How many minutes should you promise a customer for delivery? Not the average, but the 90th percentile

The "Somsa Ekspress" kitchen wants to advertise home delivery. Behruz, head of the courier service, says: "The average is 34.5 minutes, so we'll write 'delivered in 35 minutes'." Let's test that promise on 20 orders.

Key rules

• A percentile is a cut-off value in an ordered series: values less than or equal to the p-th percentile make up about p percent. The 50th percentile is the median, and the 25th and 75th percentiles are the first and third quartiles
• The average shows neither the "typical" order nor the "late" one: a few very long deliveries pull the average up, but do not tell you what percentage of customers were left waiting
• For a promise the question is different: "what percentage of orders can we deliver within this time?" A percentile answers it directly: the 90th percentile (P90) is the time that 90 percent of orders do not exceed
• Nearest-rank method: arrange the values in ascending order, find the rank with the formula p/100 × n and, if it is a fraction, round up. The result is always an actual value from the series
• There is no single standard definition of a percentile. Excel's PERCENTILE.INC function interpolates between two neighbouring values, so the result may differ slightly. State in the report which method was used
• In a small sample, high percentiles are unreliable: with 20 orders, P95 (by the nearest-rank method) turns out to be the 19th value, i.e. the one just below the largest, and depends heavily on one or two random delays. Before announcing a promise, collect at least several weeks of data
• Calculate the percentile separately for hours and days when delivery conditions differ sharply

Example: 20 orders

One day's delivery times, in minutes, in ascending order (hypothetical, compiled by the author):

22, 25, 26, 27, 28, 28, 29, 30, 30, 31, 32, 33, 34, 35, 36, 38, 41, 47, 55, 63

• Average: total 690 / 20 = 34.5 minutes
• Median: the average of the 10th and 11th values, (31 + 32) / 2 = 31.5 minutes
• P90 (nearest-rank method): 0.9 × 20 = 18, i.e. the 18th value: 47 minutes
• P90 (PERCENTILE.INC): rank 0.9 × (20 − 1) + 1 = 18.1; 47 + 0.1 × (55 − 47) = 47.8 minutes

Now let's test different promises:

With the "35 minutes" promise, three in every ten customers experience a delay. Behruz has two options: put a time based on P90 in the advert (rounding 47 minutes to "up to 50 minutes"), or find the causes of the 41, 47, 55 and 63-minute orders and shorten the "tail" of the series. The second is more useful: even if the average barely changes, customers notice the difference when P90 falls.