Math Tools
Standard Deviation Calculator
Calculate population or sample standard deviation, variance, mean, and count from a list of numbers.
Standard Deviation Calculator
Calculate population or sample standard deviation, mean, variance, and count from a list of numbers.
Example: 2, 4, 4, 4, 5, 5, 7, 9
Results
Count
8
Mean
5
Variance
4
Standard deviation
2
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Population vs sample
Use population standard deviation when your numbers represent the full group. Use sample standard deviation when your numbers are only part of a larger group.
What is a standard deviation calculator?
A standard deviation calculator is a statistics tool that measures how spread out a set of numbers is around the mean. Instead of working through each formula by hand, you can paste a list of values and instantly calculate the count, mean, variance, and standard deviation.
This is useful in statistics, probability, business analysis, finance, quality control, research, and classroom math. It helps you understand whether your data points stay close to the average or are widely dispersed.
Standard deviation formulas
Population standard deviation formula
σ = √( Σ(x - μ)² / N )
Use this when your dataset contains the entire population.
Sample standard deviation formula
s = √( Σ(x - x̄)² / (n - 1) )
Use this when your dataset is a sample from a larger population.
Mean formula
Mean = Σx / n
The mean is the average value of the dataset.
Variance formula
Variance = average squared distance from the mean
Standard deviation is the square root of variance.
Worked standard deviation example
Here is a common dataset used to explain population and sample standard deviation.
Example dataset:2, 4, 4, 4, 5, 5, 7, 9
For this list, the mean is 5. The population variance is 4 and the population standard deviation is 2. If the same list is treated as a sample, the sample variance is 4.571429 and the sample standard deviation is about 2.13809.
| Method | Count | Mean | Variance | Standard deviation |
|---|---|---|---|---|
| Population | 8 | 5 | 4 | 2 |
| Sample | 8 | 5 | 4.571429 | 2.13809 |
Standard deviation interpretation table
This table gives a simple way to understand what different standard deviation results usually suggest about a dataset.
| Result type | Meaning |
|---|---|
| Small standard deviation | Values are close to the mean, so the dataset has low spread or low variability. |
| Large standard deviation | Values are spread farther from the mean, so the dataset has more variation. |
| Standard deviation of 0 | Every value in the dataset is the same, so there is no variation. |
How to calculate standard deviation step by step
First, calculate the mean of the dataset by adding all values and dividing by the number of values. Then subtract the mean from each value to find the deviation of each point.
Next, square each deviation so all values become positive. Add those squared deviations together and divide by the correct denominator. Use N for population variance or n - 1 for sample variance.
Finally, take the square root of the variance. That result is the standard deviation, which tells you how much the data typically varies from the mean.
What standard deviation tells you about your data
A small standard deviation means the numbers are tightly clustered around the mean. A large standard deviation means the numbers are more spread out and vary more from the average.
For example, test scores with a low standard deviation are more consistent, while sales figures or measurements with a high standard deviation show greater variability. This makes standard deviation a key statistic in data analysis and decision-making.
When to use population vs sample standard deviation
Use population standard deviation when your dataset includes every value in the full group you want to study. For example, if you have the exam scores of all students in a class, that is population data.
Use sample standard deviation when your numbers are only a subset of a larger group. For example, if you survey only some customers from a full customer base, that is a sample. In that case, the formula uses n - 1 to better estimate the spread of the full population.
Where a standard deviation calculator is useful
A standard deviation calculator is useful in statistics homework, science experiments, market research, investment analysis, quality control, and performance tracking. It helps summarize the spread of data quickly and clearly.
It is especially helpful when comparing two datasets that may have the same mean but different levels of variability. In those cases, standard deviation provides insight that the average alone cannot.
Standard deviation calculator FAQ and common questions
How do you calculate standard deviation?
First find the mean, then measure each value’s distance from the mean, square those distances, average them as variance, and take the square root of the variance.
What is the difference between population and sample standard deviation?
Population standard deviation uses the full dataset and divides by N. Sample standard deviation uses a subset of a larger population and divides by n - 1.
What does a standard deviation of 0 mean?
It means every number in the dataset is exactly the same, so there is no variation around the mean.
What is variance in statistics?
Variance measures the average squared distance of data points from the mean. Standard deviation is the square root of variance.
Can I calculate standard deviation from a list of numbers?
Yes. Enter numbers separated by commas or spaces, and the calculator will return the count, mean, variance, and standard deviation.
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