Blog · Forecast and pipeline · Supply chain
The Poisson distribution describes demand counts under specific rate and independence assumptions. Compare supplied count-based forecasts with observed low-volume demand, check zero frequency and variation, and recognize when the assumptions are unsuitable.
The Poisson distribution for demand forecasting describes probabilities for whole-number demand counts during a defined period. Its main business assumption is that events occur independently at a stable rate within that period. Before relying on a supplied Poisson forecast, compare its implied zeros and variability with observed demand and check what the counted event actually means.
This is a review of count assumptions, not an inventory ordering rule. The demand forecasting overview explains the wider planning task; here the question is whether a supplied distribution is a reasonable description of a low-volume item's demand.
A Poisson distribution has one rate parameter, usually written λ, pronounced lambda. It is the expected number of events in the stated period. If the event is a request for one replacement part, the count is units requested. If the event is a customer order, the count is orders, which may contain several units each.
Probability of k events = e^(−λ) × λ^k / k!
k is a nonnegative whole number; λ is the expected count for the same exposure period.
NIST gives the Poisson probability formula and identifies λ as the average event count for the time interval. The distribution's mean is λ, and its standard deviation is √λ, so its variance is also λ. That relationship provides an important diagnostic: a supplied mean also implies a particular amount of variation.
The expected count can be fractional even though actual counts cannot. It is an average over possible future outcomes, not a promise that a partial unit will be requested. Forecasting: Principles and Practice explains why small counts require methods that respect nonnegative integer outcomes.
Bring a complete calendar of comparable demand periods, actual counts and the supplied forecast rate for each period. Include product, site, issue date, unit of measure and availability status. Keep zero-demand periods in the export; a transaction list alone omits them.
The exposure period must match the rate. A forecast stated per open business day cannot be compared directly with demand per calendar week. Store closures and partial trading periods need explicit treatment. A changing rate is possible in more elaborate count forecasts, but it must be supplied and documented rather than assumed constant for convenience.
Also separate genuine zeros from missing records. A closed site, an incomplete export and a fully stocked day with no customer request are different observations. If sales were blocked by unavailable stock, shipments do not establish unconstrained demand.
For demand with long gaps or irregular order sizes, the intermittent demand forecasting article covers the broader pattern. A low-volume label alone does not establish a Poisson process.
A US service-parts distributor receives a supplied forecast of λ = 1 unit per open day for one item at one site. The rate was issued before the following evaluation days. Each request in this example is for one unit, and the item remains available throughout.
The second site in the table is a variation with the same total demand concentrated in one day. It helps expose a limitation that a matching average can hide.
| Open day | Site A demand (units) | Site B demand (units) |
|---|---|---|
| 1 | 0 | 0 |
| 2 | 1 | 0 |
| 3 | 0 | 0 |
| 4 | 2 | 0 |
| 5 | 1 | 0 |
| 6 | 0 | 0 |
| 7 | 1 | 0 |
| 8 | 2 | 0 |
| 9 | 1 | 0 |
| 10 | 2 | 10 |
| Total | 10 | 10 |
| Sum of squared daily counts | 16 | 100 |
Both observed means are 10 / 10 = 1 unit per day. The supplied mean also implies total expected demand of 10 × 1 = 10 units. That reconciliation checks the total, but it cannot distinguish the two patterns.
Using the supplied λ = 1, the forecast probabilities and expected frequencies are:
| Daily demand | Probability calculation | Probability | Expected days out of 10 | Observed A days | Observed B days |
|---|---|---|---|---|---|
| 0 | e^(−1) | 36.8% | 3.68 | 3 | 9 |
| 1 | e^(−1) × 1 / 1 | 36.8% | 3.68 | 4 | 0 |
| 2 | e^(−1) × 1 / 2 | 18.4% | 1.84 | 3 | 0 |
| 3 or more | 1 − P(0) − P(1) − P(2) | 8.0% | 0.80 | 0 | 1 |
| Total | 100.0% | 10.00 | 10 | 10 |
Probabilities are rounded to one decimal place and expected days to two. Calculations use unrounded probabilities. Expected frequencies are averages across repeated evaluation windows, so they need not be whole numbers and need not exactly match this short sample.
Site A has 3 / 10 = 30% zero days, compared with an implied probability of 36.8%. Site B has 9 / 10 = 90% zero days and one large spike. Its matching average therefore offers little reassurance about the supplied count assumption.
Use sample variance to summarize how widely the observed daily counts spread around their mean:
Sample variance = [Sum of squared counts − (Sum of counts)² / n] / (n − 1)
For site A, the calculation is [16 − 10² / 10] / 9 = 0.67. For site B, it is [100 − 10² / 10] / 9 = 10.00. Variances are rounded to two decimals; the supplied Poisson variance is 1 because λ = 1.
Site B's variation is far above the supplied variance. Investigate whether the spike reflects batch ordering, a delayed request, a campaign or a recording error. Site A's lower observed variance does not prove a Poisson fit either. Ten days are an illustration, not enough evidence for a broad conclusion about long-run behavior.
Keep the forecast rate fixed when evaluating it. Re-estimating the rate from these same test rows and then presenting the result as independent validation would weaken the check. Extend the comparison to later, comparable periods and retain the original forecast issue dates.
The forecast bias review by SKU and site checks whether supplied mean forecasts are systematically high or low. That is distinct from checking whether a distribution describes variability. For interval evaluation, see probabilistic demand forecasting.
Demand planning owns the supplied distribution and exposure definitions. Site operations verifies that counts represent comparable open periods and that inventory availability did not constrain them. Procurement uses the uncertainty discussion alongside lead times and supplier constraints; finance checks the business consequences of shortages and excess commitments.
The main failure modes are:
Upload your demand history and supplied count forecast or Google Sheet, including the complete period calendar and availability fields. Covirage maps the columns; its tools compute and check comparisons, and the external AI model explains the results with every figure traced to its rows. Create a free Covirage account, which takes about a minute, and bring a sample of your own data.
See Covirage for supply chain teams for the supporting workflow.
It can be a useful starting assumption for nonnegative counts when events occur independently at a stable rate within the relevant period. Check the business process as well as the data. Batch orders, campaigns, stockouts and changing customer activity can make that assumption unsuitable.
Yes. The expected count is an average across possible outcomes and need not be a whole number. Individual realized counts must be whole numbers. Keep the expected count separate from a purchase quantity, which also depends on operational requirements and decision rules.
No. Low demand may come in clusters, follow customer schedules or include more zero periods than the supplied distribution anticipates. Compare observed variability and zero frequency with the assumption, then review out-of-sample performance rather than choosing a distribution from volume alone.