What Are Residuals? How to Read Residuals in SPSS Regression

What are residuals

You run a linear regression in SPSS, open the Residuals Statistics table or the Normal P-P Plot of Regression Standardized Residual, and see columns for residuals, standardized residual, and studentized residual. Residuals are the differences between the observed values of the dependent variable and the values predicted by the regression model.

The basic formula is:

Residual = Observed value - Predicted value

If the observed value is denoted by Y and the predicted value by Ŷ, the residual for one observation is e = Y - Ŷ. 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.

Residuals are not a new independent variable that you can add to the model whenever you want. They are information used to check whether the regression model fits the data or violates assumptions such as an approximately normal distribution, constant variance, and the presence of unusual observations. You can also see related topics in the Statistics category if you are working through several output tables at once.

Why residuals matter in quantitative research

In linear regression, the model explains only part of the variation in the dependent variable. The remaining part appears in the error and is reflected in the residuals. Residuals therefore help you answer a practical question: are the model's prediction errors distributed reasonably, or do they follow an unusual pattern?

If residuals are scattered randomly around the 0 line and do not form a funnel, curve, or separate cluster, the model usually shows better agreement with the assumptions of linearity and constant variance. You still need to review the related tests and plots. Do not draw a conclusion from one residual point alone.

Residuals can also help identify observations that have a strong influence on the model. A row with a very large residual may be an outlier on the dependent variable, a data-entry error, or a valid case that differs from most of the sample. When you see an unusual residual, return to the respondent ID and the original data. The procedure is different from simply searching for outliers, because an outlier is not always allowed to be deleted.

In SPSS, residuals are commonly used when checking regression, while SmartPLS has its own model-assessment procedures and indices. Do not apply a PLS-SEM threshold to OLS regression output in SPSS. You also need to keep residuals separate from concepts such as a mediator or moderator. You can review the distinction in the articles on mediators and moderators.

What residual value is acceptable

Residuals do not have one absolute threshold comparable to Cronbach's Alpha. A residual of 12 may be large or small depending on the scale, measurement unit, and spread of the dependent variable. For that reason, you usually examine the standardized residual, plots, and diagnostic statistics together.

The table below lists reference points commonly used in regression practice. Read each point in context, rather than using it as an automatic reason to delete data.

The ±3 point in the table is a practical screening signal, not a rule that is sufficient for deleting an observation. The permitted sources do not provide one single standardized residual threshold. When you write your thesis, state which index you used, which table or plot you checked, and how you handled the observation.

For distribution checks, a nearly normal plot does not mean that every point must lie exactly on the diagonal. Real survey data usually show some degree of deviation. Review the overall pattern, sample size, and other diagnostics before expecting the data to be perfect.

How to read residuals in the output

You can create residuals in SPSS while running the regression. Open Analyze > Regression > Linear, move the dependent variable to Dependent and the independent variables to Independent(s). Then select Save, tick Unstandardized or Standardized in the Residuals group, select Continue, and click OK.

SPSS will add new variables at the end of the data table, usually with names such as RES_1, ZRE_1, or SRE_1, depending on your selection and version. Rename them or add clear notes so you know which variable is the raw residual and which is the standardized residual. This prevents confusion with the original survey variables.

To check the histogram, select Plots in the regression dialog box. Move *ZPRED to the X axis and *ZRESID to the Y axis, then select Histogram and Normal probability plot if needed. SPSS will produce a scatterplot of standardized predicted values and standardized residuals, together with a distribution plot.

Illustrative output

The following table simulates the form of results you may encounter. This is illustrative output, not the result of a specific study.

Predicted Value is the dependent-variable value predicted by the model. Residual is the difference in the original measurement unit. Std. Residual has been standardized so that observations are easier to compare. Studentized Deleted Residual is calculated by leaving that observation out when assessing the residual, so it may differ from Std. Residual.

In the illustrative output, Case Number 83 has Std. Residual = -2.58. This point needs further checking, but it is not enough reason to delete it. Reopen the data row and check whether the response was entered incorrectly, whether data are missing, or whether the case violates the screening criteria. If the respondent is valid, consider keeping the observation and reporting how you checked it.

In Residuals Statistics, review the Minimum, Maximum, Mean, and Std. Deviation columns for the residual. For a standardized residual, the mean is generally expected to be close to 0. If the Histogram and P-P Plot show a strong departure, review the whole model instead of focusing on one number.

What to do when residuals are not acceptable

Start with the data and the model, and only then consider transformation or deleting an observation. This order reduces the risk of being asked why you removed a data row simply because it made the table look better.

Check data-entry and coding errors

Open the row with the large residual in Data View and compare it with the original questionnaire and collection file. Check values outside the Likert range, reverse-coded variables that have not been recoded, missing-value codes, and incorrectly entered decimal points. If you find a clear error, correct the original data, save the version before and after the correction, and rerun the model.

Check the sample criteria

Check whether the respondent met the screening question and sample-selection criteria. A response from someone outside the target population may need to be removed under the criteria stated in the methods section, but the reason must be a research condition, not the fact that the residual is large.

Check the model form

A curved pattern or a cluster of residuals may indicate that the relationship between an independent variable and the dependent variable does not fit a straight line, that an important variable is missing, or that an interaction has not been considered. Consider adding a variable, transforming a variable, or using another model only when you have a theoretical basis. Do not add variables merely to reduce the residuals.

Rerun the model after a valid decision

If you remove an observation, record the row ID, reason, criterion, and results before and after the decision. You can report a sensitivity analysis if removing the observation substantially changes the coefficient or conclusion. A committee usually cares more about a process that can be checked than about a result table that looks completely clean.

If several residuals form a funnel, check constant variance and the way the variable was measured. You may consider an appropriate transformation, robust standard errors, or another method if the objective and assumptions allow it. SPSS, JASP, and R provide different diagnostic options. Changing software does not repair a data problem by itself.

Distinguishing residuals from standardized residuals

A raw residual keeps the unit of the dependent variable. If the dependent variable is measured as a mean score from 1 to 5, the raw residual is also expressed in those score units. This measure is easy to interpret when you want to know how many points the model differs from the observed value.

A standardized residual is a residual converted to a standardized scale, usually with a mean close to 0 and a standard deviation close to 1. It is useful when comparing the relative unusualness of observations. However, a standardized residual remains a diagnostic signal, not an automatic conclusion that an observation is an outlier.

A studentized residual uses a different estimation approach and is generally intended to reflect the influence of each observation on the error more effectively. In the output, the column name may change according to the procedure and the variable-saving option. Read the exact SPSS column name instead of calling every residual a standardized residual.

A residual plot is also easy to misunderstand. It is a graph showing residuals against predicted values or an explanatory variable. It is not one single index. When reading the ZPRED and ZRESID scatterplot, look for the pattern of dispersion, a curve, a funnel, or clusters. A graph with a few points far from the center is not enough to conclude that the model is wrong.

Residuals also differ from R². R² indicates the proportion of variation in the dependent variable explained by the model, while residuals show the error for each observation and the overall shape of the error. A model with a reasonably high R² can still have a variance problem or a few highly influential observations.

Common mistakes

The first mistake is writing that “the residual must equal 0.” In survey data, a regression model rarely predicts every observation exactly. Assess the distribution and pattern of the residuals instead of expecting every residual to disappear.

The second mistake is using raw residuals to compare dependent variables measured on different scales. Use standardized residuals when the aim is relative comparison, and state clearly which index you used.

The third mistake is immediately deleting rows with large residuals. This changes the data and can create biased conclusions. Before deleting an observation, you need a reason based on the sample criteria, a data-entry error, or sufficient diagnostic evidence.

The fourth mistake is looking only at the Coefficients table and ignoring the residual plots. Coefficients and p-values answer questions about relationships between variables, while residuals help you check whether the regression assumptions are reasonably satisfied.

The fifth mistake is reporting a threshold without a source. If you state a Durbin-Watson range from 1 to 3, cite (Field, 2013). If you use another standardized residual threshold, check the methods literature required by your school or supervisor instead of attaching an unsuitable citation.

How to write this in your thesis

A paragraph you can adapt in your thesis is: “The residual diagnostics showed that the standardized residuals were dispersed around 0, the Normal P-P Plot of Regression Standardized Residual showed no serious deviation, and the scatterplot of ZPRED and ZRESID did not form a clear pattern. Therefore, the residual assumption of the [model name] model was considered acceptable.” Replace the bracketed text with the exact model name and describe what appears in your own output.

Frequently asked questions

What are residuals in SPSS regression?

Residuals are the differences between the observed value of the dependent variable and the value predicted by the model. In SPSS, you can save raw residuals or standardized residuals through the Save button in the Analyze > Regression > Linear dialog box.

What residual value is acceptable?

A raw residual has no universal threshold because it depends on the measurement unit. Review the standardized residual, Histogram, Normal P-P Plot, ZPRED and ZRESID scatterplot, and other diagnostics together. An unusual-value reference point is useful for screening, but it does not automatically decide whether data should be deleted.

Is a standardized residual an outlier?

A large standardized residual is a sign that you need to investigate, but it does not mean that the observation must be removed. Check the data row, sample criteria, data-entry errors, and the observation's influence on the model. If you keep it, record how you checked it.

What is a residual plot?

A residual plot is a graph showing residuals against predicted values or a related variable. In SPSS regression, the scatterplot of ZPRED and ZRESID is used to see whether residuals are randomly dispersed or form a funnel, curve, or cluster.

What should you do when residuals are not acceptable?

Check the original data first, then review the coding, sample criteria, and model form. Remove an observation only when you have a clear methodological reason, preserve the data versions, and report the effect of the decision. Changing software does not replace checking the cause.

Open your .sav file now, rerun Analyze > Regression > Linear, save ZRESID and ZPRED, and compare each unusual point with the original data row. If you need to run the analysis on your own file and document the handling procedure, you can use M4 data analysis.