In statistical models, a residual is the difference between the observed value and the mean value that the model predicts for that observation. Residual values are especially useful in regression and ANOVA procedures because they indicate the extent to which a model accounts for the variation in the observed data.
The formula for a residual is:
Residual = Observed value – Predicted value
In regression analysis, the predicted (or fitted) value comes from the regression equation. Residuals help identify how well the model fits individual observations, and analyzing their patterns can reveal problems like nonlinearity, unequal variance (heteroscedasticity), or outliers.
For example, suppose a regression model predicts that a student’s test score will be 85 based on their study hours, but the student’s actual score is 90. The residual for that observation is:
Residual = 90 – 85 = 5
This residual indicates the model underestimated the student’s score by 5 points.
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