Foundation · 13 min
The average that describes nobody
After this you can
- calculate a mean, a median and a range from a small set of measurements by hand.
- explain why the mean alone can mislead, using a case where the median tells a different story.
Assumes you have done Signal or noise: reading a number that moves.
The problem
A service desk published its callback performance: "average callback time 25 minutes". The team was told to get it under 20. They spent six weeks on it. What nobody had noticed was that 17 of the 20 callbacks that morning were faster than the average — most were around twelve minutes — and the number was being held up by three calls that took an hour and a half each. The target was aimed at a figure no customer had ever experienced.
The idea
Three numbers describe a set of measurements, and each one answers a different question.
The mean is the total divided by how many there are. Every value pulls on it in proportion to its size, so one very large value moves it a long way. It answers: if the total were shared out equally, how much would each get?
The median is the middle value once you sort them. If there is an even number of values, it is the average of the middle two. Only the position of an extreme value matters, not how extreme it is — moving the largest value from 118 to 1,180 does not move the median at all. It answers: what is a typical one?
The range is the largest minus the smallest. It answers: how spread out are they? It is the easiest measure of spread to compute and the least reliable, because it uses exactly two of your values and throws the rest away. The bigger your sample, the more chances it has to contain an extreme, so the range tends to grow with sample size even when nothing about the process changed.
Why this matters more than it sounds
For symmetric data the mean and the median land in nearly the same place, and it does not much matter which you quote. But almost nothing you will measure at work is symmetric.
Waiting times, handling times, costs, defect counts, days to resolve — all of them are bounded below (you cannot wait less than no time) and unbounded above (a case can always go catastrophically wrong). That shape is called right-skewed, and it has a reliable consequence: the mean sits above the median, dragged up by the long tail.
So when you read "average handling time", ask which average, because in right-skewed data:
- the mean describes the total workload honestly — it is total ÷ count, which is exactly what capacity planning needs;
- the median describes the typical experience honestly;
- and quoting the mean as if it were the typical experience is a true statement that misleads everyone who reads it.
Neither is wrong. Using one to answer the other's question is.
The test: if you cut the largest value in half, does your number move a lot? If yes, you are using the mean, and you should say so out loud.
Worked example
Twenty callback requests from one morning, in minutes:
12, 9, 14, 11, 8, 16, 13, 10, 15, 12, 9, 118, 11, 14, 96, 13, 10, 12, 87, 15
The mean. Add them: 505. Divide by 20. 25.25 minutes.
The median. Sort them:
8, 9, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 15, 15, 16, 87, 96, 118
Twenty values, so the median is the average of the 10th and 11th: (12 + 13) ÷ 2 = 12.5 minutes.
The range. 118 − 8 = 110 minutes. True, and almost useless: it tells you only that the morning contained one very fast callback and one very slow one.
Now the part that matters.
| Mean | 25.25 min |
| Median | 12.5 min |
| Callbacks at or below the median | 10 of 20 — by definition |
| Callbacks faster than the MEAN | 17 of 20 |
Seventeen of twenty customers had a better experience than "average". The mean is not describing a typical callback; it is describing what happens when you average three disasters into seventeen ordinary events.
Drop the three long calls and the mean becomes 12.00 minutes — almost exactly the median. That is the whole story in one line: three values out of twenty were carrying thirteen minutes of the "average".
So which number should the service desk have published? Both, and they answer different questions:
- "Half our callbacks happen within 13 minutes" — the median, describing the experience.
- "Three calls this morning took over an hour" — the tail, which is the actual problem.
The improvement work should have been aimed at the three long calls, not at the seventeen that were already fine. Reporting the mean alone hid the only thing worth fixing.
Dataset: ds-callback-times — the same data loads in the tool below, so you can reproduce every figure here yourself.
Your turn
The tool opens with the callback data, and shows the mean and median on the same axis.
- Confirm the mean is 25.25 and the median is 12.5. Look at where the dashed mean line sits: to the right of almost every dot.
- Change the 118 to 1180 — a typing error of the kind that happens constantly. The mean jumps to over 78. The median does not move at all. This is the single most useful demonstration in the tool: the median is what you reach for when you do not fully trust your data.
- Delete the three long calls. Watch the mean fall to 12.00 and the skew warning disappear.
- Now look at Q1 and Q3 — 10.75 and 15. The middle half of the morning sat inside a four-minute band. The range said 110. Which of those two describes the process?
Mean, Median & Range
practicen
20
Mean
25.25
Median
12.5
Range
110
- Minimum
- 8
- Q1
- 10.75
- Q3
- 15
- Maximum
- 118
Right-skewedThe mean (25.25) is above the median (12.5). A few large values are pulling it up, so "average callback time 25.25" overstates what a typical case looks like — half of them are at or below 12.5. Waiting times, handling times and costs are nearly always shaped like this, which is why "average" so often describes nobody. Quote the median, or quote both — but do not quote the mean alone.
Range is max − min: it uses exactly two of your 20 values and gets less reliable as the sample grows, because a bigger sample has more chances to contain an extreme. Q1 to Q3 covers the middle half and does not have that problem.
One number per line, or label, number.
How this is calculated
- Mean — sum ÷ n. Every value pulls on it, so one extreme value moves it.
- Median — the middle value once sorted; the average of the two middle values when n is even. Only the position of the extremes matters, not how extreme they are.
- Range — maximum − minimum.
- Q1 and Q3 — the 25th and 75th percentiles, by the R7 rule (the one R, NumPy and Excel PERCENTILE.INC use). Packages disagree about quartiles, so the method is stated rather than assumed.
- Standard deviation — the sample form, dividing by n − 1. Dividing by n understates the spread of a sample and is one of the most common errors in hand-built spreadsheets.
- The skew note fires when the mean sits more than 0.2 spreads away from the median, where a spread is (Q3 − Q1) ÷ 1.349 — 3.15 here. That divisor is deliberately not the standard deviation: one extreme value inflates the SD faster than it moves the mean, so measured that way the gap is capped at 1/√n for a single outlier and from n = 26 upward nothing could ever trip it. The middle half of the data is the part the outlier is not in.
Source: Montgomery, D.C., Introduction to Statistical Quality Control 7e, Ch. 3; Hyndman, R.J. & Fan, Y. (1996), “Sample Quantiles in Statistical Packages”, The American Statistician 50(4), for the quantile definitions.
Check yourself
No hints. Wrong answers are explained, not softened.
Six repair times, in hours: 2, 3, 3, 4, 9, 33. What are the mean, the median and the range?
A hospital reports mean length of stay of 6.2 days and median of 3 days. A manager proposes a target of 'mean stay under 5 days'. What is wrong with it?
You have 200 cycle times. One of them is 40 times the others because somebody entered seconds as minutes. You cannot correct it before the meeting. Which summary do you present?
Worth remembering
When n is even, how is the median calculated?
Sort the values and average the two middle ones. With 20 values that is the mean of the 10th and 11th.
Why does the mean sit above the median in waiting-time data?
Waiting times are bounded below by zero and unbounded above, so the distribution has a long right tail. The tail pulls the mean up; the median stays where the bulk of the data is.
Why is the range an unreliable measure of spread?
It uses only two of your values. A larger sample has more chances to contain an extreme, so the range tends to grow with sample size even when the process has not changed.
Can you do this now?
Rate yourself honestly. We compare your rating with how you actually answered — the gap is more useful than either number alone.
I can calculate a mean, a median and a range from a small set of measurements by hand.
I can explain why the mean alone can mislead, using a case where the median tells a different story.
Your rating is recorded alongside your drill results. Neither alone marks the competency as met.