If the average wait at a help desk is 9.1 minutes, give or take 9.59, how many customers waited less than zero minutes? The question sounds absurd, and it is. Yet "9.1 ± 9.59" is exactly what a spreadsheet produces when you ask for the mean and the standard deviation of twenty real-looking waiting times, and its lower end, -0.49 minutes, is a wait nobody can have.
Nothing is wrong with the arithmetic. What is wrong is the choice of summary. There are two ways to describe a list of numbers in one line. The first is the mean with the standard deviation, and its cousin the coefficient of variation. The second is a handful of positions in the sorted list, the median and other percentiles, plus the mode, the value that turns up most often. Both are correct. They are not interchangeable, and this article puts them side by side on two data sets that behave very differently, then ends with a one-line test for choosing.
Every figure below comes from the mode calculator, the percentile calculator and the coefficient of variation calculator, and the screenshots show the same inputs.
Two ways to say what a list looks like
Summary X is the mean plus the standard deviation. The mean is the total shared out evenly; the standard deviation is the typical distance of a value from that mean. Divide one by the other and you get the coefficient of variation, the spread as a share of the level, which lets you compare a list of milliliters with a list of minutes.
Summary Y describes positions instead of averages. The median is the value in the middle of the sorted list. The 90th percentile, P90, is the value that nine in ten observations stay at or below. The mode is the most frequent value. None of them uses the size of the extreme values, only their place in the queue.
Summary X is what most people learn first and what most reports print. Summary Y is what service-level agreements, growth charts and salary surveys use. The two data sets below show why each camp is right about its own kind of data.
Round one: ten bottles from a filling line
A filling machine set to 500 ml puts out ten bottles: 498, 502, 501, 499, 500, 503, 497, 500, 501, 499 ml. The mean is 500.00 ml, the standard deviation 1.83 ml, the coefficient of variation 0.37%. Two standard deviations either side of the mean give a band from 496.35 to 503.65 ml, and all ten bottles sit inside it.
Now the other summary. The median is 500.00 ml, the same as the mean. P5 is 497.45 and P95 is 502.55, a band only slightly narrower than the two-SD one, because ten bottles cannot fill out the tails. The picture is identical, described at greater length.
The mode is where summary Y stumbles. Three values tie at two bottles each: 499, 500 and 501. Excel's MODE.SNGL picks 501, simply because it comes first in the list. For a measurement recorded to the milliliter the mode is an accident of rounding, not a description.
Round one goes to X. On symmetric data, where the values bunch around the middle and thin out evenly on both sides, the mean with its standard deviation says everything in two numbers, and a reader can rebuild the likely range from them.
Round two: twenty waits at a help desk
Twenty support requests, time until a person answered, in minutes: 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 8, 9, 11, 14, 18, 26, 41. Most are short, a few are long, and one is very long. This is what waiting times, response times, incomes and file sizes usually look like: a floor at zero and a long tail on one side.
The mean is 9.1 minutes and the standard deviation 9.59. One standard deviation either side runs from -0.49 to 18.69 minutes. The lower end is impossible, and the band is also wrong in a quieter way: in bell-shaped data about two thirds of the values fall within one standard deviation of the mean, while here 18 of the 20 do, 90%. The mean itself is misleading as a "typical" wait. Fifteen of the twenty customers, 75%, waited less than the average.
| Statistic | Value | What it tells a reader |
|---|---|---|
| Mean | 9.1 min | the sum spread evenly; 15 of the 20 waits are shorter |
| Standard deviation | 9.59 min | larger than the mean itself |
| Mean minus one SD | -0.49 min | a negative wait, impossible |
| Mode | 3 min | the most common wait, 4 of 20 |
| Median (P50) | 5.5 min | half wait less, half wait more |
| P75, third quartile | 9.5 min | three in four wait this long or less |
| P90 | 18.8 min | the wait that only one in ten goes past |
Summary Y reads this list without strain. The most common wait is 3 minutes, four customers out of twenty. Grouped into five-minute classes, 8 of the 20 waits fall between 0 and 5 minutes, and the grouped estimate of the mode is 4.4 minutes. The median is 5.5 minutes.
The tail is where percentiles earn their keep. Three quarters of the customers were answered within 9.5 minutes. The 90th percentile is 18.8 minutes: nine in ten were answered within that time, and one in ten waited longer. That second sentence is the one a help desk manager actually needs, and nothing in "9.1 ± 9.59" produces it.
One caution about percentiles: the definition matters on short lists. The same twenty waits give a P90 of 18.8 minutes in Excel's PERCENTILE.INC, 25.2 in PERCENTILE.EXC and 18.0 by the nearest-rank rule. When a percentile is written into a contract, the method should be written next to it.
Round two goes to Y, and not narrowly. On skewed data the mean describes almost nobody and the standard deviation describes a shape the data do not have.
Round three: one extreme value
Suppose the slowest request had taken 90 minutes instead of 41, a ticket forgotten over a lunch break. One value out of twenty changes. The table shows what happens to each summary.
| Statistic | Slowest wait 41 min | Slowest wait 90 min |
|---|---|---|
| Mean | 9.1 | 11.55 |
| Standard deviation | 9.59 | 19.40 |
| Coefficient of variation | 105.3% | 168.0% |
| Mode | 3 | 3 |
| Median | 5.5 | 5.5 |
| P90 | 18.8 | 18.8 |
| P95 | 26.8 | 29.2 |
The standard deviation more than doubles, from 9.59 to 19.40, and the mean rises by 2.45 minutes. The mode, the median and even the 90th percentile do not move at all, because the slow ticket was already the slowest and stays in the same place in the queue. Only the top percentiles feel it: P95 goes from 26.8 to 29.2.
Whether that is good depends on the question. If you are budgeting total staff time, the forgotten ticket really did cost 49 more minutes, and the mean is right to report it. If you want to tell customers how long they will probably wait, a single outlier should not change the answer, and the percentiles are right to ignore it.
What the coefficient of variation adds
The coefficient of variation puts both data sets on one scale. The bottles vary by 0.37% of their mean; the waits by 105.34%. Written as a ratio that is 0.0035, and the Feltz and Miller test for equal coefficients of variation gives a statistic of 12.965 with p below 0.001. Nobody needed a test to see that difference, but the same test is useful when two lines or two teams are closer together.
The more useful thing the coefficient of variation does is warn you. For data that cannot go below zero, a coefficient above 50% has a precise consequence: the mean minus two standard deviations is negative. At 50% it lands exactly on zero, and above it the "mean ± 2 SD" band reaches into values the data cannot take. For the waits it reaches -10.07 minutes. At that point the bell-shaped reading of mean and standard deviation has already failed, whatever the histogram looks like.
The single outlier makes the point sharper. With the 90-minute ticket the coefficient of variation jumps from 105.3% to 168.0%, while nothing about the typical customer's experience has changed.
The scorecard
| Situation | X: mean and standard deviation | Y: percentiles and mode |
|---|---|---|
| Symmetric measurements (fills, weights, test scores) | Wins: compact, and the 2 SD band is honest | Works, but says the same thing at greater length |
| Skewed data (waits, incomes, file sizes) | Loses: implies negative values, hides the tail | Wins: P50 and P90 describe the typical and the bad case |
| One extreme value | Moves a lot (SD doubled here) | Barely moves (P90 unchanged) |
| Categories or rounded counts | Meaningless for words | The mode is the only average that works |
| Comparing spread across units | The CV does it, when the mean is well above zero | Ratios of percentiles do it, less familiar to readers |
| Spreadsheet formula | =AVERAGE and =STDEV.S | =MEDIAN, =PERCENTILE.INC and =MODE.SNGL |
The pattern is not "one is modern and one is old-fashioned". Summary X assumes the data spread evenly around a center, and when they do, it is the more efficient description. Summary Y assumes nothing about shape, which costs a few more numbers and buys robustness.
The verdict, and a test that takes ten seconds
Before choosing, compute two things: the coefficient of variation and the gap between the mean and the median. If the coefficient is well under 50% and the mean sits close to the median, as with the bottles (0.37%, 500.00 against 500.00), report the mean and the standard deviation. If the coefficient is near or above 50%, or the mean sits clearly above the median, as with the waits (105.3%, 9.1 against 5.5), report the median and a high percentile such as P90, and add the mode when the values are counts or categories.
Two refinements keep the rule honest. First, for data that can be negative, such as temperature changes or profit and loss, the coefficient of variation means nothing, so compare the mean and the median alone. Second, when the audience needs totals, such as hours of staff time or liters of product shipped, report the mean as well, because only the mean multiplies back to a total. The waits add up to 182 minutes, which is 20 times 9.1, and no percentile can tell you that.
The help desk, then, should not say "9.1 minutes on average". It should say: half of requests are answered within 5.5 minutes, nine in ten within 18.8. The filling line can keep saying 500 ml, give or take 1.83.
Tools discussed in this article
- Mode calculator: the most frequent value in numbers or words, bimodal and no-mode lists, Excel MODE.SNGL and MODE.MULT, and the modal class for grouped data.
- Percentile calculator: any percentile or percentile rank by linear, exclusive and nearest-rank methods, with the matching Excel results.
- Coefficient of variation calculator: CV and %RSD from raw data or from a mean and standard deviation, and a test for whether two CVs differ.
More statistics tools
Interquartile range (quartiles, outlier fences and a box plot) · Wilcoxon test (comparing skewed groups by ranks) · Fisher's exact test (small 2x2 tables of counts) · Median (the middle value of a list) · Average (arithmetic, weighted and geometric means) · Standard deviation (spread around the mean, step by step)