A first-order autoregressive series. The ACF decays geometrically — 0.667, 0.397, 0.128, 0.008 — and crosses zero by lag 4. The PACF is 0.667 and then nothing: −0.086, −0.185, 0.039. That pattern, a decaying ACF with a PACF that cuts off after lag 1, is the signature of an AR(1) and is what the two plots are read together for.
60 observations · 12 lags · mean 48.5300
Lag 1 is the first to clear its band
The lag-1 autocorrelation is 0.6670 against a white-noise band of ±0.2530. 2 of 12 lags clear their band. The pattern across lags is what identifies the structure, not any single bar.
Lag 1 autocorrelation
0.6670
band ±0.2530
Lag 1 partial
0.6670
identical to the above, by construction
Lags clearing the band
2 of 12
first at 1
Observations
60
enough to read
Autocorrelation by lag
The correlogram: correlation, partial correlation and significance band at each lag
Lag
ACF
PACF
Band
Bar
1
+0.6670 *
+0.6670
±0.2530
████████████████
2
+0.3971 *
-0.0860
±0.3478
██████████
3
+0.1284
-0.1849
±0.3757
███
4
+0.0075
+0.0391
±0.3785
█
5
-0.0716
-0.0463
±0.3785
−██
6
-0.1747
-0.1780
±0.3794
−████
7
-0.1821
+0.0458
±0.3845
−████
8
-0.2138
-0.0997
±0.3900
−█████
9
-0.2446
-0.1484
±0.3974
−██████
10
-0.2253
+0.0254
±0.4070
−█████
11
-0.2003
-0.0547
±0.4149
−█████
12
-0.0423
+0.1517
±0.4210
−█
The band is not a constant. At lag 1 it is 1.96/√n = 0.2530; Bartlett’s formula widens it at each subsequent lag as the earlier correlations accumulate, reaching ±0.4210 by lag 12. A plot drawn with one flat band overstates significance at every lag but the first.
Reading the two together
How ACF and PACF shapes identify a model
ACF
PACF
Suggests
decays geometrically
cuts off after lag p
an autoregression of order p
cuts off after lag q
decays
a moving average of order q
decays
decays
both components together
falls very slowly
cuts off after lag 1
non-stationarity — difference it first
a wave at period m
spikes at m
seasonality of period m
ACF and PACF together Bartlett bands, not a flat line Durbin-Levinson recursion Assumes stationarity
What this tool shows
An AR(1) series and a linear trend both have a partial autocorrelation that cuts off after lag 1 — 0.667 then −0.086, against 0.915 then −0.041. On that plot alone they are indistinguishable. Their autocorrelations are not: the AR(1) decays through 0.397 and 0.128 and crosses zero by lag 4, while the trend falls only from 0.915 to 0.579 across eight lags. Both presets are here, which is why the two plots belong side by side.
Autocorrelation at every lag, with the series mean and sample size stated
Partial autocorrelation by the Durbin–Levinson recursion, printed beside it
Bartlett significance bands that widen with each lag rather than one flat line
A text correlogram, so the shape is visible without a chart library
The ACF and PACF shapes that identify an AR, an MA, a seasonal cycle and non-stationarity
A white-noise preset showing what one near-miss bar looks like
ACF and PACF Bartlett bands Up to 24 lags Model shapes
A correlogram on a trending series describes the trend, not the structure.
Updated 13 September 2026 · Works in any browser, no installation
The autocorrelation at lag k is how strongly the series resembles itself k steps earlier. The partial autocorrelation at lag k is what is left of that once every shorter lag has been accounted for — so if today resembles yesterday and yesterday resembled the day before, the ACF at lag 2 will be large and the PACF at lag 2 near zero. Reading the two together is what identifies the structure, and neither on its own is enough.
At a glance
Formula shown
rₖ = Σₜ(xₜ − x̄)(xₜ₊ₖ − x̄) / Σₜ(xₜ − x̄)², with the full sum of squares in the denominator at every lag — the biased-but-stable form that keeps the correlogram positive-definite. The partial correlations come from the Durbin–Levinson recursion on those rₖ. The band at lag k is z·√((1 + 2Σⱼ₌₁ᵏ⁻¹ rⱼ²)/n), Bartlett’s formula, which equals z/√n at lag 1 and widens thereafter.
Scenario support
Identifying the order of an ARIMA model, detecting seasonality and its period, deciding whether a series needs differencing, checking whether model residuals still carry structure, and diagnosing why a forecast keeps missing in the same direction.
Educational estimate
Planning support from the values you enter — not professional advice.
Two series with the same PACF and nothing else in common
The partial autocorrelation is usually presented as the plot that identifies an autoregression. The first two presets show what it cannot do alone.
The AR(1) preset: PACF 0.667, then −0.086, −0.185, 0.039. A clean cut-off after lag 1.
The trend preset: PACF 0.915, then −0.041, 0.217, −0.014. Also a cut-off after lag 1. On this plot the two series look like the same model.
The ACF separates them completely. The AR(1) decays 0.667, 0.397, 0.128, 0.008 and is through zero by lag 4. The trend runs 0.915, 0.831, 0.789, 0.749, 0.701, 0.657, 0.612, 0.579 — still above 0.5 at lag 8.
An ACF that decays that slowly is the signature of non-stationarity, and the response is to difference the series rather than to fit an AR(1) to it. Fitting one would produce a model that forecasts the past.
The significance band is not a straight line
Most correlogram plots draw two horizontal dashed lines at ±1.96/√n. That is right at lag 1 and wrong at every lag after it.
The variance of rₖ depends on the correlations before it. Bartlett’s formula adds 2∑rⁱ² over the earlier lags into the band.
On the AR(1) preset the band starts at 0.2530 and reaches 0.3900 by lag 8 and 0.4210 by lag 12. A bar of 0.30 at lag 8 is inside the real band and outside a flat one.
So a flat-band plot overstates significance at every lag but the first, and does it worst on exactly the series where the early correlations are large.
The widening bands are printed per lag here for that reason. On white noise the two conventions agree, because there are no earlier correlations to accumulate.
What pure noise looks like
The white-noise preset is the most useful one on the page, because it calibrates what a “significant” bar is worth.
Sixty independent draws, twelve lags, and nothing clears the band. But lag 4 reaches −0.252 against a band of 0.2530 — a hair inside.
At twelve lags and 5%, roughly one bar crossing the line is the expected outcome. Not an anomaly: the expectation.
Which is why a single crossing at an unremarkable lag means very little, and a crossing at lag 12 on monthly data means a great deal.
For a yes-or-no answer across all the lags at once, use the Ljung-Box test, which pools them into one p-value instead of twelve separate looks.
Seasonality has a shape, not a spike
Seasonal structure is the thing a correlogram finds most readily, and it looks different from what people expect.
The seasonal preset’s ACF runs 0.747, 0.433, 0.058, −0.418, −0.702, −0.761. That is a wave, not a decay.
The trough sits half a period out — six months, for an annual cycle — and the correlation returns to a peak at lag 12.
So seasonality shows up across the whole correlogram, not as an isolated spike at the period. The negative lobe in the middle is often the clearest sign.
You need at least two full cycles of data to see it, and the lag count has to reach past the period — twelve lags on monthly data will show the descent and miss the return.
Difference first, then read
Everything a correlogram means assumes the series is stationary. On a trending series the plot describes the trend and nothing else.
A trend makes every pair of observations correlated simply because both are rising, which is why the trend preset stays above 0.5 out to lag 8.
The tell is an ACF that decays slowly and linearly rather than geometrically, with a PACF that cuts off after lag 1 and a lag-1 value close to 1.
The fix is to difference the series and redraw. On a differenced trend the correlogram usually collapses to noise, which is the confirmation that the trend was the whole structure.
Over-differencing leaves its own mark: a large negative autocorrelation at lag 1, which is also what a Durbin-Watson above 2.5 reports.
Why the denominator does not change with the lag
At lag k only n − k pairs exist, and there are two ways to divide by that. This calculator uses the one that looks wrong.
The full sum of squares stays in the denominator at every lag. So rₖ is computed from n − k products over an n-term denominator, which biases it toward zero.
That bias is deliberate. The alternative — dividing by n − k — is unbiased at each lag and can produce a set of correlations no real process could have.
The biased form keeps the autocovariance matrix positive-definite, which is what every estimation method downstream requires, and it has lower variance at long lags where the pair count is small.
At short lags relative to n the two barely differ. At long lags the biased form is smaller, which is another reason to distrust the far right of any correlogram.
Reporting a correlogram
Four items, and the second is the one that decides whether the first means anything.
Give n and the lag count. A correlogram on forty points and one on four hundred are different objects, and the band depends entirely on n.
Say whether the series was differenced or detrended first. An undifferenced correlogram on a trending series is a picture of the trend.
Say which band convention you used. Flat 1.96/√n and Bartlett’s widening band disagree at every lag past the first, and the flat one is more generous.
And pair it with a pooled test. Twelve bars is twelve looks; one Ljung-Box p-value is one.
Sources and methodology
References for the correlogram, its bands and the recursion used here.
Method. Autocorrelations use the full sum of squares in the denominator at every lag, which is the biased form; the alternative is unbiased at each lag and can produce a set of correlations no real process could generate, so it is not used here. Partial autocorrelations come from the Durbin–Levinson recursion on those correlations, and the suite asserts the identity that makes the recursion checkable — the lag-1 partial correlation must equal the lag-1 correlation exactly, on 150 generated series. Significance bands follow Bartlett’s formula and therefore widen with each lag rather than sitting flat at 1.96/√n; the flat convention is more generous everywhere past lag 1. The same routine backs the Ljung-Box test, so the two pages cannot report different correlations for the same series. The suite also asserts that every autocorrelation stays within ±1, that shifting and rescaling the series leaves the whole correlogram unchanged, and that an AR(1) leaves no partial correlation past lag 1 — the identification pattern this page is built around, checked rather than claimed. That engine is verified on every change against 157 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Ljung-Box TestThe Ljung-Box portmanteau test with Box-Pierce beside it, the degrees-of-freedom correction for fitted models and a lag-by-lag table.
Durbin-WatsonThe Durbin-Watson statistic with dL and dU computed by Imhof inversion rather than tabled, so the bounds exist at any n and k.
Moving AverageSimple, exponential, weighted and centred moving averages with the lag each one carries — exactly (n−1)/2 periods for a simple average, which is also the reason α = 2/(n+1) is in every EMA formula.
Correlation CoefficientReports Pearson, Spearman and Kendall together with the scatter plot, and ships Anscombe's quartet built in — four datasets with an identical r of 0.816 that Spearman tells apart.
ResidualResiduals, standardised and studentised residuals, leverage and Cook's distance, with Anscombe's quartet built in — including the dataset every residual rule scores cleanest and which one point controls entirely.
Runs TestThe Wald-Wolfowitz test with every run boundary drawn: ten 1s then ten 0s gives z = −3.9055 and perfect alternation +3.9055 — identical counts, mirror-image results, which no frequency test can see.
An educational tool. A correlogram assumes the series is stationary: on a trending series it describes the trend rather than any structure, and every bar will look significant. The significance bands are asymptotic and unreliable below about fifty observations, and reading twelve bars is twelve separate looks — one bar crossing the line is what pure noise produces.