Describing a distribution
Use individual turnaround times to check whether a business can support the deadline its customers care about.
Summarize a business dataset using location, spread, percentiles, and a suitable chart, explaining when the average misleads.
What can the sales team promise?
A distributor takes an average of three business days to send a custom quote. Its sales director wants to tell customers, “Nine times out of ten, you’ll have your quote within three business days.” The operations manager asks to see the individual turnaround times before she agrees. The director has a number for the average. His promise needs a different number: how many customers actually got their quotes by day three?
Both businesses and all records in this lesson are fictional. Use paper and a calculator. Recall Lesson 042: the print shop expected $640 from holding a press slot for a rush job. How many dollars could the shop actually earn from that choice on a single Friday?
Start with the individual quotes
The distributor reviewed all ten requests received during one week and followed every request until it was quoted. Each entry below represents one customer request. Turnaround is the number of business days from receipt to sending the quote, measured the same way for every request. Here are the results from shortest to longest:
1, 1, 2, 2, 2, 2, 3, 3, 4, 10 business days.
A distribution tells you which values occurred and how often. To describe its location, or center, start with the mean and median. The mean gives each observation equal weight: add the ten turnaround times and divide by ten. The sum is 30 business days, so the mean is 30 ÷ 10 = 3 business days per quote.
The median is the middle of the sorted list. With an odd count, use the single middle observation. With an even count, average the two middle observations. The fifth and sixth quotes both took two days, so the median is (2 + 2) ÷ 2 = 2 days. Six customers got their quotes within two days. The ten-day wait pulls the mean upward because every extra day adds to the sum. It does not move the two middle observations. NIST explains this distinction when comparing measures of location for data with extreme values. NIST: Measures of Location.
Neither number is wrong. The mean uses every day customers waited and lets you recover the total: 10 × 3 = 30 customer-days of waiting. Those are elapsed waits, not 30 days of staff labor. The median tells you about the customer in the middle. Neither tells you directly how long the slower quotes took.
See where the waits cluster
Spread describes how far apart the observations are. The simplest measure is the range: largest minus smallest. Here it is 10 − 1 = 9 business days. That captures the distance between the extremes but does not tell you whether one quote took ten days or half of them did.
Group the whole-day measurements into equal-width intervals. This frequency chart uses one square per quote. Read down the intervals to see the shape, including the empty stretch before the longest wait.
| Turnaround in business days | Quotes | One square per quote |
|---|---|---|
| 0–1 | 2 | ■■ |
| 2–3 | 6 | ■■■■■■ |
| 4–5 | 1 | ■ |
| 6–7 | 0 | |
| 8–9 | 0 | |
| 10–11 | 1 | ■ |
The counts add back to ten. Eight quotes cluster in the first two intervals. A single slow quote stretches the distribution to the right, toward larger values. Check its timestamps and what happened on that request. If the ten days are real, keep them: that customer experienced a delay even if the other nine did not.
Match the measure to the promise
A percentile gives a cutoff for a stated share of observations. For service deadlines, use the smallest observed time by which at least that share finished. To find the 90th percentile, or P90, sort the times, multiply the count by 0.90, round the position up to a whole number if needed, and read that entry. This is the nearest-rank convention. Other conventions interpolate between entries, so state the method when comparing reports.
Here, 10 × 0.90 = 9. The ninth time is four days. Thus nine of ten quotes arrived within four days, while only eight of ten arrived within three. The mean was three days; the three-day completion rate was 80%. To check a fixed deadline, count what finished by that deadline. To find the deadline that covered a given share, read the percentile.
Tell sales what happened in this group. Before making a continuing promise, check more weeks and whether the mix of standard and complex requests has changed. With ten quotes, a single additional late finish among the same ten would reduce the on-time percentage by ten points.
Exercise: Compare two repair shops
Allow 8–10 minutes. You are considering buying a repair shop and can examine one of two sellers more closely. Each advertises a five-day average. Your intended customers care about getting at least 90% of jobs back within six calendar days. Both sellers supplied ten consecutive jobs received during the same period, followed through completion. The clock runs from drop-off to ready for pickup. The figures are already sorted.
| Shop | Turnaround in calendar days |
|---|---|
| A | 4, 4, 4, 4, 4, 6, 6, 6, 6, 6 |
| B | 2, 2, 2, 2, 3, 3, 3, 4, 9, 20 |
- Check each mean. Calculate the median, range and nearest-rank P90 for each shop.
- Sketch a frequency chart for each using the same intervals: 1–5, 6–10, 11–15 and 16–20 days. Which shop usually finishes sooner? Which met the six-day deadline more consistently?
- Choose which seller to investigate first for your stated customers. Write one sentence with the evidence and one identifying a record you need before relying on it.
Try the comparison before reading the answer.
Answer
Shop A totals (5 × 4) + (5 × 6) = 50 days. Shop B totals (4 × 2) + (3 × 3) + 4 + 9 + 20 = 50 days. Dividing by ten gives five days per job at both shops. Here is what the identical averages leave out:
| Measure | Shop A | Shop B |
|---|---|---|
| Median | (4 + 6) ÷ 2 = 5 days | (3 + 3) ÷ 2 = 3 days |
| Range | 6 − 4 = 2 days | 20 − 2 = 18 days |
| P90: ninth job in sorted order | 6 days | 9 days |
| Ready within six days, inclusive | 10 of 10, or 100% | 8 of 10, or 80% |
For the four chart intervals, A’s counts are 5, 5, 0, 0. B’s are 8, 1, 0, 1. B finishes the middle job sooner, but its two slowest jobs took nine and twenty days. Notice that the 6–10 interval straddles your deadline: a six-day job passes and a nine-day job misses. Use the individual records to count on-time jobs, even when the grouped chart makes the shape easier to see.
Investigate A first: all ten observed jobs met the six-day deadline, compared with eight at B. Ask for several months of dated work orders, including the age of jobs still open, and compare similar repairs before judging whether A can sustain 90%. These ten jobs support a choice about where to investigate. They do not establish either shop’s future service level or what the business is worth.
Recall answer: the rush slot paid either $1,600 or $0. The $640 was a probability-weighted average. Likewise, Shop A’s five-day mean and median both fall on a time that no job actually took.
Start your evidence audit
Keep the repair-shop comparison as the first entry in this week’s evidence audit. Record the unit, which jobs were counted, the cutoff customers care about, and what the ten records cannot establish. Next you’ll examine how the choice of records can change the conclusion even when every calculation is correct.
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