fit from influence on one estimate.
Educational estimate
Planning support from the values you enter — not professional advice.
Some influence is holding your standard errors down
Influence is usually discussed as a hazard. The covariance ratio makes it clear that half the time it is the opposite, and the second preset puts the two side by side.
Row 25 has a covariance ratio of 3.58480. It sits far out along x and exactly on the line. Deleting it takes the slope’s standard error from 0.11751 to 0.21428 — an 82% rise.
Row 9 has a covariance ratio of 0.34384. Ordinary predictors, large residual. Deleting it takes the same standard error down to 0.08922.
Their Cook’s distances are 0.18398 and 0.24561. Nearly the same, and on a non-negative scale that cannot distinguish the two situations at all.
So “this point is influential” is an incomplete sentence. One of these rows is the reason the slope has a usable confidence interval; the other is the reason it does not have a better one.
Which coefficient, and in which direction
Cook’s distance is a squared quantity summed over every fitted value. Two pieces of information are lost in that step and DFBETAS recovers both.
On the first preset, a row with Cook’s distance 1.759588 has DFBETAS of −1.67223 and +1.84258. Intercept down by 1.67 standard errors, slope up by 1.84.
Opposite directions, which a squared summary cannot express. A point far out along a predictor almost always does this: it pivots the line rather than shifting it.
And it names the target. If only the slope matters to the conclusion, the intercept’s movement is irrelevant and the relevant number is 1.84258.
On the fourth preset that difference changes the count. Five rows move a coefficient past 2/√n while sitting below Cook’s 4/n cutoff — diluted out of the summary by the coefficients they leave alone.
Everything is in standard errors, which is why it is comparable
The scaling on these measures is what makes them readable, and it is a deliberate choice rather than a convention.
A raw coefficient change is uninterpretable. Moving a slope by 0.3 is enormous if its standard error is 0.02 and invisible if it is 5.
DFBETAS divides by that standard error, so 1.84 means “the estimate moves by 1.84 of its own standard errors” regardless of units.
The divisor comes from the reduced fit, not the full one — the standard error is recomputed with the row already gone, which is what makes it a genuine comparison.
Same idea for DFFITS, which is the change in one fitted value in units of its own standard error, and which is related to Cook’s distance closely enough that the two rankings rarely disagree by much — the sign is the thing DFFITS adds.
The cutoffs are calibrated to return candidates
All three thresholds fire more readily than people expect, and reading them as verdicts generates a great deal of pointless investigation.
2/√n for DFBETAS is 0.36515 at thirty observations. On the clean fourth preset it flags seven rows out of thirty.
2√(p/n) for DFFITS is 0.63246 there, and flags two. Neither dataset has anything structurally wrong with it.
These are size-adjusted screening rules, designed to surface roughly a fixed share of any dataset for a human to look at — the same design as Cook’s 4/n, and with the same consequence.
Read the magnitudes. A DFBETA of 0.4 on a cutoff of 0.365 is a row worth a glance; one of 1.84 is a row the conclusion depends on, and the flag column says the same thing about both.
Three measures, one shared blind spot
Adding measures does not buy robustness when they all share an assumption, and every quantity on this page shares the same one.
All of them delete exactly one row. DFFITS, DFBETAS, the covariance ratio and Cook’s distance alike.
Two similar influential points defeat all four at once. On the third preset their Cook’s distances are 0.62290 and 0.25306, and their DFFITS and DFBETAS are correspondingly modest.
Remove one and the other’s Cook’s distance becomes 10.08012. The pair moves the slope from 1.76451 to 2.45769.
The fix is not more single-row statistics. It is deleting the top two or three rows together and refitting, which takes a moment and is the only thing that finds this.
Closed forms, checked against the refits they replace
Every quantity here has an exact expression in the residual, the leverage and the inverse of the cross-product matrix. None of them needs a refit, and that is where implementations go wrong.
The formulas are compact and easy to transcribe subtly wrong, in ways that look plausible on ordinary data and only fail at the extremes that matter.
So the suite does the refits. Every row deleted, the model refitted, and each coefficient’s actual movement compared against the closed-form DFBETA.
Agreement is required to better than 1e-9 across twelve generated datasets, every row within them and every coefficient within those — several thousand comparisons.
The deletion variance is checked the same way, which matters because it appears in all three measures and an error there would move every number on the page together, consistently, and invisibly.
Reporting influence properly
Four items, and the second is what separates a useful report from “observation 12 was influential”.
Name the coefficient. Influence on the intercept and influence on the slope of interest are different findings, and only DFBETAS distinguishes them.
Give the direction. A row that inflates the estimate and one that deflates it are both “influential” on a squared scale, and mean opposite things.
Say what happens to the standard error. A covariance ratio well above 1 means deleting the row costs precision, which is the direction usually left unmentioned.
And report the fit with the row, not without it, unless there is a reason about the observation itself to exclude it. Both fits, side by side, is the honest form.
Method. DFFITS, DFBETAS, the covariance ratio and the deletion variance are all computed from closed forms in the residual, the leverage and the inverse cross-product matrix — no refitting. The verification suite performs the refits anyway: every row is deleted, the model refitted, each coefficient’s actual movement measured and compared against the closed-form DFBETA, with agreement required to better than 1e-9 across twelve generated datasets, every row and every coefficient, which is several thousand comparisons. The least-squares core underneath is separately checked against this site’s other, independently written regression engine. The suite also asserts the structural properties that would catch a sign error: DFFITS always carries the sign of the externally studentised residual, the covariance ratio is always positive, and there is exactly one DFBETA per coefficient with the intercept included — each on 100 generated datasets. 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:
Cook's DistanceCook's distance for every row against both cutoffs in circulation, with the leverage and studentised residual it is built from and the F percentile it corresponds to.
LeverageHat values for every observation with the 2p/n and 3p/n cutoffs, the studentised residuals alongside, and the Σh = p identity printed as a check.
Multiple RegressionFits several predictors with a VIF on every term, and names the configuration people misread: a model significant at p = 0.0103 where neither predictor reaches 0.05, at a VIF of only 7.11.
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.
VIFVariance inflation factors with Belsley-Kuh-Welsch condition indices and variance-decomposition proportions, so a near-dependency is named rather than only detected.
Breusch-PaganThe Breusch-Pagan test in both forms — Koenker's studentised statistic and the original 1979 chi-square — with White's test alongside and a residual plot.
An educational tool. DFFITS, DFBETAS and the covariance ratio all delete one observation at a time, so a pair or small cluster of similar influential points will defeat every one of them at once. The conventional cutoffs are size-adjusted screening rules that return roughly a fixed share of any dataset — they identify rows to examine, not rows to remove.