Residual Plot on TI-84 Calculator

Residual Plot on TI-84 Calculator, A residual plot is one of the most useful diagnostic tools for evaluating a regression model. It shows the differences between the observed values and the values predicted by the regression equation.

A residual is calculated as:

Residual = Observed value − Predicted value

A residual plot can help you determine whether a regression model is appropriate for your data. In particular, it can reveal patterns such as nonlinearity, unequal variance, and potential outliers.

What Is a Residual Plot?

Suppose a regression model predicts a value of 15 when the actual observed value is 18.

The residual is:

Residual = 18 − 15 = 3

If the model predicts 20 when the actual value is 18:

Residual = 18 − 20 = −2

A residual plot displays these residuals against the corresponding x-values or predicted values.

For a well-fitting linear regression model, the residuals should generally appear randomly scattered around zero.

A good residual plot should ideally show:

• Residuals distributed around zero
• No obvious pattern
• Approximately constant spread
• No strong curvature
• No unusually influential observations

A visible pattern in the residual plot may indicate that the regression model is not appropriate.

Why Use a Residual Plot?

A high R² value does not necessarily mean that a regression model is appropriate. A residual plot provides additional information about the quality of the model.

For example, if residuals form a curved pattern, the relationship may not be linear.

If the residuals form a funnel-shaped pattern, the variability may change as x increases. This is commonly associated with heteroscedasticity.

If one or more residuals are substantially larger than the others, those observations may deserve further investigation.

Step 1: Enter the Data

Start by entering your dataset into the TI-84 calculator.

1. Press STAT.
2. Select 1:Edit.
3. Enter the x-values into L1.
4. Enter the corresponding y-values into L2.

For example:

Make sure each x-value is paired with the correct y-value.

Step 2: Create a Scatterplot

Before fitting the regression model, it is useful to examine the original data.

Press:

2nd → Y=

This opens the STAT PLOT menu.

Select Plot1 and turn it On.

Choose the scatterplot option.

Set:

Xlist = L1

Ylist = L2

Then press:

ZOOM → 9:ZoomStat

The TI-84 will display the scatterplot.

This allows you to examine whether a linear relationship appears reasonable before creating the residual plot.

Step 3: Perform Linear Regression

Next, calculate the linear regression equation.

Press:

STAT → CALC

Select:

LinReg(ax+b)

Then press ENTER.

You can also enter:

LinReg(ax+b) L1,L2

and press ENTER.

The calculator will display the regression coefficients.

For example, you might obtain an equation similar to:

ŷ = 7.397 + 1.389x

Here:

• 7.397 is the intercept
• 1.389 is the slope
• ŷ represents the predicted value of y

The exact coefficients will depend on your dataset.

Step 4: Turn On Regression Diagnostics

For additional regression statistics such as r and R², turn on diagnostics on the TI-84.

Press:

2nd → 0 (CATALOG)

Find:

DiagnosticOn

Press ENTER, then press ENTER again.

Run the linear regression again.

The calculator can now display additional regression statistics when available.

Step 5: Create the Residual Plot

After running the regression, the TI-84 automatically stores the residuals in RESID.

To create the residual plot:

1. Press 2nd → Y=.
2. Select Plot1.
3. Turn the plot On.
4. Select the scatterplot.
5. Set Xlist to L1.
6. Set Ylist to RESID.

To enter the residual list:

Press:

2nd → STAT

Then select:

RESID

The setup should look approximately like:

Xlist: L1

Ylist: RESID

Step 6: Display the Residual Plot

Once the plot is configured, press:

ZOOM

Then select:

9:ZoomStat

The TI-84 will automatically choose an appropriate viewing window and display the residual plot.

The x-axis represents the x-values, while the y-axis represents the residuals.

A horizontal reference around y = 0 is especially important because residuals should ideally be distributed around zero.

How to Interpret the Residual Plot

The appearance of the residual plot provides important information about the regression model.

1. Random Scatter Around Zero

A desirable residual plot looks approximately like a random cloud of points centered around zero.

For example:

Residual   |  +|       •     •   |   •       •  0|-------------------------   |     •   •      •  -|  •       •   |   +------------------------- x

This suggests that a linear model may be appropriate.

However, the plot should be interpreted together with other diagnostics rather than using a rigid visual rule.

2. Curved Pattern

If the residuals form a noticeable curve, the relationship between x and y may not be linear.

For example:

Residual   |  +| •             •   |   •         •  0|------•---•--------------   |        •  -|          •   |   +------------------------- x

This suggests that a nonlinear model, such as polynomial, logarithmic, or exponential regression, may provide a better fit.

3. Funnel-Shaped Pattern

A funnel-shaped residual plot may indicate that the variability of the residuals changes as x increases.

For example:

Residual   |  +|             •   |          •     •  0|------•------------------   |    •       •       •  -|  •   |   +------------------------- x

This can indicate heteroscedasticity, meaning that the variance of the errors is not constant.

4. Large Individual Residuals

If one observation has a residual substantially larger than the others, investigate that observation.

It may represent:

• An unusual observation
• A data-entry error
• A genuine outlier
• A potentially influential observation

Do not automatically remove an observation simply because its residual is large. Investigate the underlying reason first.

How to View the Actual Residual Values

The TI-84 stores the calculated residuals in the RESID list.

To view them:

1. Press STAT.
2. Select 1:Edit.
3. Scroll across the lists.

Depending on the calculator state and setup, you can access the residual list through:

2nd → STAT → RESID

The residual values correspond to the observations in your dataset.

For example:

The residual is always:

Observed y − Predicted y

How to Store Predicted Values

The TI-84 can also store predicted values so you can examine the fitted values alongside the original observations.

When running the regression, you can specify a list for storing predicted values.

For example:

LinReg(ax+b) L1,L2,Y1

stores the regression equation in Y1.

You can then use the regression equation to calculate fitted values and compare them with the observed values.

Residual Plot vs Scatterplot

These two plots serve different purposes.

Scatterplot

A scatterplot displays:

Observed y vs x

It helps you understand the relationship between the original variables.

Residual plot

A residual plot displays:

Residuals vs x

It helps you determine whether the regression model adequately describes the relationship.

Therefore, you should generally examine both plots when evaluating a regression model.

What Does a Good Residual Plot Look Like?

For a linear regression model, you generally want to see residuals:

• Scattered randomly
• Centered around zero
• Without obvious curvature
• Without systematic trends
• With reasonably consistent spread

A residual plot that satisfies these conditions provides support for the use of a linear model.

It does not, however, prove that every assumption of regression has been satisfied.

Important Regression Assumptions

Residual plots are particularly useful for evaluating several aspects of regression.

Linearity

The relationship between the predictor and response should be appropriately represented by the model.

A curved residual pattern can indicate that the relationship is not adequately modeled by a straight line.

Constant Variance

The variability of residuals should remain reasonably stable across the range of predicted values.

A funnel-shaped pattern can indicate unequal variance.

Independence

Residual plots can sometimes reveal patterns associated with the order of observations, particularly when observations are collected over time. However, independence generally requires knowledge of how the data were collected and should not be assessed from a residual plot alone.

Normality

A residual plot can help identify unusual patterns, but a standard residual-vs-x plot is not a direct test of normality.

If normality is important for your analysis, consider using an appropriate graphical or statistical normality assessment, such as a histogram or Q-Q plot of the residuals.

Residuals and R²

R² measures the proportion of variation in the response variable explained by the regression model.

A high R² can indicate that the model explains a large amount of variation, but it does not guarantee that the model is correctly specified.

For example, a model could have a relatively high R² while its residuals still display a clear curved pattern.

Therefore, use:

R² + scatterplot + residual plot + subject-matter knowledge

rather than relying on R² alone.

Common Mistakes When Creating a Residual Plot on a TI-84

Mistake 1: Using L2 as the YList

For the residual plot, the YList should contain the residuals.

Use:

YList = RESID

not:

YList = L2

Mistake 2: Forgetting to run the regression first

The RESID values are generated from the regression model. Perform the regression before trying to plot the residuals.

Mistake 3: Using the wrong x-list

Use the same x-list used in the regression.

For example:

XList = L1

YList = RESID

Mistake 4: Interpreting every pattern as proof of a problem

Residual plots are diagnostic tools. A small amount of visual variation is normal. Look for meaningful and systematic patterns rather than expecting a perfectly random arrangement.

Mistake 5: Assuming residuals must be normally distributed

A residual-vs-x plot is primarily useful for identifying patterns such as nonlinearity and changing variance. Normality should be assessed separately when required.

Frequently Asked Questions

What is a residual?

A residual is the difference between an observed value and its predicted value:

Residual = Observed y − Predicted y

How do you make a residual plot on a TI-84?

First enter the x-values in L1 and y-values in L2. Run LinReg(ax+b). Then open STAT PLOT, turn on a plot, set XList = L1, and set YList = RESID. Finally, use ZOOM → ZoomStat.

Where are residuals stored on a TI-84?

After performing a regression, the TI-84 stores the calculated residuals in RESID.

What should a good residual plot look like?

For a linear model, the residuals should generally be randomly scattered around zero without an obvious systematic pattern or substantial change in spread.

What does a curved residual plot mean?

A curved pattern can indicate that a linear regression model does not adequately describe the relationship. A nonlinear regression model may be more appropriate.

What does a funnel-shaped residual plot mean?

A funnel-shaped pattern can indicate that the variance of the errors changes across the range of x-values, which is commonly referred to as heteroscedasticity.

Can a residual plot identify outliers?

Yes. An unusually large residual can identify an observation that deserves investigation. However, a large residual alone does not necessarily mean that the observation should be removed.

Online Residual Plot Calculator

If you do not have a TI-84 calculator, you can also use an online residual plot calculator to calculate regression residuals and visualize the model’s errors.

This can be useful when working with larger datasets or when you want to quickly compare different regression models.

Conclusion

A residual plot is an important diagnostic tool for evaluating a regression model. On a TI-84 calculator, you can create one by first fitting a regression model and then plotting the RESID values against the predictor variable.

The basic workflow is:

Enter data → Fit regression → Open STAT PLOT → XList = L1 → YList = RESID → ZoomStat

A residual plot with points that are reasonably randomly distributed around zero provides support for a linear model. Curvature, funnel-shaped patterns, or unusually large residuals may indicate that the model requires further investigation.

Residual plots should be used alongside R², scatterplots, subject-matter knowledge, and other appropriate statistical diagnostics rather than as a standalone test.

For more calculator-based statistics tutorials, explore the TI-84 Archives » FINNSTATS.