Tutorial: Assembly Line Dimension Error Analysis

The Problem

An automotive parts assembly plant has three production lines — A, B, and C — with finished parts undergoing dimensional inspection. The shop floor reports that Line B may be producing parts with larger dimension errors, but environmental conditions like temperature and humidity also vary day to day. It is unclear whether the line itself or the environmental conditions are driving the differences.

Using 300 inspection records, we estimate the magnitude of line-to-line differences and analyze the contribution of environmental factors.

Load the Data

Click Assembly Line in the Sample Data section of the launcher screen. This loads a dataset with 300 rows and 7 columns.

Open this state in MIDAS

Column	Description
line	Assembly line (A / B / C)
shift	Shift (Day / Night)
operator	Operator ID (Op1 -- Op5)
temperature	Ambient temperature (°C)
humidity	Humidity (%)
cycle_time	Cycle time (seconds)
dimension_error	Deviation from target dimension (mm)

dimension_error is the response variable. Smaller values indicate parts closer to the target dimension.

Check Distributions

Click the dimension_error column in the Data Table tab. The Statistics tab displays a histogram and summary statistics. The overall mean is approximately 0.116 mm with a standard deviation of approximately 0.039 mm.

Select line from the Show stats by dropdown to view statistics broken down by line. Line B's mean is visibly higher than A or C.

Explore with Graphs

Compare Line Distributions

Select Analysis > Graph Builder... from the menu bar.

1. Select Histogram as the Graph Type
2. Set Column to dimension_error
3. Set Group By to line and change the display to Faceted

Line B's histogram is shifted to the right. Lines A and C have similar distributions.

Examine the Temperature Relationship

Create a new graph in Graph Builder.

1. Select Scatter Plot as the Graph Type
2. Set X-Axis to temperature
3. Set Y-Axis to dimension_error
4. Set Color to line

Higher temperatures tend to coincide with larger dimension errors. At the same time, Line B's points sit higher than the others at any given temperature.

Two patterns have emerged: Line B has larger errors, and temperature is positively associated with dimension error. We now quantify these patterns.

Estimate Line Differences (ANOVA)

Select Analysis > ANOVA... from the menu bar.

1. Set Dataset to Assembly Line
2. Set Analysis Type to One-Way
3. Set Factor A to line
4. Set Response Variable to dimension_error
5. Click Run Analysis

Read Group Statistics

The table shows each line's mean and 95% confidence interval.

Line B's mean is approximately 0.144 mm, compared to about 0.098 mm for Line A and 0.105 mm for Line C. The confidence intervals for Line B do not overlap with those of A or C, while A and C largely overlap.

Read the ANOVA Table

The ANOVA table decomposes the total variance in dimension_error into between-line variation and residual variation.

η² is approximately 0.24, meaning about 24% of the total variance in dimension_error is attributable to differences between lines. ω² (the degrees-of-freedom-adjusted estimate) is similar. The remaining 76% is within-line variation — environmental conditions, measurement variability, and other factors.

Read Tukey HSD

Tukey HSD provides simultaneous confidence intervals for the mean difference of every pair of lines.

Pair	Difference (mm)	Direction
A -- B	approx. −0.046	B is larger
A -- C	approx. −0.007	Nearly equal
B -- C	approx. +0.039	B is larger

The confidence intervals for A--B and B--C do not include zero, indicating that the estimated differences are stable within the precision of these data. The A--C interval straddles zero — the two lines have a small and imprecisely estimated difference.

Check Assumptions

The Assumption Diagnostics section displays pooled-residual and group-level Q-Q plots. Points close to the diagonal line support the normality assumption for these data.

Verify that the SDs in the Group Statistics table are similar across lines (approximately 0.035--0.038). This serves as a rough check for equal variances.

Analyze Environmental Contributions (Linear Regression)

ANOVA estimated the line-to-line differences, but even within a single line, environmental conditions may affect dimension error. We use linear regression to evaluate how much temperature, humidity, and cycle time contribute to dimension_error.

Select Analysis > Linear Regression (OLS)... from the menu bar.

1. Set Dataset to Assembly Line
2. Set Response Variable to dimension_error
3. Check temperature, humidity, and cycle_time as Predictor Variables
4. Click Run Analysis

Read Model Summary

R-squared is approximately 0.46. The three environmental variables alone explain about 46% of the variance in dimension_error. The remainder includes line-to-line differences and measurement variability.

Read the Coefficients Table

The table shows each variable's estimated effect size and precision.

Variable	Estimate	95% CI	Std. Coef.
temperature	≈ 0.012	excludes zero	≈ 0.52
humidity	≈ 0.001	near zero	≈ 0.09
cycle_time	≈ −0.003	excludes zero	≈ −0.40

temperature: approximately 0.012 mm/°C. A 1°C increase in temperature is associated with a 0.012 mm increase in dimension error on average. The confidence interval excludes zero, so the direction of the effect is stable.

cycle_time: approximately −0.003 mm/s. Longer cycle times are associated with smaller errors, suggesting that slowing down the assembly process improves precision.

humidity: the coefficient is near zero and the confidence interval straddles zero. The contribution of humidity is not clear from these data.

Std. Coef. (standardized coefficients) compare the effect of a one-standard-deviation change in each variable. Temperature has the largest standardized coefficient, indicating the largest contribution in these data.

Check VIF

Inspect the VIF column in the coefficients table. Large VIF values indicate strong correlations between predictors, which destabilize coefficient estimates. All three variables have low VIF in this dataset, so multicollinearity is not a concern.

Check Diagnostics

Enter a model name, click Save Model, and then click View Diagnostics to display four diagnostic plots.

Residuals vs Fitted: residuals scattered randomly around zero support the linearity assumption. A funnel pattern (residuals fanning out as fitted values increase) suggests heteroscedasticity.

Normal Q-Q: standardized residuals following the diagonal line support the normality assumption.

Scale-Location: $\sqrt{|r_i^*|}$ evenly scattered horizontally indicates that the variance does not depend on fitted values.

Residuals vs Leverage: no points beyond the Cook's Distance contours (0.5 and 1.0) indicates no single observation has an outsized influence on the model estimates.

Create a Report

To organize your analysis results, click the Add to Report button in Graph Builder. Graphs are added to the Report tab, where you can reorder items by dragging and add text blocks for commentary.

Save the project with File > Save to Browser. The report is included in the saved project. Export as an MDS file to share with others.

The Exploratory Nature of This Analysis

This tutorial is an exploratory analysis: we noticed Line B looked different in the graphs and then estimated the difference with ANOVA; we spotted a temperature trend in the scatter plot and then fitted a regression.

Confidence intervals from exploratory analyses can be optimistic about estimation precision. Because patterns were selected from the data itself, they may partly reflect chance variation.

When reporting these results in practice, state explicitly that the findings are exploratory. Include effect sizes and confidence intervals as a gauge of estimation precision, and let the decision on how to act depend on the cost of collecting new data and the stakes of the decision. To confirm that an effect is real, collecting new data is the standard approach.

Related Pages

• ANOVA — ANOVA tab details, two-way ANOVA, sum of squares types
• Linear Regression — Coefficients table, diagnostic plots, prediction intervals
• Creating Graphs — Graph types available in Graph Builder
• Report — Report tab operations

Also available as a Markdown file.