MAPE says how far off the forecast is. Bias says which way. A forecast with a large bias is not noisy, it is wrong in a direction you can correct.
Forecast bias is the average of the signed percentage errors, (forecast − actual) / actual; MAPE is the average of the same errors with the sign removed. Over eight illustrative quarters the bias is 1.87 per cent and the MAPE 2.15 per cent: almost the whole error is one-directional. Dividing each forecast by 1.0187 cuts the MAPE to 1.23 per cent, a 42.8 per cent improvement with no new information.
Worked in full in Financial Planning and Analysis by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A finance team logs the revenue forecast it submits at the start of each quarter and the actual that follows. Eight quarters, in millions, all illustrative. The error is measured against the actual, and a positive error means the forecast was too high.
| Quarter | Forecast | Actual | Forecast less actual | Error | Absolute error |
|---|---|---|---|---|---|
| Q1 | 100.0 | 97.5 | 2.5 | 2.56% | 2.56% |
| Q2 | 104.0 | 102.8 | 1.2 | 1.17% | 1.17% |
| Q3 | 98.0 | 99.1 | −1.1 | −1.11% | 1.11% |
| Q4 | 110.0 | 106.2 | 3.8 | 3.58% | 3.58% |
| Q5 | 106.0 | 103.0 | 3.0 | 2.91% | 2.91% |
| Q6 | 108.0 | 107.4 | 0.6 | 0.56% | 0.56% |
| Q7 | 112.0 | 108.3 | 3.7 | 3.42% | 3.42% |
| Q8 | 115.0 | 112.9 | 2.1 | 1.86% | 1.86% |
| Mean | 1.87% | 2.15% |
Error for each quarter = (forecast − actual) / actual
Bias = AVERAGE(error) = 1.87%
MAPE = AVERAGE(ABS(error)) = 2.15%
Noise = STDEV.S(error) = 1.60%
Is the bias real? t = bias / (noise / √n) = 1.868% / 0.567% = 3.30
Excel: =AVERAGE(Err)/(STDEV.S(Err)/SQRT(COUNT(Err)))
Seven of the eight quarters were over-forecast. The ratio of bias to noise is 1.17: the systematic part of the error is larger than the random part. The t-statistic of 3.30 is well above the 2.365 that a two-sided test at 95 per cent needs with seven degrees of freedom, so with eight observations this is not bad luck.
A second test, common in demand planning, gives the same verdict in money. The tracking signal is the cumulative error divided by the mean absolute deviation. Actuals totalled 837.2M against forecasts of 853.0M, a cumulative shortfall of 15.8M, against a mean absolute miss of 2.25M a quarter. The tracking signal is minus 7.02, far outside the plus or minus 4 that practitioners commonly use as a trigger.
A forecast that is merely noisy cannot be improved by arithmetic. One whose mean error is not zero can. Divide each historical forecast by 1 plus the bias, 1.0187, and recompute: the mean error falls to zero and the MAPE falls from 2.15 to 1.23 per cent. The 42.8 per cent improvement in accuracy came from one division.
Applied forward, a ninth-quarter forecast of 120.0M becomes 117.80M, a correction of 2.20M. Report the correction with its uncertainty: one standard error either side of the bias puts the corrected figure between 117.15M and 118.46M. That band, 1.31M wide, is the precision of the correction, not of the quarter. The quarter's own error, one standard deviation either side, spans 115.97M to 119.68M.
The correction assumes the cause of the bias persists. If the team knows why it over-forecasts, a pipeline weighted too generously or a sales target leaking into the forecast, fixing the cause beats adjusting the number.
The same MAPE can describe two very different forecasts. The table holds the size of each error and changes only its direction or its centre.
| Pattern | Bias | MAPE | Noise | t-statistic |
|---|---|---|---|---|
| As reported | 1.87% | 2.15% | 1.60% | 3.30 |
| Same sizes, signs alternating | 0.35% | 2.15% | 2.53% | 0.40 |
| Half the bias | 0.93% | 1.54% | 1.60% | 1.65 |
| Bias removed | 0.00% | 1.23% | 1.57% | 0.00 |
The first two rows have an identical MAPE. The first is a correctable forecast; the second is an honest one with nothing to correct. A pack that reports MAPE alone cannot tell them apart, and a target set on MAPE alone rewards neither the team that fixes its bias nor the one that never had any.
Three errors recur. The first is reporting accuracy without direction, as above. The second is letting positive and negative errors cancel and calling the result accuracy: a mean error of 0.35 per cent in the alternating case says nothing about how far off any quarter was. The third is concluding that a forecast is biased from too few points. With three or four quarters the standard error is wide enough that a t-statistic of 3.30 would be unusual; build the log before drawing the conclusion, and keep it running. Which direction of error costs more is a separate question, answered in which costs more, a forecast too low or too high.
The error log with room for twelve quarters, the de-biasing and the reforecast are live in the free workbooks for the book's company.
Accuracy, usually MAPE, measures the size of the errors regardless of direction. Bias measures their average direction. Two forecasts can share a MAPE of 2.15 per cent: one with errors that alternate in sign has a bias of 0.35 per cent, the other, missing the same way seven quarters out of eight, has a bias of 1.87 per cent and can be corrected.
Divide the mean error by its standard error, the standard deviation over the square root of the number of observations. Here 1.87 per cent over 0.57 gives a t-statistic of 3.30, above the 2.365 needed at 95 per cent with seven degrees of freedom. With fewer quarters or noisier errors the same bias would not pass.
The cumulative forecast error divided by the mean absolute deviation. Over these eight quarters actuals fell 15.8M short of forecasts against a mean absolute miss of 2.25M, a tracking signal of minus 7.02. A common rule of thumb flags anything outside plus or minus 4 as a biased forecast.
This article is one calculation from Financial Planning and Analysis. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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