Standard Deviation Calculator

What standard deviation measures

Standard deviation measures typical distance from the mean. It is reported in the same units as the data, so a standard deviation of 5.89 minutes means roughly six minutes of wobble around the average, not six percent and not six dollars.

Three measures describe spread, and they answer different questions:

• Range: the largest value minus the smallest. For 28, 31, 35, 29, 44, 30 and 27 the range is 44 minus 27, which is 17 minutes. Easy to read, but it uses only two of the seven numbers.
• Variance: the average of the squared deviations from the mean. Squaring removes the signs, and it also inflates the units, so variance is reported in square minutes and nobody can picture it.
• Standard deviation: the square root of the variance, which puts the number back into minutes. This is the figure people quote.

The square root is what makes the measure usable. A list that sits within a few minutes of its mean gets a small standard deviation, and a list with one wild value gets a large one, because that value is squared before it is averaged.

The National Institute of Standards and Technology (NIST), 2019 describes standard deviation as a measure of dispersion and gives the sample divisor of n minus 1.

Population or sample: the n and n-1 question

There are two formulas and they differ in one place, the divisor.

Dividing by the smaller number makes the sample result slightly larger, and that is deliberate. A sample mean sits closer to its own data than the true population mean does, so the squared deviations you can see are a little too small. Dividing by n minus 1 corrects for that, and the correction has a name: Bessel's correction. The NIST/SEMATECH e-Handbook of Statistical Methods defines the sample standard deviation with that divisor and notes the n minus 1 degrees of freedom.

The size of the gap depends on the count. With seven values the sample figure is 8 percent above the population figure. With 100 values the gap is about half a percent, which is why the choice matters for small lists and barely matters for large ones. If you truly have evalue, the population formula is the right one. If the list is a sample of something larger, use n minus 1.

Step by step: seven delivery times

Take seven deliveries that took 28, 31, 35, 29, 44, 30 and 27 minutes.

Step 1, the mean. 28 + 31 + 35 + 29 + 44 + 30 + 27 = 224, and 224 / 7 = 32 minutes.

Step 2, the deviations. Subtract 32 from each value: -4, -1, 3, -3, 12, -2 and -5. They add to zero, which is a useful check that the mean is right.

Step 3, the squares. 16, 1, 9, 9, 144, 4 and 25.

Step 4, the total. 16 + 1 + 9 + 9 + 144 + 4 + 25 = 208.

Step 5, the variance. For a sample, 208 / 6 = 34.67. For a population, 208 / 7 = 29.71.

Step 6, the square root. The square root of 34.67 is 5.89 minutes, and the square root of 29.71 is 5.45 minutes.

Look at the 44 minute run. It is 12 minutes from the mean, so it contributes 144 to a total of 208, about 69 percent of all the spread in the list. A dispatcher who only looked at the average of 32 would miss the fact that one route went badly wrong.

Reading the answer: one, two and three standard deviations

When data follows a bell curve, about 68 percent of values fall within one standard deviation of the mean, about 95 percent within two, and about 99.7 percent within three. The rule is a description of a shape, not a law that any list obeys.

For the delilist with mean 32 and sample standard deviation 5.89, one standard deviation covers 26.11 to 37.89 minutes. Six of the seven runs land inside that band, which is 86 percent, well above the 68 percent the rule predicts. Two standard deviations reach 20.22 to 43.78, and the 44 minute run sits just outside it. A single outlier is enough to break the pattern.

That gap is the honest part of the story. A small list with one slow day is not a bell curve, so the 68 and 95 figures are a rough sanity check at best. What the standard deviation does tell you without any assumption about shape is how far the values sit from the mean on average, and that is enough for most practical work.

Comparing things that are not on the same scale

A standard deviation of 5.89 minutes sounds calm on a 32 minute route and frantic on a 6 minute one. To compare spread across lists with different averages, divide the standard deviation by the mean to get the coefficient of variation.

Route A averages 32 minutes with a standard deviation of 5.89, so 5.89 / 32 = 18.4 percent. Route B averages 50 minutes with a standard deviation of 6.5, so 6.5 / 50 = 13 percent. Route B is the longer drive but the more predictable one, and the raw standard deviations alone would have said the opposite.

Investment returns use the same idea. A fund that averages 8 percent a year with a standard deviation of 12 points has a coefficient of variation of 1.5, while a fund with the same 8 percent average and an 18 point standard deviation comes in at 2.25. Same reward, more turbulence, and the second fund has to be justified by something other than its average.

Standard deviation in money and in the news

Daily volatility quoted for a stock or a fund is a standard deviation of daily returns. To put it on an annual footing, multiply by the square root of the number of trading days, which is conventionally 252.

A fund with a daily standard deviation of 1.2 percent has an annual figure of 1.2 x 15.87 = 19.04 percent. The square root scaling comes from the way variances add when returns are independent; it is a convention, and it understates the real figure when big down days cluster together.

Do not confuse the standard deviation of the data with the standard error of the mean. The standard error is the standard deviation divided by the square root of the count, and it describes how well the mean is pinned down. For the deliveries, 5.89 / 2.646 = 2.23 minutes, so the average of seven runs carries a margin of roughly two minutes either way. The standard deviation of the runs themselves stays at 5.89; more deliveries shrink the standard error and leave the spread alone.

Measurement data uses the same arithmetic. The NIST Special Publication 1500-28 on general statistics defines the sample standard deviation with the n minus 1 divisor and treats it as the usual measure of dispersion next to the mean.

Mistakes that inflate or deflate the answer

The arithmetic is not hard, and most wrong answers come from four habits.

• Mixing units. Minutes and seconds, or dollars and cents, in the same list produce a standard deviation that means nothing. Convert first.
• Using the wrong divisor. Dividing by n when the list is a sample understates the spread, and the error is largest on short lists.
• Rounding the mean before squaring. A mean rounded to 32.9 instead of 32.857 shifts edeviation, and the squares amplify it.
• Reporting the standard deviation of a skewed list as if it were symmetric. For incomes or house prices, one standard deviation below the mean can be a negative number, which is a sign the measure does not fit the shape.

One last habit: report the count with the standard deviation. A spread of 5.89 from seven values and a spread of 5.89 from 700 values are different claims about the world, and the reader cannot tell them apart without the number of observations.