The Standard Deviation Calculator helps you measure how spread out your data values are. Enter your numbers to calculate the mean, variance, standard deviation, median, and range. It also displays a graph to help you understand your data distribution at a glance.
Calculator Inputs
Separate values with commas, spaces, semicolons, or new lines.
Choose population when your data contains the entire group, or sample when it represents part of a larger group.
Try an Example
Calculation Results
Standard Deviation Formula
Standard deviation measures how far data values typically spread from their mean. The formula depends on whether you are working with an entire population or a sample of a larger population.
1. Population Standard Deviation Formula
Use this formula when your data contains the entire population.
Where:
- σ = Population standard deviation
- xᵢ = Each data value
- μ = Population mean
- N = Total number of population values
Example: Population Standard Deviation
Suppose the data values are 2, 4, 6, 8, 10.
Step 1: Calculate the mean.
Step 2: Calculate the squared differences.
| Value | Difference from mean | Squared difference |
|---|---|---|
| 2 | −4 | 16 |
| 4 | −2 | 4 |
| 6 | 0 | 0 |
| 8 | 2 | 4 |
| 10 | 4 | 16 |
| Sum of squared differences | 40 | |
Step 3: Apply the formula.
Population standard deviation = 2.828
2. Sample Standard Deviation Formula
Use this formula when your data represents a sample from a larger population. The denominator is n − 1 instead of n.
Where:
- s = Sample standard deviation
- xᵢ = Each data value
- x̄ = Sample mean
- n = Number of values in the sample
Example: Sample Standard Deviation
Use the same data: 2, 4, 6, 8, 10.
Step 1: Calculate the sample mean.
Step 2: Find the sum of squared differences.
Step 3: Apply the sample formula.
Sample standard deviation = 3.162
3. Population vs. Sample Standard Deviation
| Feature | Population | Sample |
|---|---|---|
| Symbol | σ | s |
| Denominator | N | n − 1 |
| Data | Entire population | Sample of a population |
| Example result | 2.828 | 3.162 |
4. How to Use the Standard Deviation Calculator
- Enter your numbers, separated by commas or spaces.
- Select population or sample standard deviation.
- Choose your preferred decimal precision.
- Click Calculate to see the standard deviation and other statistics.
- Review the graph to understand how your data values are distributed.
5. What Does Standard Deviation Tell You?
- Low standard deviation: Values are closer to the mean.
- High standard deviation: Values are more spread out.
- Zero standard deviation: All values are identical.
Standard deviation is commonly used in statistics, education, research, finance, quality control, and data analysis. Always interpret the result in the context of your dataset.