How to use this standard deviation calculator
Paste your numbers into the box above: commas, spaces or one value per line all work, and you can paste a column straight out of Excel or Google Sheets. Pick Sample if your numbers are a subset of something bigger (some students from a school, some days from a year). Pick Population if your numbers are the whole group you care about. Hit Calculate and you get the standard deviation together with the full working, so you can see exactly where the answer came from.
What is standard deviation, in plain words?
Standard deviation measures how spread out numbers are around their average. Two classes can have the same mean score of 70: in one class everyone scored between 65 and 75, in the other scores ran from 30 to 100. The averages match, but the classes are completely different. Standard deviation is the number that tells them apart: small means the values huddle close to the mean, large means they are scattered far from it.
Think of it as the typical distance from the average. It is measured in the same units as your data (centimetres, marks, rupees), which is why teachers and researchers prefer it over the variance, whose units are squared.
The population formula (σ) divides by N. The sample formula (s) divides by n − 1 instead. That small change, called Bessel's correction, fixes the bias you get when you estimate a whole population's spread from just a sample of it. A sample almost always looks a little less spread out than the population it came from.
A bakery's daily sales. A small bakery records its sales (in hundreds of rupees) over eight days: 42, 38, 45, 51, 44, 39, 47, 50. These eight days are a sample of the bakery's trading year.
Mean = 356 ÷ 8 = 44.5. Squaring each day's distance from 44.5 and adding gives 158. Dividing by n − 1 = 7 gives a variance of 22.57, and the square root gives a sample standard deviation of ≈ 4.75. A typical day lands within about Rs 475 of the Rs 4,450 average. If those eight days were the entire period you cared about (population), you would divide by 8 instead and get σ ≈ 4.44.
How to calculate standard deviation by hand
- Find the mean. Add all values and divide by how many there are.
- Subtract the mean from each value. These are the deviations: positive above the mean, negative below.
- Square every deviation. Squaring removes the minus signs and punishes big distances more than small ones.
- Add the squared deviations to get the sum of squares.
- Divide: by N for a population, or by n − 1 for a sample. This is the variance.
- Take the square root. That is your standard deviation.
The calculator above shows this exact table for your own data, which makes it a good way to check homework: do the working by hand first, then compare.
Sample or population: which one do I choose?
Ask one question: is there more data out there that I didn't collect? If your dataset is everyone or everything in the group you are describing (all 30 students in this class, all 12 months of last year), it is a population. If it is a slice standing in for a bigger group (30 students from a school of 900, the last 20 days of trading), it is a sample. When in doubt, exam questions and published research almost always want the sample version.
What is a "good" standard deviation?
There is no universal good or bad value. It only makes sense next to the mean and the units. A standard deviation of 4.75 on daily sales averaging 44.5 is about 11% of the mean (that ratio is the coefficient of variation), which most shopkeepers would call fairly steady. The same 4.75 on exam marks averaging 44.5 would mean something quite different. Always interpret the number inside its own context.
Frequently asked questions
What is the difference between sample and population standard deviation?
The population standard deviation (σ) describes the spread of an entire group and divides by N. The sample standard deviation (s) estimates a population's spread from a subset and divides by n − 1 (Bessel's correction), which corrects the slight under-estimate a sample usually gives.
Can standard deviation be negative?
No. It is the square root of an average of squared values, so the smallest possible value is 0, which happens only when every value in the dataset is identical.
What does a standard deviation of 0 mean?
Every value in your data is exactly the same, so there is no spread at all around the mean.
Is standard deviation the same as variance?
No. The variance is the average squared distance from the mean, and the standard deviation is its square root. Standard deviation is in the original units; variance is in squared units.
Which standard deviation does Excel calculate?
Both. STDEV.S gives the sample standard deviation and STDEV.P gives the population standard deviation. This calculator matches those functions.