Quick Answer: Autocorrelation Result

The worked Autocorrelation example checks whether the ordered values of G3 final grade behave independently across row order or whether earlier observations help explain later observations. The visible output does not show a completely random sequence. The ordered plot contains several large negative shocks, the ACF plot shows notable positive spikes at lags 2, 3, 7 and 9, the Ljung-Box p-value plot drops below .05 from the early cumulative lags onward, and the Durbin-Watson context value is about 1.857.

The main interpretation is balanced. The Durbin-Watson value of 1.857 is close to the no-autocorrelation reference value of 2, so the first-order residual warning is not severe by itself. However, the ACF/PACF and Ljung-Box charts show that dependence is not only a one-lag question. The series has small but visible serial structure across selected later lags, especially around lag 2 and lag 9.

What Is Autocorrelation?

Autocorrelation is the correlation of a variable with lagged versions of itself. Instead of asking whether two different variables are related, autocorrelation asks whether a value at one position in an ordered series is related to values that came before it. If today’s value is related to yesterday’s value, or if row 50 is related to row 49, the series may contain autocorrelation.

Autocorrelation is common in time series data, repeated measurements, financial prices, weather records, educational progress data, sensor data and regression residuals. Positive autocorrelation means high values tend to be followed by high values and low values tend to be followed by low values. Negative autocorrelation means high values tend to be followed by low values and low values tend to be followed by high values.

The worked example is useful because it shows a realistic result. The lag 1 relationship is weak, but several later ACF and PACF bars are visible. That means the analyst should not stop at a single Durbin-Watson value. The complete interpretation needs the ordered plot, lag scatterplot, ACF, PACF and Ljung-Box p-values together.

Autocorrelation matters because many statistical methods assume independent errors or independent observations. If the independence assumption is violated, confidence intervals, p-values, model selection and forecasting decisions can become unreliable. Related diagnostics include the Durbin-Watson test, Ramsey RESET test, Goldfeld-Quandt test, Q-Q plot normality check and P-P plot normality check.

Autocorrelation Formula

The lag k autocorrelation compares each value in a series with the value k positions earlier. The sample autocorrelation formula can be written as:

Here, r_k is the autocorrelation at lag k, y_t is the current observation, y_t-k is the lagged observation and ȳ is the series mean. Values near +1 show strong positive autocorrelation. Values near -1 show strong negative autocorrelation. Values near 0 suggest weak or no linear autocorrelation at that lag.

Null and Alternative Hypotheses for Autocorrelation

The hypothesis depends on the method being used. For a lag-specific autocorrelation, the null states that the lag correlation is zero. For a Ljung-Box test, the null states that the series has no meaningful autocorrelation up to the tested lag.

Dataset and Ordered Series Used

The worked example uses the student performance dataset and focuses on G3 final grade as the ordered series. When a true date, semester, month, timestamp or measurement-order column is available, that column should define the time sequence. When no time column is available, row order can be used only as a documented observation sequence.

Before interpreting Autocorrelation, it is useful to understand the series using descriptive statistics, frequency distributions, histograms, box plots, and the five-number summary.

SPSS Output Interpretation for Autocorrelation

The SPSS output PDF provides the official SPSS verification for the Autocorrelation workflow. In SPSS, the analyst should first prepare the ordered series, compute lagged values when a lag scatterplot is needed, run ACF/PACF output, and then use Durbin-Watson when the question is about regression residual independence.

Open the SPSS Autocorrelation output PDF

SPSS Autocorrelation Output Summary

SPSS Reporting Decision

Python Chart-by-Chart Interpretation

The Python charts show the complete Autocorrelation diagnostic workflow. They move from the raw ordered series to lag-based visual evidence, then to ACF/PACF structure, Ljung-Box p-values and Durbin-Watson context. Each chart below is interpreted directly from the visible output.

Python Chart 1: Ordered Series Plot

The ordered series is centered around the zero line, and most observations stay within about two units of zero. This means the usual movement of the series is fairly compact. The chart is not dominated by one smooth trend from left to right, but it does show visible local runs and several strong downward shocks. Two of the deepest drops appear around the middle-left section of the sequence, near cases around 160 to 175, where the series falls close to -9. Later in the sequence, additional negative drops appear around the 520 range, the 585 range and the final 630 to 650 range.

The positive side is smaller than the negative side. One early positive spike rises close to +6, and several later positive movements reach roughly +2 to +3. This uneven pattern explains why the lag tests are needed: the series is mostly centered, but the repeated negative shocks and local clusters suggest that the ordered values may not be fully independent. The chart supports a cautious autocorrelation report rather than a simple “random sequence” statement.

Python Chart 2: Lag 1 Scatterplot

The lag 1 scatterplot places the previous value on the horizontal axis and the current value on the vertical axis. Most points form a dense central cloud around previous values from about -1 to +2 and current values from about -1 to +2. The fitted line tilts slightly upward, so the chart suggests weak positive first-order autocorrelation: larger previous values are associated with slightly larger current values, but the relationship is not steep.

The outlying pairs are important. Several points show current values far below zero even when the previous value is near zero, and one point on the far right shows a high previous value followed by a lower current value. These off-center points weaken the visual strength of the lag 1 relationship. Therefore, the chart should be reported as weak positive first-order serial dependence, not as a strong lag 1 pattern. This also explains why Durbin-Watson is close to 2 while the wider ACF/Ljung-Box checks still detect dependence at later lags.

Python Chart 3: Autocorrelation Function ACF

The ACF chart shows positive autocorrelation at several lags. Lag 1 is positive but remains close to the approximate 95% reference limit. Lag 2 is the first clear spike and rises to roughly .13, which is visibly above the dashed limit. Lag 3 is also positive at about .10, and lag 4 sits close to the upper limit. The later pattern is not flat: lag 7 is again positive, and lag 9 is another clear peak around .12 to .13.

This chart is the strongest visual evidence that the series has selected lag dependence. The dependence is not a smooth decay across every lag; instead, it appears at specific positive lags. After lag 9, the bars become smaller, and later lags mostly stay within the reference limits, with only a small negative value near lag 16. The correct chart-specific conclusion is that the G3 ordered series has notable positive autocorrelation at lags 2, 3, 7 and 9, while most other lags are weak.

Python Chart 4: Partial Autocorrelation PACF

The PACF chart clarifies which lag relationships remain after controlling for earlier lags. Lag 2 is the clearest positive partial autocorrelation and rises above the upper reference line. Lag 3 remains positive and is near or above the same limit. Lag 7 is also positive and close to the reference boundary, while lag 9 is a clear positive direct lag signal. Several middle lags, such as lags 4 and 5, remain positive but smaller.

The negative bars are limited and less dominant. Lag 6 and lag 10 fall below zero, and lag 16 is the largest negative bar, but it is not the main visual message of the chart. The PACF therefore supports the ACF conclusion: the series is not driven only by adjacent observations. Lag 2 and lag 9 remain important after shorter lag effects are accounted for, which is why the article should discuss selected-lag autocorrelation rather than only first-order autocorrelation.

Python Chart 5: Ljung-Box P-Values

The Ljung-Box chart begins with a p-value slightly above the .05 reference line at lag 1. After the early lag range, the p-values drop below the alpha line and remain very close to the bottom of the plot across the remaining cumulative lags. This means the first lag alone is not the strongest evidence, but the combined lag structure quickly becomes statistically meaningful.

This is the chart that turns the ACF pattern into a formal decision. Since the p-value line stays below .05 after the early cumulative lags, the no-autocorrelation null is not supported for the tested lag set. In a report, this chart should be written as evidence of joint serial correlation across lags, especially when interpreted together with the ACF spikes at lags 2, 3, 7 and 9.

Python Chart 6: Durbin-Watson Context

The Durbin-Watson chart displays a value of approximately 1.857. The no-autocorrelation reference is 2, and the chart also shows warning guidance around the lower and upper regions. Because 1.857 sits close to 2, the chart does not indicate severe first-order positive autocorrelation. It suggests only a mild positive first-order tendency.

This chart is important because it prevents overstatement. If the report looked only at Durbin-Watson, the conclusion would be that first-order autocorrelation is weak. If the report looked only at Ljung-Box and ACF, it would emphasize serial dependence across selected later lags. The correct combined interpretation is that the series has selected-lag autocorrelation even though the first-order Durbin-Watson warning is mild.

R Chart-by-Chart Validation

The R charts validate the same interpretation with a separate statistical workflow. R repeats the ordered series, lag scatterplot, ACF, PACF, Ljung-Box and Durbin-Watson context checks, which gives software-to-software confirmation. The color version also makes the important bars easier to separate from ordinary within-limit bars.

R Chart 1: Ordered Series Plot

The R ordered series chart confirms the same sequence shape as the Python chart. Most centered G3 values remain close to the zero line, but the line has several sharp downward movements. The deepest negative points again appear in the middle-left part of the sequence and near the later case ranges, while the positive movements are smaller and less frequent. This creates an asymmetric pattern where negative shocks stand out more strongly than positive shocks.

The colorful R version makes the same report decision clearer: the series is not a smooth trend, but it is not completely pattern-free either. The repeated dips and local runs justify checking lag dependence. This chart supports the explanation that autocorrelation should be interpreted from multiple diagnostics, not only from the ordered line plot.

R Chart 2: Lag 1 Scatterplot

The R lag 1 scatterplot again shows a dense cluster around the center and a slightly upward fitted line. Most observations are concentrated near zero on both axes, so the main body of the data does not show a strong first-order relationship. The positive slope is visible, but it is shallow.

The chart also contains several unusual lag pairs, including points with current values far below zero and points where a high previous value is followed by a much lower current value. These pairs explain why the lag 1 relationship remains weak. The R chart therefore validates the Python conclusion: lag 1 dependence exists only mildly, and the stronger evidence comes from the broader ACF/PACF and Ljung-Box results.

R Chart 3: Autocorrelation Function ACF

The R ACF plot uses color to highlight the important bars. The strongest positive spikes are visible at lag 2 and lag 9, and positive bars at lags 3, 4 and 7 also stand out. Lag 1 is positive but not the dominant feature. After lag 9, most bars are smaller, and the final negative bar near lag 16 remains modest compared with the positive peaks.

This R chart confirms that the autocorrelation pattern is not a Python-only result. The same lag structure appears again: selected positive lag relationships are present, especially at lag 2 and lag 9. The report should therefore say that autocorrelation is present across selected lags, not that every lag is correlated and not that the whole sequence is governed by a single first-order process.

R Chart 4: Partial Autocorrelation PACF

The R PACF plot shows that lag 2 remains the clearest direct positive lag. Lag 3 is also positive, lag 7 reaches the reference area, and lag 9 remains an important positive spike. The smaller green bars around lags 4, 5, 8, 11, 13 and 15 suggest that many later partial relationships are weak after earlier lags are controlled.

The negative side is not the main result. Lag 16 is negative and visible, but the strongest repeated signal is still positive at selected lags. The R PACF chart supports the same interpretation as Python: the ordered series has direct lag structure at a few specific positions, especially lag 2 and lag 9, so the diagnostic conclusion should mention selected-lag autocorrelation.

R Chart 5: Ljung-Box P-Values

The R Ljung-Box chart mirrors the Python p-value pattern. The first point is close to or just above the .05 line, but the p-values quickly fall under the alpha threshold and remain near zero through the later cumulative lags. This means the formal evidence becomes strong once more than the first lag is tested.

This chart is especially useful for the final decision paragraph. The ACF and PACF show where the lag relationships occur; the Ljung-Box plot shows that the combined lag evidence is statistically meaningful. The R output therefore validates the conclusion that the ordered G3 series should not be described as fully independent across the tested lag range.

R Chart 6: Durbin-Watson Context

The R Durbin-Watson chart again reports approximately 1.857. The bar sits close to the horizontal reference at 2 and between the broad warning guide lines. This confirms that the first-order autocorrelation warning is mild rather than severe.

The value is still slightly below 2, so the direction is toward positive autocorrelation, but it is not far enough away from 2 to be the main evidence by itself. The final report should combine this with the ACF/PACF and Ljung-Box charts: Durbin-Watson gives weak first-order context, while Ljung-Box and the lag plots show selected-lag serial dependence.

SPSS, R, Python and Excel Workflows for Autocorrelation

The same autocorrelation workflow can be reproduced in SPSS, R, Python and Excel. The software differs, but the logic is the same: define the order, create lagged values, inspect lag relationships, calculate ACF/PACF, run Ljung-Box or a similar portmanteau test, and use Durbin-Watson when regression residual independence is the question.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Autocorrelation

SPSS Syntax for Autocorrelation

* Autocorrelation test in SPSS using native SPSS commands.
* Put this syntax inside the Autocorrelation folder and keep dataset.csv in the same folder.

SET PRINTBACK=ON MPRINT=ON.
OUTPUT CLOSE ALL.
OUTPUT NEW NAME=Autocorrelation_Output.

GET DATA
  /TYPE=TXT
  /FILE="dataset.csv"
  /ENCODING="UTF8"
  /DELCASE=LINE
  /DELIMITERS="," 
  /ARRANGEMENT=DELIMITED
  /FIRSTCASE=2
  /VARIABLES=G3 F8.2.
EXECUTE.

COMPUTE case_order=$CASENUM.
COMPUTE G3_lag1=LAG(G3).
COMPUTE G3_centered=G3-MEAN(G3).
EXECUTE.

TITLE "Autocorrelation Test for Ordered G3 Series".
GRAPH /LINE(SIMPLE)=VALUE(G3_centered) BY case_order.
GRAPH /SCATTERPLOT(BIVAR)=G3_lag1 WITH G3.

ACF VARIABLES=G3
  /NOLOG
  /MXAUTO=16
  /SERROR=IND
  /PACF.

REGRESSION
  /DEPENDENT G3
  /METHOD=ENTER case_order
  /STATISTICS COEFF OUTS R ANOVA DW.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="Autocorrelation-Test-SPSS-Output.pdf".

Python Code for Autocorrelation

import pandas as pd
import numpy as np
from statsmodels.tsa.stattools import acf, pacf
from statsmodels.stats.diagnostic import acorr_ljungbox
from statsmodels.stats.stattools import durbin_watson

# Load data. Use a real time/order column if available.
df = pd.read_csv("dataset.csv")
series = pd.to_numeric(df["G3"], errors="coerce").dropna().reset_index(drop=True)
centered = series - series.mean()

# Lag 1 autocorrelation
lag1 = centered.autocorr(lag=1)
print("Lag 1 autocorrelation:", lag1)

# ACF and PACF values
acf_values = acf(centered, nlags=16, fft=False)
pacf_values = pacf(centered, nlags=16, method="yw")

# Ljung-Box test
lb = acorr_ljungbox(centered, lags=list(range(1, 17)), return_df=True)
print(lb)

# Durbin-Watson context
dw = durbin_watson(centered)
print("Durbin-Watson context:", dw)

R Code for Autocorrelation

# Autocorrelation analysis in R

df <- read.csv("dataset.csv")
series <- as.numeric(df$G3)
series <- na.omit(series)
centered <- series - mean(series)

# Lag 1 autocorrelation
lag1 <- cor(centered[-1], centered[-length(centered)])
print(lag1)

# ACF and PACF plots
acf(centered, lag.max = 16, main = "Autocorrelation Function for G3")
pacf(centered, lag.max = 16, main = "Partial Autocorrelation Function for G3")

# Ljung-Box test for selected lags
for (k in 1:16) {
  print(Box.test(centered, lag = k, type = "Ljung-Box"))
}

# Durbin-Watson context using centered series
dw <- sum(diff(centered)^2) / sum(centered^2)
print(dw)

Excel Formulas for Autocorrelation

Assume ordered G3 values are in B2:B650.

Lag 1 column in C3:
=B2

Lag 1 autocorrelation:
=CORREL(B3:B650,C3:C650)

Lag k autocorrelation pattern:
Create a lagged column for each k and use CORREL(current_range, lagged_range).

Durbin-Watson context for residuals in D2:D650:
=SUMXMY2(D3:D650,D2:D649)/SUMSQ(D2:D650)

Decision flag for Ljung-Box or p-value output:
=IF(p_value_cell<0.05,"Autocorrelation detected","No strong autocorrelation evidence")

APA Reporting Wording for Autocorrelation

When reporting an Autocorrelation diagnostic, describe the ordered variable, how the order was defined, the lag evidence, the ACF/PACF pattern, the Ljung-Box decision and the Durbin-Watson context when regression residual independence is relevant. Do not report only “autocorrelation was checked” without explaining which lags were important and whether the result changes the independence assumption.

APA-Style Full Report

Short APA-Style Version

Independence and Modeling Decision Wording

Common Mistakes in Autocorrelation Interpretation

When to Use Autocorrelation

Use an Autocorrelation test when observations have a natural order and independence matters. This includes time series, repeated measures, sensor logs, economic indicators, stock prices, weather records, monthly sales, learning progress and regression residuals sorted by time. They are especially useful when the goal is residual independence checking, serial dependence screening, forecasting preparation or transparent regression reporting.

Do not use autocorrelation as a general correlation test for unrelated rows. If the order is arbitrary, use Pearson correlation, Spearman correlation, regression diagnostics or group comparison methods instead. The key question is always whether earlier observations can predict later observations in the same series.

For related assumption and regression topics, see Ordinary Least Squares Regression, Regression Residual Analysis, Generalized Least Squares Regression, Levene’s Test, Brown-Forsythe Test, Kolmogorov-Smirnov Test, and Cramer-von Mises Test.

Downloads and Resources for Autocorrelation

The resources below verify the chart outputs and provide worked files for readers who want to reproduce the same Autocorrelation workflow in Python, R, SPSS and Excel. Use the PDF reports for visual interpretation and the Excel workbook for manual lag-correlation practice.

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