What Is a Residual? How to Read Residuals in SPSS Regression
What Is a Residual
If you are running a linear regression in SPSS, a residual is the difference between the observed value of the dependent variable and the value predicted by the model. In simple terms, suppose the model predicts a person's Y score as 4.2, while the observed score is 4.8. The residual for that observation is 0.6.
The basic formula is:
Residual_i = Y_i - Ŷ_i
Here, Y_i is the observed value and Ŷ_i is the value predicted by the regression equation. A positive residual means that the model predicted a value below the observed value. A negative residual means that the model predicted a value above the observed value. A residual of 0 means that the prediction matches that observation exactly.
In SPSS, you may see related names such as RES_1, ZRESID, SRESID, SDRESID, or the Residual Statistics column. These are ways of storing or standardizing residuals, not new research variables. Before deciding that your model has a problem, identify which type of residual you are reading.
A residual is not a scale such as what is a scale, and it is not a variable used to test the effect of a mediator or moderator. It describes the model's prediction error after the independent variables have been entered into the equation.
Why Residuals Matter in Quantitative Research
Residuals help you answer a practical question: has the regression model missed a clear pattern of error? When residuals are distributed fairly randomly around 0, their variance does not change sharply, and no small group of observations looks unusually different, the regression assumptions have stronger support.
You normally use residuals to check three groups of issues.
The first is linearity. When you plot ZPRED against ZRESID, the points should be scattered fairly randomly around the 0 line. A curve, U-shape, or clear trend may indicate that the relationship between the independent and dependent variables is not linear.
The second is homoscedasticity, meaning that the error variance is relatively stable. If the points form a funnel, with small residuals at one end and increasingly large residuals at the other, the model may have heteroscedasticity. The regression coefficients can still be calculated, but you need to interpret the standard errors and statistical significance carefully.
The third is outliers and influential observations. A large absolute residual indicates that the model predicts that observation poorly. A large residual alone is not enough to conclude that you should delete the row. You also need to examine leverage, Cook's Distance, and the context of the response. This check is directly related to identifying outliers in your dataset.
Residuals also help you check whether the error distribution is approximately normal. In regression, the focus is on the residuals. Every individual observed variable does not have to follow a perfectly normal distribution. Combine a histogram, Normal P-P Plot, and scatterplot instead of relying on one number.
What Residual Value Is Acceptable
Raw residuals do not have one fixed threshold that applies to every study. If the dependent variable is measured on a 1 to 5 Likert scale, a residual of 1 means something different from a residual of 1 in a model predicting revenue measured in millions. Do not decide that a raw residual is acceptable or unacceptable from its absolute value alone.
In practice, researchers often examine standardized residuals or studentized residuals to put errors on a comparable scale. The table below summarizes commonly used reference points and the source for each criterion. These points are diagnostic guidelines. They do not replace examining the charts and the data context.
| Check | Reference point | Interpretation | Source |
|---|---|---|---|
| Durbin-Watson for residuals | From 1 to 3 | A range commonly used when considering first-order autocorrelation | (Field, 2013) |
| Standardized residual | Approximately -3 to +3 | An observation outside this range needs further checking as a possible outlier | (Field, 2013) |
| Regression sample-size check | n from 50 + 8m | m is the number of independent variables in one regression model | (Tabachnick and Fidell, 2013) |
| Harman single factor | Below 50% of variance | A reference point when checking common method bias, not a residual threshold | (Podsakoff et al., 2003) |
The standardized residual range from -3 to +3 is commonly used for initial screening. If a point falls outside this range, open its record and check whether the observed value was entered incorrectly, whether the respondent completed the questionnaire validly, whether the observation belongs to the target population, and whether the point changes the regression coefficient.
Durbin-Watson is related to autocorrelation among residuals, especially when the data have a time order or observation order. With cross-sectional survey data, you can still report this statistic if your department requires it, but do not treat it as the only evidence for every regression assumption.
How to Read Residuals in the Output
Creating Residuals in SPSS
Open Analyze > Regression > Linear. Move the dependent variable to Dependent and the independent variables to Independent(s). In the main window, select Statistics and tick Durbin-Watson if you need this statistic. Select Plots, move *ZPRED to the X-axis and *ZRESID to the Y-axis, then select Normal probability plot if you want to view the P-P Plot.
To save residuals, select Save in the regression dialog box. In the Residuals group, tick Standardized or Studentized, depending on what you need to check. SPSS will add another column in Data View. The column name can change according to how many times you have run the model, such as ZRE_1, SRE_1, or a similar name.
Reading the Residual Statistics Table
The Residual Statistics table usually contains the columns Minimum, Maximum, Mean, Std. Deviation, and N. For raw residuals, the Mean is usually very close to 0 when the model includes a constant. Check Minimum and Maximum to see the error range, but interpret them alongside the measurement unit of the dependent variable.
The example below is illustrative output, not the result of a real study:
| Residual Statistics | Minimum | Maximum | Mean | Std. Deviation | N |
|---|---|---|---|---|---|
| Predicted Value | 2.18 | 4.71 | 3.56 | 0.49 | 156 |
| Residual | -1.42 | 1.36 | 0.000 | 0.58 | 156 |
| Std. Predicted Value | -2.81 | 2.34 | 0.000 | 1.000 | 156 |
| Std. Residual | -2.91 | 2.76 | 0.000 | 0.997 | 156 |
In this illustrative table, the standardized residual ranges from -2.91 to 2.76. No observation exceeds the -3 to +3 range, but the final decision still requires the charts and influence statistics. The residual Mean being close to 0 is consistent with how residuals are calculated in a model with a constant.
Reading the ZPRED and ZRESID Plot
In the scatterplot, the X-axis is ZPRED and the Y-axis is ZRESID. Look for a cloud of points distributed fairly evenly around the horizontal 0 line. A funnel, curve, or separate cluster of points is a sign that needs further checking.
If the chart is not perfectly clean, you do not need to rerun the model immediately. Real survey data rarely produce a perfect cloud of points. Save the chart, record the unusual feature, and then check outliers, variable coding, missing data, and variables that may have been reverse-coded incorrectly.
Reading Studentized Residuals
A studentized residual is adjusted using an estimate of the standard deviation and is often useful when comparing observations in models where observations have different levels of influence. In SPSS, you can save a Studentized residual through the Save button. Do not combine studentized and standardized residuals as if they were one column. Use the correct statistic name in the results chapter.
What to Do When Residuals Are Not Acceptable
Check Data Errors First
Open Data View and find the row with a large standardized or studentized residual. Check the respondent code, the variables used to calculate the mean scores, and the reverse-coding rules. An unprocessed reverse-coded item can distort the model, but deleting a row simply because its residual is large removes the evidence of the underlying problem.
You should also check whether the value falls outside the valid range. With a 1 to 5 Likert scale, a value of 55 may be a data-entry error. When an extreme value is valid, record why you kept or removed the observation in your data-processing log.
Check the Observation's Influence
A large residual only tells you that the model predicts one row poorly. Also examine Cook's Distance, leverage, and how the coefficient changes when you run the model with and without that observation. If removing one row changes the conclusion completely, you need a clearer explanation for your supervisor or the committee.
Reconsider the Model Form
If the scatterplot has a curve, the linear model may not describe the relationship correctly. Review the theoretical basis, how you created the composite variables, and whether a moderator may be present. In moderation research, the interpretation differs from direct regression, so you can also consult the guidance on moderating and moderator.
If the variance clearly changes, discuss an appropriate way to report standard errors or consider another model specification. Do not delete points with large residuals simply to make the chart look better. The committee usually asks why data were removed, rather than looking only at the final output table.
Distinguishing Residuals, Standardized Residuals, and Studentized Residuals
A residual is the error in the original unit of the dependent variable. If Y is measured from 1 to 5, the residual is also expressed on the scale of Y. This makes it easy to understand in practical terms, but difficult to compare across models that use different units.
A standardized residual is a residual that has been standardized, usually with a mean close to 0 and a standard deviation close to 1. It helps you identify observations with relatively large errors. This is the column used when reading the ZPRED and ZRESID scatterplot.
A studentized residual is adjusted further according to residual variability and the influence of each observation. The displayed name can differ across references or SPSS versions. Check the exact label in the Save dialog and record the name of the SPSS variable that was created.
Residuals are also easy to confuse with predicted value. A predicted value is the value produced by the model. A residual is the distance between the observed value and the predicted value. The two columns belong together, but they answer different questions.
Finally, a residual is not the same as an outlier. An observation with a large residual may be an outlier in the Y direction, but concluding that it is an unusual and influential point requires several indicators. An observation with high leverage can sometimes have a residual that is not very large and still exert a strong effect on the equation.
Common Errors
The first error is looking for one raw-residual threshold that applies to every dataset. Use standardized or studentized residuals when you need a relative comparison, and state clearly which type of residual you are reading.
The second error is seeing one point beyond ±3 and deleting it immediately. Check data-entry errors, whether the response is valid, Cook's Distance, and the results before and after the decision. If you keep the point, you can state that the data were valid and that there was no methodological basis for removing the observation. You can also report a sensitivity analysis if keeping or removing it changes the result.
The third error is looking only at the Coefficients table and ignoring the residual charts. A coefficient with a small p-value does not prove that every regression assumption is satisfied. Read Model Summary, ANOVA, Coefficients, Residual Statistics, and the diagnostic charts as one body of evidence.
The fourth error is applying CB-SEM criteria to every model. CFI, TLI, RMSEA, and SRMR are fit indices commonly used in CB-SEM according to (Hu and Bentler, 1999). They are not thresholds for evaluating residuals in ordinary linear regression or PLS-SEM.
The final error is writing an overly strong conclusion. A sentence such as “the residuals are acceptable, so the model is completely accurate” is not appropriate. Residuals provide evidence about model error and assumptions. They do not prove causality or guarantee that the model has omitted no variables.
Frequently asked questions
What is a residual in SPSS regression?
A residual is the difference between the observed value of the dependent variable and the value predicted by the regression equation. The formula is Y - Ŷ. A residual is positive when the model predicts below the observed value and negative when the model predicts above the observed value.
What residual value is acceptable?
A raw residual has no universal threshold because its meaning depends on the measurement unit of the dependent variable. For standardized residuals, the range from -3 to +3 is often used to screen for unusual observations, but you should also check the chart, leverage, and Cook's Distance before taking action.
Does a standardized residual above 3 mean that I should delete the data?
You cannot make that decision from the threshold alone. Check whether the row contains a data-entry error, violates the sample criteria, or contains an invalid response. If the data are valid, you may keep the observation and explain the decision, while reporting a sensitivity analysis if keeping or removing it changes the result.
How are standardized and studentized residuals different?
Both are forms of residual adjusted to make comparison easier, but their adjustment and diagnostic meaning are not identical. Standardized residuals are commonly used in the ZPRED and ZRESID scatterplot. Studentized residuals include an adjustment related to residual variability and the influence of each observation.
Which SPSS table shows residuals?
Use the Residual Statistics table to read Minimum, Maximum, Mean, and Std. Deviation. Use the ZPRED and ZRESID scatterplot to check the shape of the distribution. If you need to check unusual observations, save standardized or studentized residuals through the Save button in Analyze > Regression > Linear.
Is a residual the same as an outlier?
A large residual can be related to an outlier, but the two concepts are not identical. Examine the residual together with leverage, Cook's Distance, and the data context. Do not delete an observation because a single indicator exceeds a reference point.
How to write this in your thesis
Open the .sav file again, run Analyze > Regression > Linear, save ZRESID and Studentized residuals, and compare each unusual point with the original data before writing your comments in Chapter 4. If you need to run the complete procedure on your actual data, use M4 data analysis.