Key takeaways
- Standard deviation measures how far values typically sit from the mean.
- Population SD: σ = √(Σ(x − μ)² ÷ n). This calculator divides by n.
- The sample SD divides by n − 1 instead and gives a slightly larger answer.
- Standard deviation is the square root of the variance, in the same unit as the data.
In this guide
What is standard deviation?
Standard deviation puts a number on how spread out your data is. Roughly, it is the typical distance of a value from the mean. A small standard deviation means the values are bunched close together. A large one means they are scattered.
Why does standard deviation matter?
An average alone hides a lot. A bus that is on time on average, but sometimes 30 minutes early and sometimes 30 minutes late, is useless. Standard deviation shows how reliable or consistent something is.
The data sets 10, 20, 30 and 60, 70, 80 have very different means, but exactly the same standard deviation. Spread and centre are separate ideas.
The formula
This calculator finds the population standard deviation, which divides by n:
| Symbol | What it means | Example |
|---|---|---|
| σ (sigma) | The standard deviation | 1.41 |
| x | Each value in your list | 4, 8, 6, 5, 7 |
| μ (mu) | The mean of the list | 6 |
| Σ | Add up for every value | 4 + 4 + 0 + 1 + 1 |
| n | How many values there are | 5 |
The sample standard deviation divides by n − 1 instead of n. Use it when your data is a small sample taken from a bigger group. It always gives a slightly larger answer.
How to calculate standard deviation step by step
- 1Find the mean. Add the values and divide by n.
- 2Find each deviation. Subtract the mean from every value.
- 3Square each deviation. Now every number is positive.
- 4Average the squares. Add them and divide by n. This is the variance.
- 5Take the square root. That is the standard deviation.
Now paste your own numbers into the calculator below and follow each step.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Forgetting the square root. Stopping at step 4 gives the variance, not the standard deviation.
- Not squaring the deviations. If you just add them, the positives and negatives always cancel to 0.
- Using the wrong mean. Recalculate the mean if you add or remove a value.
Tips and tricks
- Adding a constant to every value (like +10 marks for everyone) does not change the standard deviation.
- Converting sample to population: multiply the population variance by
n ÷ (n − 1), then take the square root. - Measure how unusual one value is with the z-score calculator, which divides by the standard deviation.
Where you'll use it in real life
- Sport: how consistent a cricket player's scores are from match to match.
- Factories: keeping product sizes close to the target.
- Money: comparing how risky two investments are.
- Science: in a normal distribution, about 68% of values lie within one standard deviation of the mean.
Quick summary
To find the standard deviation, find the mean, square each distance from it, average those squares and take the square root. This calculator divides by n (population). Use it above to see every step for your own data.
Worked example
A small café counted how many cakes it sold on five days: 4, 8, 6, 5 and 7. The owner wants to know how much daily sales usually move away from the average.
- 01Data set: 4, 8, 6, 5, 7
- 02Mean (μ) = 4 + 8 + 6 + 5 + 7 ÷ 5 = 6.00
- 03Variance = Σ(x - μ)² ÷ n = 2.00
- 04Standard deviation = √2.00 = 1.41
These numbers are pre-filled in the calculator above.
Frequently asked questions
Is this population or sample standard deviation?
This calculator gives the population standard deviation, which divides by n. For the sample version, divide by n − 1 instead; the answer will be a little larger.
What does a standard deviation of 0 mean?
All the values are exactly the same, so there is no spread at all. For 10, 10, 10, 10 the standard deviation is 0.
What is the difference between variance and standard deviation?
Variance is the average of the squared distances from the mean. Standard deviation is the square root of the variance, so it is back in the same unit as your data.
Is a high standard deviation bad?
Not always. It simply means the values are spread out. For a factory making bolts that is bad, but for a list of ages in a whole town it is expected.
