Standard Deviation Calculator

How to calculate standard deviation step by step

Take the eight numbers the page loads by default, 2, 4, 4, 4, 5, 5, 7, 9: they add up to 40 across eight values, which puts the mean at 5, the pivot every later step measures against.

Subtracting 5 from each number gives the deviations, running from −3 for the lowest value up to +4 for the 9. Those signed deviations always add back to zero, which is the reason each one gets squared first. The squares total Σ(x − x̄)² = 32. The single value 9 carries half of that total on its own.

Dividing 32 by n − 1 = 7 gives a sample variance of 4.571429, whose square root is the sample standard deviation s ≈ 2.13809. Dividing the same 32 by n = 8 gives a population variance of 4 and a population standard deviation of exactly 2. The variance is that same figure one step earlier, still in squared units; the square root brings it back into the units of the data.

Sample or population standard deviation, which one applies?

A teacher holding the marks of all 24 pupils in one class already has every number that matters, so the divisor is n. An inspector who measures 30 bolts out of a run of 4,000 holds a fragment, which calls for n − 1. The length of the list settles nothing here. A list of 8 numbers can be a sample and a list of 800 can be a population; the divisor follows how far the conclusion reaches.

Most real data sets turn out to be samples, which is why n − 1 is the default here, as it is in Excel's STDEV.S and in the sd() function of R. That smaller divisor has a name worth knowing, Bessel's correction. A sample rarely contains the true extremes of the population it came from, so its raw squared distances understate the real spread; dividing by one less value pushes the answer back up.

Picking the wrong divisor changes the number without usually changing the decision. On the eight default values the two answers sit at 2.13809 and 2, close to 7 % apart. On the seven test scores 85, 92, 78, 88, 95, 71 and 90, the sample answer is 8.423324 against 7.798482 for the population, roughly 8 % apart. That gap narrows as the list grows.

What does a standard deviation of 1.5 mean?

On its own, 1.5 means nothing until it is read against the mean it came from. Exam marks averaging 8 out of 20 with a standard deviation of 1.5 describe a group that spreads far enough to rank; a set averaging 150 with the same 1.5 describes values that are all but identical. As a proportion, 1.5 against a mean of 8 is 18.75 % of the mean, the ratio known as the coefficient of variation.

On data with a roughly bell-shaped distribution, close to 68 % of the values fall within one standard deviation of the mean and about 95 % within two. A mean of 8 with a standard deviation of 1.5 therefore puts two thirds of the observations between 6.5 and 9.5, almost all of them between 5 and 11. Skewed data with a long tail breaks that rule; the deviation table is where that shows.

Two sets with the same mean can describe different situations. Delivery times averaging 30 minutes with a standard deviation of 3 minutes arrive when promised; the same average with a standard deviation of 15 minutes covers a band running from 15 to 45 minutes.

What is the standard deviation of a single value or of identical values?

Entering one number leaves the sample answer blank. With n = 1 the divisor n − 1 falls to zero, so the sample standard deviation is undefined instead of 0. The page prints that word in place of a figure. The population version returns 0 in the same case, since one point cannot be spread out.

Five identical values behave differently. Every deviation is zero and the sum of squares is zero, so both answers come out at 0 while n = 5 still leaves a valid divisor of 4. Zero is a genuine result there; a range of 0 between the minimum and the maximum confirms it at a glance.

The page keeps six decimals and drops trailing zeros, which is why the sample answer on the default set reads 2.13809 while the population answer reads a bare 2. Reports rarely need that much, so round the figure to the precision of the measurements behind it.