Example: electricity use in drying batches
72 independent batches, with electricity use and three numerical explanatory variables.
Download the example data (CSV)This is the same example used in Help. Import the CSV to follow the steps, or use Open example in a practice copy in Help to prefill the inputs.
1. Create the workspace
In the sidebar, open Basic Statistics and choose Linear Regression. Create an analysis, name it, select New blank worksheet, then choose Create analysis.
2. Import the example data
Download the CSV above and import it into the worksheet. Choose Comma as the delimiter, Dot as the decimal separator and First row contains column names.
3. Assign the input columns
On Inputs, assign electricity_kwh to Response (Y). Open Predictors and select batch_mass_kg, inlet_moisture_pct and ambient_temperature_c in that order. Results label them X1, X2 and X3 respectively.
4. Set the model
Enter 0.95 in Confidence level (0–1). Check Include intercept and Show variance inflation factors (VIF). Leave Standardize predictors unchecked to retain the entered units.
5. Run analysis
Choose Run analysis, then open Results.
6. Interpret results
Identify the fitted response
The model uses 72 independent batches to describe electricity use from mass, inlet moisture and ambient temperature. R² is about 0.8938. This describes fit to these batches; the recorded relationships do not establish causes.
Read a coefficient in its units
Holding the other predictors fixed, an extra 10 kg is associated with about 3.42 kWh more use. Moisture adds about 2.65 kWh per percentage point and temperature about −0.753 kWh per °C. All three 95% coefficient intervals exclude zero. The all-zero intercept lies outside the study conditions.
Read the residual plot
A residual is measured electricity minus fitted electricity, in kWh. Observation 49 is about 11.09 kWh below its fit, the largest absolute residual. Check its records and inspect the whole cloud for curves or changing spread. Do not assume an error from that point alone.
Residuals vs fitted View full-size chart (opens in a new tab) Check the residual tails
The QQ plot compares ordered residuals with normal reference positions. Look for systematic curves and unusual tails. The residual Shapiro–Wilk test has p = 0.4018 and does not reject normality at 0.05; neither result proves the error model. This concerns residuals, not the raw electricity values.
Residual QQ plot View full-size chart (opens in a new tab) Check use on new batches
VIFs of about 1.003–1.045 show little inflation from linear dependence among predictors; they do not establish the right model form. Keep predictions within comparable conditions: observed mass 80.1–217.1 kg, moisture 6.05–13.57% and temperature 15.2–27.9°C. Staying within those separate ranges is not enough if a new combination differs from those studied. Validate predictions on new batches before planning from them.
More details: read each result section
Complete result tables use six significant digits, as in Help.
Linear regression fit
R² describes fitted variation; Adjusted R² accounts for predictor count, and Raw adjusted R² retains the value before the stated display convention. Model F and p-value test the predictors together. Model DF and Residual DF record available information. The remaining fit settings document confidence, intercept and scaling. These are study-fit results, not held-out predictive performance.
| Statistic | Value |
|---|---|
| R² | 0.893774 |
| Adjusted R² | 0.889088 |
| Raw adjusted R² | 0.889088 |
| Adjusted R² convention | Negative values reported as zero (validation dataset) |
| Model F | 190.715 |
| Model p-value | 4.9105e-33 |
| Model DF | 3 |
| Residual DF | 68 |
| AIC | 438.592 |
| BIC | 447.699 |
| Observations | 72 |
| Predictors retained | 3 |
| Parameters | 4 |
| Input rows | 72 |
| Used rows | 72 |
| Dropped rows | 0 |
| Fitted rows | 72 |
| Predictors | X1, X2, X3 |
- Model is statistically significant. Significant predictors: X1, X2, X3
Regression coefficients
X1 is batch mass, X2 inlet moisture and X3 ambient temperature. β is the estimated coefficient in response units per predictor unit. SE is coefficient uncertainty; t and p-value test a zero coefficient; CI low and CI high bound its 95% interval. Each slope holds the other predictors fixed. The intercept assumes all predictors are zero.
| Term | β | SE | t | p-value | CI low | CI high |
|---|---|---|---|---|---|---|
| Intercept | 28.8348 | 5.20051 | 5.5446 | 5.20972e-7 | 18.4573 | 39.2122 |
| X1 | 0.342474 | 0.0147858 | 23.1623 | 5.58816e-34 | 0.312969 | 0.371979 |
| X2 | 2.64786 | 0.278707 | 9.50051 | 4.24797e-14 | 2.09171 | 3.20401 |
| X3 | -0.752958 | 0.161902 | -4.65069 | 0.0000157443 | -1.07603 | -0.429887 |
Variance inflation factors
Predictor labels follow the selected order. VIF measures coefficient-variance inflation from linear dependence with other predictors. Values near 1 suggest little inflation; they do not validate linearity, independence or future predictions.
| Predictor | VIF |
|---|---|
| X1 | 1.04507 |
| X2 | 1.04345 |
| X3 | 1.00284 |
Regression ANOVA
Model is variation fitted by the predictors; Residual is unexplained variation. SS is sum of squares, DF degrees of freedom and MS is SS divided by DF. F compares the model and residual mean squares; p-value tests the predictors together. Dashes mark inapplicable entries.
| Source | SS | DF | MS | F | p-value |
|---|---|---|---|---|---|
| Model | 14032.3 | 3 | 4677.45 | 190.715 | 4.9105e-33 |
| Residual | 1667.76 | 68 | 24.5259 | — | — |
Residual summary
Residual is observed minus fitted electricity in kWh. Mean, SD, minimum, quartiles, median and maximum summarize the 72 study residuals. These summaries are not prediction intervals for a new batch.
| Statistic | Value |
|---|---|
| mean | -3.75009e-14 |
| sd | 4.8466 |
| min | -11.088 |
| q1 | -3.95264 |
| median | 0.706102 |
| q3 | 3.15057 |
| max | 10.0148 |
Residual normality
Test sw means Shapiro–Wilk on model residuals. Statistic and p-value assess a normal working error model. Read them with the QQ plot; p above 0.05 does not prove normal errors.
| Statistic | Value |
|---|---|
| Test | sw |
| Statistic | 0.982202 |
| p-value | 0.401846 |
Fitted values and residuals
Index locates an analyzed batch in source order. Fitted is model electricity use in kWh; Residual is measured minus fitted kWh. All 72 rows are included. Positive means use exceeded its fit and negative means it was lower. These fits use the study’s own measurements.
| Index | Fitted | Residual |
|---|---|---|
| 1 | 76.8375 | -3.81753 |
| 2 | 111.443 | -2.13264 |
| 3 | 75.7204 | 1.14955 |
| 4 | 73.4656 | 3.88444 |
| 5 | 100.348 | -2.22753 |
| 6 | 95.3359 | -7.58587 |
| 7 | 61.8309 | 8.55906 |
| 8 | 106.66 | 7.07966 |
| 9 | 74.3682 | 0.981848 |
| 10 | 99.8278 | 7.0722 |
| 11 | 97.5272 | 2.16277 |
| 12 | 104.194 | 0.365946 |
| 13 | 94.7032 | 1.26677 |
| 14 | 85.201 | -6.62098 |
| 15 | 79.8409 | -8.93085 |
| 16 | 80.5206 | 3.12941 |
| 17 | 84.9566 | -6.02665 |
| 18 | 93.7188 | 1.90123 |
| 19 | 90.0313 | -3.88125 |
| 20 | 94.0478 | -1.20783 |
| 21 | 114.851 | -6.16149 |
| 22 | 92.2695 | -4.23952 |
| 23 | 96.6762 | 6.52381 |
| 24 | 66.6088 | -2.5788 |
| 25 | 76.9681 | -3.57805 |
| 26 | 90.0441 | 1.4559 |
| 27 | 95.2115 | 2.02854 |
| 28 | 108.518 | 7.58219 |
| 29 | 88.2918 | 9.85815 |
| 30 | 77.943 | 1.38699 |
| 31 | 112.115 | -0.604855 |
| 32 | 80.9672 | -0.84725 |
| 33 | 93.5637 | 1.61629 |
| 34 | 75.9464 | 6.92363 |
| 35 | 81.2173 | 3.64274 |
| 36 | 67.877 | -6.48696 |
| 37 | 83.1836 | 1.18643 |
| 38 | 83.2826 | -4.13263 |
| 39 | 62.5775 | 1.63248 |
| 40 | 77.0167 | -1.19671 |
| 41 | 74.7452 | 10.0148 |
| 42 | 62.3518 | -6.91183 |
| 43 | 72.1913 | -6.30129 |
| 44 | 103.086 | 4.32423 |
| 45 | 77.018 | 2.35204 |
| 46 | 106.022 | -5.8115 |
| 47 | 101.478 | 2.57173 |
| 48 | 78.4989 | 1.67105 |
| 49 | 97.358 | -11.088 |
| 50 | 90.0492 | -0.529165 |
| 51 | 104.944 | -0.953625 |
| 52 | 91.5789 | 5.68115 |
| 53 | 71.2582 | 1.67181 |
| 54 | 84.7296 | 0.430356 |
| 55 | 103.353 | -3.93251 |
| 56 | 70.0678 | 3.20216 |
| 57 | 99.8221 | 4.06795 |
| 58 | 78.7868 | 4.27315 |
| 59 | 102.688 | 7.0117 |
| 60 | 96.268 | -7.12799 |
| 61 | 89.2849 | 4.64514 |
| 62 | 71.7717 | -1.67166 |
| 63 | 77.974 | 2.30597 |
| 64 | 85.5329 | -6.52289 |
| 65 | 72.9776 | -0.0575804 |
| 66 | 102.704 | -4.1637 |
| 67 | 74.6012 | -6.77125 |
| 68 | 106.379 | -4.28947 |
| 69 | 103.197 | 3.13337 |
| 70 | 116.943 | -4.01301 |
| 71 | 68.0824 | -1.86244 |
| 72 | 102.321 | 5.51867 |
7. Create a report
Keep Linear regression fit, Regression coefficients, Variance inflation factors, Regression ANOVA, both residual plots, residual summaries and normality test, and every Fitted values and residuals row. Record predictor order and units; describe associations and check future predictive performance separately.
Choose Add to report from the completed run. Open Choose individual items and advanced options to select tables and charts, then choose Add to draft. Hidden items stay in the saved analysis.
Name the report and add the purpose of the study and your interpretation. Choose Save, review Preview, then use Export → Download HTML. The example below contains the full selected content, including results beyond one screen.