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Why mean fails · Binomial model · Stockouts · 95% reorder point · Index
Steady and spiky demand can average fourteen units a day while requiring different inventory protection over a three-day replenishment lead time. Every demand outcome and service threshold is shown.
Chapter index
Advanced worked companion: full demand distribution
Chapters 1–3
1. The same daily average can conceal different replenishment risks · 2. Define a single lead-time protection event · 3. Compute the two equal daily means
Chapters 4–6
4. Calculate daily variance and standard deviation · 5. Derive three-day demand for S · 6. Express V’s total through the number of spike days
Chapters 7–9
7. Calculate all four lead-time probabilities · 8. Verify the expected lead-time demand · 9. A 42-unit inventory position produces different stockouts
Chapters 10–12
10. Find the smallest point achieving at least 95% protection · 11. Distinguish mean lead-time demand and safety stock · 12. Stockout probability differs from expected shortage units
CHAPTER 1 OF 12
1. The same daily average can conceal different replenishment risks
Two fictional products each average demand of fourteen units per day. A purchasing dashboard might recommend the same stock position for each when replenishment takes three days. But a steady item that sells fourteen every day and a variable item that sells ten on most days and fifty occasionally need not have equal stockout risk.
Define Item S with exactly fourteen units of demand every day. Define Item V with daily demand ten with probability 0.9 and fifty with probability 0.1. We will assume the V draws are independent from day to day and that replenishment lead time is exactly three days.
All numbers are invented inventory mathematics, not store records, real consumer demand or procurement advice. The example demonstrates that mean demand does not determine a lead-time demand distribution, and an inventory threshold must be based on the service question rather than an isolated average.
Suppose the relevant inventory position just before a three-day lead time is R units and no additional stock arrives during that interval. Define a cycle as protected when total demand D over those three days satisfies D≤R. Equality counts as meeting demand without shortage.
This is a one-cycle mathematical stockout event, not a complete continuous-review inventory process with order quantities, backorders and repeated replenishment. Actual reorder policies have extra timing and inventory-position rules that must be defined.
The MIT Logistics and Supply Chain Management lecture on stochastic inventory identifies lead time, demand variability and service level as inputs to reorder-point decisions.
For S, every day’s demand is fourteen, so mean μ_S=14 and variance zero in the stipulated deterministic model. A three-day total is exactly 42 units without uncertainty.
For V, expected demand E[D_day]=0.9×10+0.1×50=9+5=14. The mean matches S exactly, even though a realisation can be much smaller or much larger.
This agreement is not a data mistake. The same average can be generated by steady outcomes or a mixture of frequent quiet demand and occasional spikes, and inventory must protect against the relevant cumulative demand.
V’s deviations from mean fourteen are −4 when demand is ten and +36 when demand is fifty. Its population daily variance is 0.9×(−4)²+0.1×36²=14.4+129.6=144 units².
The daily standard deviation is sqrt144=12 units. S’s standard deviation is zero under its exact daily-demand model. Mean equality therefore coexists with a large difference in variability.
Variance is helpful but is not itself a stockout probability. We will calculate the exact discrete distribution of three-day demand rather than substitute an unjustified bell-shaped approximation.
Over exactly three days, S requires 14+14+14=42 units. Thus P(D_S≤42)=1 and P(D_S>42)=0. A starting inventory position of 42 units exactly covers the deterministic lead-time requirement.
An actual replenishment process may have order-processing delays, spoilage, reservations or uncertain lead time; the toy model includes none of those sources of uncertainty. Its zero stockout risk is conditional on the stipulated assumptions.
This benchmark makes the variable item comparison clean. Both expected totals are 42, but only one is certain to use exactly that amount.
Let K be the count of fifty-unit days during the three-day lead time. Under the explicitly assumed independent daily model, K has a binomial distribution with n=3 and p=0.1.
Every day consumes at least ten units, so the three-day baseline is thirty. Each spike day adds forty units relative to that baseline. Therefore D_V=30+40K.
Consequently the only possible three-day totals are 30, 70, 110 and 150 units. The lead-time demand distribution can be computed exactly from the probabilities for K=0,1,2 and3.
With K=0, P=0.9³=0.729, giving D=30. With exactly one spike, P=3×0.1×0.9²=0.243, giving D=70.
With two spikes, P=3×0.1²×0.9=0.027, giving D=110. With three spikes, P=0.1³=0.001, giving D=150. These probabilities sum to exactly one.
None of the four possible totals equals the expected value 42. The expected total is a weighted average, not a guaranteed or even possible realised lead-time demand for Item V.
Multiply each possible V total by its probability: E[D_V]=0.729×30+0.243×70+0.027×110+0.001×150=42 units. This matches three times the daily expected demand fourteen.
Under independent days, lead-time variance is three times daily variance, or 3×144=432 units². Standard deviation is sqrt432≈20.785 units. The same mean has much wider lead-time outcomes.
This independent check guards against mistakenly treating the average daily demand as the only required input to a reorder decision.
For steady S, all three-day demand is exactly 42, so a position R=42 is sufficient with probability one. For V, only the 30-unit outcome can be covered at R=42; the totals 70,110 and150 all exceed it.
Therefore P(D_V≤42)=0.729 and P(D_V>42)=0.271, or 27.1 percent of the hypothetical independent three-day demand windows.
Both items have identical mean three-day demand 42. The unequal stockout probabilities follow from the distribution of possible totals and the explicit inventory threshold, not a change in the arithmetic mean.
The cumulative probabilities for V are P(D≤30)=0.729, P(D≤70)=0.972, P(D≤110)=0.999 and P(D≤150)=1. A stock position of 42 reaches only 72.9% protection, below a stipulated 95% cycle threshold.
The smallest integer R that reaches at least 95% protection is R=70, because the next attainable demand level after 30 is seventy. At R=70 the covered probability is 97.2%.
This is a mathematical service requirement chosen for illustration. A real service-level target must be connected to shortage consequences, cost, lead-time uncertainty and operational policy.
Both items have expected lead-time demand 42. Under the convention safety stock equals a selected reorder level minus expected lead-time demand, the steady item protected at R=42 has zero safety stock.
The variable item protected at R=70 has safety stock 70−42=28 units. That difference arises even though the two means are equal, because the chosen 95% cycle-protection event needs a higher quantile for V.
Safety stock is not the same as all inventory physically on a shelf; a full continuous-review policy also tracks pipeline units, outstanding orders and position definitions.
At R=42, V’s shortage amounts are 28 units if D=70, 68 if D=110 and 108 if D=150. Their expected unmet units are 0.243×28+0.027×68+0.001×108=8.748 units per defined lead-time experiment.
At R=70, shortages occur only for D=110 and150. Expected unmet units become 0.027×40+0.001×80=1.16. Probability of any shortage is 2.8%, but the expected number of short units is a different statistic.
Do not label an arbitrary ratio of expected shortage to mean demand as a universal inventory fill rate without specifying the replenishment policy and its demand cycle. The MIT supply-chain summary distinguishes cycle service from fill-rate measures.
Advanced worked companion: exact service levels and inventory risk
1. An exact frequency table that can be verified by hand
For three independent days, count the number of high-demand days K. K=0 has probability 0.9³=0.729, K=1 has probability 3×0.1×0.9²=0.243, K=2 has probability 3×0.1²×0.9=0.027, and K=3 has probability 0.1³=0.001. Together they sum to one.
The total demand D=30+40K can therefore be 30, 70, 110 or 150 with those probabilities. A reader can reproduce the entire distribution with a four-row table rather than relying on a fitted curve. In this discrete model the expected total 42 is not a possible realised value; it is a valid probability-weighted average of possible outcomes.
2. An independent check of mean and variance
Compute expected lead-time demand from the exact table: 0.729×30+0.243×70+0.027×110+0.001×150=42. The value also follows directly from adding three daily expected demands, 3×14=42. Agreement of these routes verifies the mean.
For one day of volatile demand, variance equals 0.9(10−14)²+0.1(50−14)²=144. Because the three daily draws are explicitly independent, total variance is 3×144=432 and standard deviation is about 20.785 units. Without independence, covariances would enter and the three-day variance could differ even with an unchanged daily marginal distribution.
3. Why the expected stock requirement can fail a service objective
For the steady item, three-day demand is always 42. Position R=42 exactly covers the requirement; P(D≤42)=1. For the volatile item, the only possible three-day total no greater than 42 is 30, which occurs with probability 0.729.
The probability of a stockout for the volatile item is therefore 1−0.729=0.271 or 27.1 percent under the stipulated one-window model. The two products share the same mean of fourteen units per day and expected total of forty-two, but their chances of crossing the inventory threshold differ because their cumulative distributions differ.
4. Derive the smallest service-level reorder position
Suppose the target is to cover at least 95 percent of three-day demand realisations, with a cycle counted successful when D≤R. For the variable item, cumulative probabilities at the possible total-demand values are 0.729 at 30, 0.972 at 70, 0.999 at 110 and one at 150.
Thus the smallest integer R satisfying P(D≤R)≥0.95 is 70 units. The nearest lower demand-support level 30 gives only 72.9 percent service. No reorder point between 31 and 69 produces an intermediate service probability because there is no possible demand total in that range in this exact model.
5. Safety stock is defined relative to the expected lead-time demand
Under the simple convention R=expected lead-time demand+safety stock, the steady item requires 42+0 and the volatile item under the 95-percent rule requires 42+28. Their safety-stock difference is twenty-eight units.
This calculation does not imply that a real warehouse must hold exactly seventy physical units whenever an order is placed. Continuous-review inventory position can include on-hand stock plus outstanding replenishment minus backorders, and the timing of orders depends on a chosen replenishment policy. Our R is a threshold for one clean three-day protection experiment.
6. Cycle-service probability versus expected units short
At R=70, stockout occurs for the two-spike outcome with probability 0.027 and the three-spike outcome with probability 0.001. Thus probability of any shortage is 0.028 or 2.8 percent, while the chance of covering demand is 97.2 percent.
Expected shortage is not 2.8 units or 2.8 percent by definition. It is 0.027×(110−70)+0.001×(150−70)=1.08+0.08=1.16 units per modelled three-day demand window. A binary stockout event and quantity short are different statistics, even when they use the same demand scenarios.
7. Compare an even smaller reorder point
If R=42, expected short units for the volatile item are 0.243×28+0.027×68+0.001×108=6.804+1.836+0.108=8.748. Its probability of any shortage is 27.1 percent. Both quantities improve when R rises to seventy, but not in the same units or by the same arithmetic operation.
A meaningful cost model might attach a fixed service penalty to any stockout event, an additional cost per missing unit, or a longer-term customer consequence. Different objectives can therefore prefer different protection levels. The example identifies demand risk; it does not itself specify the economically best inventory position.
8. Protecting 99 percent needs another step in this distribution
Increase the hypothetical target to at least 99 percent coverage. At R=70, coverage is 97.2 percent, not enough. At R=110, coverage is 99.9 percent because only the three-spike outcome at 150 remains uncovered.
Thus the smallest integer threshold meeting 99 percent service is 110. Relative to expected lead-time demand 42, the illustrative safety-stock amount becomes 68 units. The discrete jump from 70 to 110 occurs because the possible demand totals are separated by forty units; it is not a rounding anomaly in the service-level definition.
9. Independent days are an assumption, not a property of high-demand dates
Our binomial calculation assumes the daily high-demand events are independent and occur with constant probability 0.1. Actual sales can cluster around holidays, local events, promotions or temporary supply disruption. Demand correlation changes the probability of multiple spikes over three days.
Under a deliberately different common-shock model in which all three days are high together with probability 0.1 and all three low together otherwise, daily marginal mean remains fourteen. But total three-day demand is now thirty with probability 0.9 or 150 with probability 0.1. R=70 would protect only ninety percent of windows, not 97.2 percent.
This alternative demonstrates why a daily histogram alone cannot justify the independence assumption. A real service-level estimate should examine sequences, seasonality and known demand drivers rather than mechanically raise a one-day variance to a multi-day forecast.
10. Uncertain lead time creates another source of variability
So far replenishment lead time is exactly three days. If orders sometimes arrive after two days and sometimes after five, the distribution of demand during replenishment changes even if daily demand retains its original one-day law.
A lead-time demand model needs the joint distribution of daily demand and the number of days until replenishment, or a defensible simplification. Lead time and demand may also be dependent: congestion or emergency conditions can raise demand while delaying shipments.
The MIT stochastic inventory lecture identifies lead time and demand variability as inputs to reorder-point planning. It does not establish those inputs for our invented products.
11. Cycle service and fill rate are not identical
Cycle service is usually concerned with whether an entire replenishment cycle avoids a stockout under the policy’s definitions. A fill rate concerns the proportion of demand units supplied immediately from stock, often requiring information about order quantities and how shortages are handled across repeated cycles.
Our P(D≤R) calculation is an exact one-window coverage probability. The expected shortage of 1.16 at R=70 is an expected quantity under that same toy experiment. It would be misleading to call P(D≤R) the percentage of units filled without specifying a full inventory process.
The MIT supply-chain summary distinguishes cycle stock, safety stock and service measures. That distinction keeps a clean mathematical example from becoming an incorrect warehouse KPI.
12. Different reorder levels can have different operational costs
At the start of one lead-time experiment, R=42, 70 and 110 hold different amounts available to protect demand. The larger levels reduce modelled shortage risk but can tie up more working capital and storage space. Actual holding cost depends on time profiles and replenishment policy, not just the point-in-time difference R−42.
Suppose a purely illustrative cost of one unit is paid per extra unit placed above the 42-unit expected demand level at the moment of review. Then R70 implies 28 extra units of the metric and R110 implies 68. These are comparable counts, not a complete long-run inventory cost calculation.
A real decision should combine service consequences, ordering behaviour, spoilage, warehousing, supplier terms and forecast uncertainty rather than simply choose the highest mathematically possible protection threshold.
13. Quantiles should not be confused with the mean
For the volatile three-day demand, expected demand is 42 but the median and many lower percentiles equal 30. The 95th-percentile service threshold is seventy by our inclusive coverage rule, and an even higher threshold may be needed for more stringent coverage.
This is a discrete cumulative distribution. The reported quantile depends on the precise inverse-CDF convention; our inventory question defines the minimum R satisfying P(D≤R)≥target. That operational definition is clearer than relying on a spreadsheet’s unnamed interpolation option.
If a simulation or dashboard returns an apparently precise fractional reorder point between support values, check whether it represents interpolated descriptive quantiles or an actual decision that changes the stockout event. A change from R=42 to R=69 does not alter service probability in this exact demand model.
14. A reorder point should not be inferred from variance alone
It would be possible to approximate a safety stock by multiplying lead-time standard deviation by a normal z-score. But our demand distribution has only four possible totals and is notably skewed; a normal model would introduce approximation errors without any need for it.
Because the full discrete distribution is known, exact cumulative probabilities give a stronger result. If the true demand law were unknown, analysts might estimate it from data or use a model supported by business and statistical evidence, with uncertainty rather than a false exact answer.
The direct lesson mirrors other Gold Standard examples: an average and standard deviation can be valid summaries but still insufficient to establish the probability of an economically important threshold event.
15. Audit the data before calculating safety stock
Define demand units, calendar day, lead-time start and end, cancellations, backorders, unmet demand and promotional effects. A stockout can censor actual sales because customers unable to buy may not appear in recorded transaction counts.
A recorded sales average is not necessarily the average underlying customer demand. Replenishment delays and out-of-stock periods can distort both the mean and its apparent variability, especially when the most important high-demand days were poorly observed.
A credible report labels whether the demand inputs are theoretical, measured, estimated or forecast. Our two items are hypothetical, so their probabilities are known by definition; a live supply chain requires source data, validation and periodic revision.
16. Exercises for a changed spike probability
Let the high-demand day probability be p rather than 0.1, with low demand ten and high demand fifty. Daily mean becomes 10+40p. To preserve a daily mean of fourteen, p must equal 0.1. If p rises to 0.2, daily mean becomes eighteen and the three-day expected demand becomes fifty-four.
Under independent days with p=0.2, probability of no high day is 0.8³=0.512. A threshold R=42 covers only the no-high-day outcome, so stockout probability becomes 48.8 percent. This is a sensitivity scenario with a changed mean as well as changed distribution; state both changes.
The exercise shows how an uncertain spike probability affects service, but it does not imply that any product actually has a stable high-demand probability across seasons or market conditions.
17. Connected Logistics and Statistics reading
The exact conclusion is that both fictional items average fourteen units per day and have expected three-day demand forty-two. But R42 gives zero stockout risk for the deterministic item and 27.1 percent for the independent variable-demand item. To protect at least 95 percent of its modelled three-day windows, V needs R70.
For further work, follow The Gold Standard of Logistics, Supply Chain Management and MIT Logistics Systems readings. They provide inventory-planning foundations rather than actual demand data for these constructed items.
Return to the Gold Standard hub and How X Works. State the stockout event, lead-time and demand law before treating a reorder-point recommendation as evidence based.
| Spike days K | Three-day demand | Probability | Cumulative probability |
|---|---|---|---|
| 0 | 30 | 0.729 | 0.729 |
| 1 | 70 | 0.243 | 0.972 |
| 2 | 110 | 0.027 | 0.999 |
| 3 | 150 | 0.001 | 1.000 |
| Stock position R | Steady protection | Variable protection | Expected variable units short |
|---|---|---|---|
| 42 | 100% | 72.9% | 8.748 |
| 70 | 100% | 97.2% | 1.160 |
| 110 | 100% | 99.9% | 0.040 |
| 150 | 100% | 100% | 0 |