Calculating $R^2$ (the Coefficient of Determination) determines how much variation in cost is explained by a candidate cost driver. Identifying and handling outliers ensures these data points do not distort the regression model.
Step-by-Step Calculation of $R^2$
$R^2$ is defined as the proportion of total variance in cost ($Y$) that is explained by the regression model:
$$R^2 = \frac{\text{Explained Variation (SSR)}}{\text{Total Variation (SST)}} = 1 - \frac{\text{Unexplained Variation (SSE)}}{\text{Total Variation (SST)}}$$
Where:
- $\text{SST} = \sum (Y_i - \bar{Y})^2$ (Total Sum of Squares)
- $\text{SSE} = \sum (Y_i - \hat{Y}_i)^2$ (Sum of Squared Errors / Residuals)
- $\hat{Y}_i = a + bX_i$ (Predicted Cost from Regression)
Numerical Walkthrough
Using the previous dataset ($n = 6$, $\bar{Y} = \$3,333.33$, and model $\hat{Y} = 248.67 + 19.08X$):
| Month | Driver (Xi) | Actual Cost (Yi) | Predicted Cost (Y^i) | Residual (Yi−Y^i) | Squared Residual (Yi−Y^i)2 | Total Variance (Yi−Yˉ)2 |
| Jan | 100 | $2,200 | $2,156.67 | +$43.33 | 1,877.49 | 1,284,437.11 |
| Feb | 120 | $2,500 | $2,538.27 | -$38.27 | 1,464.59 | 694,439.11 |
| Mar | 150 | $3,100 | $3,110.67 | -$10.67 | 113.85 | 54,443.11 |
| Apr | 180 | $3,700 | $3,683.07 | +$16.93 | 286.63 | 134,443.11 |
| May | 200 | $4,000 | $4,064.67 | -$64.67 | 4,182.21 | 444,449.11 |
| Jun | 220 | $4,500 | $4,446.27 | +$53.73 | 2,886.91 | 1,361,115.11 |
| Sum | — | — | — | — | $\text{SSE} = 10,811.68$ | $\text{SST} = 3,973,326.54$ |
$$R^2 = 1 - \frac{\text{SSE}}{\text{SST}} = 1 - \frac{10,811.68}{3,973,326.54} = 1 - 0.00272 = \mathbf{0.99728 \quad (99.73\%)}$$
- Interpretation: $99.73\%$ of the variation in maintenance cost is explained by machine hours.
Handling Data Outliers in Cost Driver Validation
An outlier is a data point that deviates significantly from the overall trend, pulling the regression line away from true operating conditions.
Cost ($)
^
| * (Normal point)
| *
| * (Outlier: Equipment Breakdown)
| *
| *
+--------------------------------------------> Driver (X)
1. Identification Methods
- Standardized Residuals: Calculate $e_i = \frac{Y_i - \hat{Y}_i}{s_e}$. Any observation where $\vert{}e_i\vert{} > 2.5$ or $3.0$ standard deviations is classified as an outlier.
- Scatter Plots: Visual inspection of driver volume vs. activity cost readily reveals isolated data points.
2. Root-Cause Analysis Criteria
Before removing or adjusting an outlier, investigate the cause:
| Outlier Cause | Example | Action Required |
| Data Entry Error | Typo ($25,000 recorded instead of $2,500) | Correct the error using primary invoices/logs. |
| One-Off / Abnormal Event | Major machine breakdown, flood, supply shock | Remove the observation from the baseline model. |
| Capacity Structural Shift | Factory added a second shift or new line | Split dataset into pre-expansion and post-expansion models. |
| Non-Linear Cost Behavior | Step-fixed costs or tiered pricing | Keep point, but switch from simple linear to non-linear or multi-segment models. |
3. Impact Assessment
Compare the regression parameters before and after removing the outlier:
$$\text{If } R^2 \text{ increases significantly (e.g., from } 0.55 \to 0.92\text{) after removing a verified abnormal event, the cost driver is validated.}$$