Standard Deviation Calculator

Standard deviation of some example lists

Population or sample: which standard deviation?

What the results mean

How to calculate a standard deviation by hand

1. Find the mean. Add the numbers and divide by how many there are: 40 ÷ 8 = 5 for 2, 4, 4, 4, 5, 5, 7, 9.
2. Find each deviation. Subtract the mean from every number. For 2, 4, 4, 4, 5, 5, 7, 9 the deviations are −3, −1, −1, −1, 0, 0, 2, 4. They always add up to zero.
3. Square the deviations. Square each one so that they are all positive: 9, 1, 1, 1, 0, 0, 4, 16.
4. Add the squares. The sum of squared deviations is 32.
5. Divide by n or n − 1. Divide by n = 8 for the population variance (4), or by n − 1 for the sample variance (≈ 4.571429).
6. Take the square root. The square root of the variance is the standard deviation: 2 for the population and ≈ 2.13809 for the sample.

Frequently asked questions

What is the difference between sample and population standard deviation?

The population standard deviation divides the sum of squared deviations by n and is right when your numbers are every value in the group. The sample standard deviation divides by n − 1 and is right when your numbers are a sample used to estimate a larger group. For 2, 4, 4, 4, 5, 5, 7, 9 they are 2 (population) and ≈ 2.13809 (sample). If you are not sure, the sample version is the usual choice, and it is what Excel’s STDEV.S and a calculator’s s button give.

Why do you divide by n − 1 for a sample?

A sample’s values sit closer to the sample’s own mean than to the true mean of the whole group, so dividing by n would on average understate the variance. Dividing by n − 1 (Bessel’s correction) removes that bias for the variance. The effect fades as n grows: with 8 numbers it matters, with 8,000 it hardly does.

How do I calculate standard deviation step by step?

Find the mean, subtract it from every value, square those deviations, add them up, divide by n (population) or n − 1 (sample), and take the square root. For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, the squared deviations add up to 32, and 32 ÷ 8 = 4, whose square root is 2. The calculator writes this working out for lists of up to 20 numbers.

What is the difference between variance and standard deviation?

The variance is the average squared deviation, and the standard deviation is its square root. The standard deviation is in the same units as the data (dollars, centimetres), while the variance is in squared units. For 2, 4, 4, 4, 5, 5, 7, 9 the population variance is 4 and the population standard deviation is 2.

Is a high or low standard deviation good?

Neither on its own: it only says how spread out the numbers are, and what counts as large depends on the scale. The coefficient of variation (sample SD ÷ mean) helps compare spread across different scales: ≈ 42.761799% for 2, 4, 4, 4, 5, 5, 7, 9. It is only meaningful when the mean is not near zero, and the calculator reports it as undefined when the mean is exactly 0 (as for -2, -1, 0, 1, 2).

What is a z-score?

A z-score says how many standard deviations a value is above (positive) or below (negative) the mean: z = (value − mean) ÷ SD. In 2, 4, 4, 4, 5, 5, 7, 9 the value 9 has a z-score of ≈ 1.870829 using the sample standard deviation, and the mean itself has a z-score of 0. You can switch the z-scores between the sample and the population standard deviation. When all the values are equal the standard deviation is 0 and z-scores are undefined.

How does an outlier affect the standard deviation?

Strongly, because deviations are squared. 10, 20, 30, 40 on their own have a sample standard deviation of ≈ 12.909944; adding 1000 raises it to ≈ 436.17657. The interquartile range is much less affected (20 for 10, 20, 30, 40, 1000), which is why it is often reported beside the median for skewed data.

How are the quartiles calculated?

Quartiles use the inclusive method (the same as Excel’s QUARTILE.INC): Q1 and Q3 sit at positions (n − 1) × ¼ and (n − 1) × ¾ of the sorted list, counting from 0, and a fractional position is interpolated between its two neighbours. Other methods, such as QUARTILE.EXC, give slightly different quartiles for short lists. For 2, 4, 4, 4, 5, 5, 7, 9, Q1 is 4, Q3 is 5.5 and the interquartile range is 1.5.

What is the standard error of the mean?

It is the sample standard deviation divided by the square root of n, and estimates how much a sample mean would vary from sample to sample. For 2, 4, 4, 4, 5, 5, 7, 9 it is ≈ 0.755929. It shrinks as the sample grows, while the standard deviation does not. It needs at least 2 numbers.

What if I enter only one number?

The population standard deviation of a single number is 0. The sample standard deviation, variance, standard error and coefficient of variation divide by n − 1 = 0, so the calculator says they need at least 2 numbers instead of showing a made-up value. The same goes for the list of equal numbers 5, 5, 5, 5, whose standard deviation is 0.

Is the result exact, even for decimals?

Yes, as far as it can be. Every number is read as an exact decimal and the sums are whole-number sums, so there is no floating-point drift or cancellation: the variance of 0.1, 0.2, 0.3 is exactly ≈ 0.006667 (population) and 0.01 (sample), and the sample standard deviation is 0.1. Only square roots can be endless decimals; those are computed to 12 places and shown rounded to 6 with a ≈ in front. Results that end within 12 decimals are shown in full.

How many numbers can I paste, and how are they read?

Up to 200,000 numbers of up to 30 digits each are used, and the page says so if more were pasted. Numbers can be separated by commas, spaces, tabs, semicolons or new lines, so a spreadsheet column works. A comma between groups of exactly three digits with no space (1,234) is one number; write 1, 234 with a space for two. Text that is not a number is ignored and listed. The z-score table shows the first 100 values.