Standard Deviation Calculator
The population standard deviation of 2, 4, 4, 4, 5, 5, 7, 9 is exactly 2, because the mean is 5 and the variance is 4. This standard deviation calculator measures how far your numbers spread from their average. Enter your values, pick sample or population, and read the standard deviation along with the variance, mean, count, and sum.
Quick answer
Standard deviation tells you how spread out a set of numbers is around the mean.
What this tells you
- •Standard deviation tells you how spread out a set of numbers is around the mean.
- •A small standard deviation means the values sit close to the average, and a large one means they are spread wide.
- •Use population when your numbers are the entire group, and use sample when they are a subset you are using to estimate a larger group.
- •The variance is the square of the standard deviation, so the two always move together.
How to Use
- 1Type or paste your numbers into the input field.
- 2Separate values with commas, spaces, or new lines.
- 3Choose Sample if your data is a subset, or Population if it is the whole group.
- 4Click Calculate to see the standard deviation, variance, mean, count, and sum.
How It Works
Formula
mean = sum of values / count. variance = sum of (value - mean)^2 divided by n for a population or by n - 1 for a sample. standard deviation = square root of the variance.Start by finding the mean of every value. Subtract the mean from each value, square the result so positives and negatives both count, then add those squared differences together. Divide that total by the count n for a population or by n - 1 for a sample. The square root of that figure is the standard deviation, expressed in the same units as your original data.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
Population standard deviation of 2, 4, 4, 4, 5, 5, 7, 9
The sum is 40 and the count is 8, so the mean is 5. The squared differences from the mean add up to 32, and dividing by 8 gives a variance of 4. The square root of 4 is 2, so the population standard deviation is exactly 2.
Sample standard deviation of 1, 2, 3, 4, 5
The mean is 3. The squared differences are 4, 1, 0, 1, and 4, which sum to 10. Dividing by n - 1 = 4 gives a variance of 2.5, and the square root of 2.5 is about 1.5811.
Sample vs Population Standard Deviation
How the two formulas differ and when to use each one.
| Type | Divisor | When to use |
|---|---|---|
| Population | n | Your data covers the entire group you care about. |
| Sample | n - 1 | Your data is a subset used to estimate a larger group. |
Dividing a sample by n - 1 instead of n corrects the tendency to underestimate the true spread.
Common mistakes
- Confusing sample with population. Population divides by n while a sample divides by n - 1, so picking the wrong one changes the result.
- Dividing by n when you should divide by n - 1 for a sample. This is the most common error and makes the standard deviation look smaller than it should.
- Forgetting to square each deviation before adding them. Without squaring, positive and negative differences cancel out and the spread reads as zero.
- Reporting the variance as the standard deviation. The standard deviation is the square root of the variance, not the variance itself.