Executive Summary
Key findings from the OLS price-volume regression.
The OLS regression of average retail price on total weekly sales volume analysed 500 weekly observations. The volume–price relationship is negative and highly significant (p < 0.001), with a slope of -1.89e-08 USD per unit. Total volume alone explains 4.2% of the week-to-week variation in average price (R² = 0.0424), with a root mean squared error of $0.3923.
Descriptive Statistics
Summary statistics for weekly average price and total sales volume.
| Statistic | Total Volume | Average Price |
|---|---|---|
| Mean | 1.06e+06 | 1.391 |
| Median | 1.092e+05 | 1.34 |
| Std Dev | 4.371e+06 | 0.4013 |
| Min | 796 | 0.51 |
| Max | 4.341e+07 | 2.9 |
| Observations | 500 |
Across 500 weekly observations, average retail price ranged from $0.51 to $2.90 with a mean of $1.39 (SD $0.40). Total weekly volume spanned from 796 to 43,409,836 units with a mean of 1,059,689 (SD 4,370,936), indicating substantial week-to-week variability in market supply.
OLS Regression Coefficients
Slope and intercept estimates with standard errors, t-statistics, and confidence intervals.
| Term | Estimate | Std Error | T Statistic | P Value | CI Lower | CI Upper |
|---|---|---|---|---|---|---|
| Intercept | 1.411 | 0.01809 | 77.98 | 0 | 1.375 | 1.446 |
| Total Volume | -1.89e-08 | 4.026e-09 | -4.695 | 0 | -2.681e-08 | -1.099e-08 |
The volume slope is -1.89e-08 USD per unit (highly significant (p < 0.001)), indicating that each additional unit of weekly volume is associated with a -1.89e-08 USD shift in average retail price. The model’s R² is 0.0424 (adjusted 0.0405), meaning total volume accounts for 4.2% of the week-to-week price variation. The model RMSE is $0.3923.
Price vs. Sales Volume (Raw Data)
Raw scatter of weekly average retail price against total weekly sales volume.
The scatter of 500 weekly observations shows a negative association between total volume and average price (Pearson r = -0.206). The cloud confirms the directional intuition from the OLS slope: weeks with higher sales volumes tend to have lower retail prices, consistent with the basic demand-supply mechanism.
Fitted Regression Line with Prediction Interval
OLS regression line with 95% prediction interval across the observed volume range.
The OLS line falls from a fitted price of $1.41 at low weekly volumes (~796 units) to $0.59 at high volumes (~43,409,836 units). The 95% prediction interval width is $1.55 at low volume and $1.68 at high volume — if the interval is appreciably wider at one extreme, this may indicate heteroscedasticity. Note: the OLS model is linear and may extrapolate to physically implausible values (e.g., negative price) at volumes outside the training range; interpret fitted values near the extremes of the volume axis as model extrapolation artefacts.
Residuals vs. Fitted Values
Scatter of OLS residuals against fitted values to check constant-variance assumption.
The residual-vs-fitted plot shows 500 observations with residuals ranging from $-0.87 to $1.49. A random scatter around zero indicates that the constant-variance (homoscedasticity) assumption holds; any funnel shape or systematic curvature would signal a violation. The residual standard deviation is $0.3927 (model RSE: $0.3931).
Residual Distribution — Normality Check
Histogram of standardised OLS residuals to verify the normality assumption.
95.4% of standardised residuals fall within ±2 standard deviations (expected ~95.4% under normality). The histogram is approximately bell-shaped and centred on zero, confirming that the normality assumption required for valid OLS confidence intervals and p-values is broadly met. Severe skew or heavy tails would suggest a log-price or log-volume transformation may improve inference.