Standard Deviation

The habit of collapsing a pile of numbers into one line stopped at the mean. Pagedeck's support response time was reported the same way — one line saying the monthly average was 32 minutes, comfortably inside the 60-minute target.

Complaints about slow replies went up that same month. Opening the ticket records showed that half had been answered within ten minutes while a handful of the rest took over two hours. The 32-minute average was a blend of those two groups and described neither. Almost no customer actually waited about 32 minutes.

A month where every ticket lands between 28 and 36 minutes and a month where ten-minute and two-hour replies mix to the same average can't be told apart from the average alone. The first supports a promise of an answer within the hour and the second doesn't, yet both get written up as 32 minutes.

So spread started getting summarized as its own number. Variance came first, but it squares the values along the way, which leaves it in squared minutes rather than minutes and makes it unreadable as reported. Standard deviation is that variance with a square root applied, which returns the unit to minutes. Being in the same unit as the average, it can be read beside it.

Square the distance of each value from the average, take the average of those squares, then apply a square root at the end. Four steps.

1. Find the average. Five response times of 30, 34, 32, 28 and 36 minutes sum to 160, so the average is 32 minutes.
2. Subtract the average from each value and square the result. The deviations are -2, 2, 0, -4 and 4, and squared they are 4, 4, 0, 16 and 16. Squaring is necessary because deviations above and below the average would otherwise cancel out to zero.
3. Average the squares. They sum to 40 across five observations, which gives 8. That value is the variance.
4. Apply the square root. The square root of 8 is about 2.8, so the standard deviation is 2.8 minutes. Reported that way it sits next to the 32 and can be compared against it directly.

Measure a different month with the same average and the difference appears. Times of 10, 55, 32, 20 and 43 minutes also sum to 160, so the average is still 32, but the squared deviations sum to 1,278 and the standard deviation comes out at 16 minutes.

One variation exists. If the numbers are a sample rather than everything there is, step 3 divides by one less than the count. Samples tend to understate spread and that adjustment corrects for it. A tool does the arithmetic either way, so the work in practice is reading the result paired with the average.

There is no absolute threshold where smaller becomes good; the value gets read as a ratio against the average. A standard deviation of 2.8 minutes against an average of 32 is 9 percent of the average, and 16 minutes against the same average is 50 percent. The first is a number you can make promises with, the second is context only.

A practical test is to turn the range from average minus deviation to average plus deviation into a sentence. The first month is 29 to 35 minutes, which supports a promise of an answer within the hour. The second is 16 to 48 minutes, while two-hour replies are actually in the mix. When the sentence comes out awkward, that data isn't well served by a standard deviation.

Data skewed to one side is the case where that happens. Response times and payment amounts cluster low and trail off with a long tail upward, and there the number distorts what people actually experience. An average of 32 minutes with a deviation of 16 tells you nothing about how many customers waited two hours. For data shaped like that, the median and percentiles get read alongside.

The sensitivity to a single outlier is worth knowing as well. Because deviations are squared, one distant value weighs more than ten ordinary ones. If one of those five tickets had taken five hours instead of 32 minutes, the standard deviation would jump from 2.8 minutes into the hours. So when the number rises suddenly, check whether one record caused it before concluding that consistency got worse. The average barely shifts in that situation while the deviation moves a great deal.

"The two teams have the same average and three times the standard deviation. We can't hold them to the same bar."

One line of average makes the teams look alike, but only one of them is predictable.

"Average of 40 minutes, deviation of 5, so let's schedule 45."

The average plus one deviation becomes the committed figure, which works because the spread is tight enough to hold.

"With a deviation that large we shouldn't promise on the average. Let's use the 95th percentile."

The spread has cost the average its representativeness, so the worse end of the experience should set the bar.

At Pagedeck the value earned its place in deployment lead time. Theo measures the hours from code merge to production release each week, and for half a year the average sat around four hours without much movement. Adding the standard deviation showed some weeks at one hour and others at nine. Looking into a high week turned up a deploy pushed from Friday afternoon to Monday, and that single record had pulled the whole week up. Mia has had the average and the deviation written on the same line ever since.

The most common misreading is that a small standard deviation means a good state. Less fluctuation is usually better, so the belief sets easily. But the number says nothing about where the values cluster. A team whose replies all land near three hours also has a small deviation — consistently bad. The value has to be read paired with the average, and a report carrying only one of the two has said half of what it needed to. Together the two describe a shape; alone, neither does.

The other frequent move is applying the memorized rule that 95 percent of values fall within two deviations of the average. Statistics classes teach it that way, so the application is natural. But that proportion holds only when values spread out in a bell shape. For data that clusters low and trails upward, subtracting two deviations from the average can produce a negative number. Compute an upper bound from the rule anyway and you end up promising something more optimistic than the data supports.