Standard Deviation Calculator
Calculate standard deviation, variance, and mean for a set of numbers.
Data Input
Enter numbers separated by commas
Parsed Data:
5, 10, 15, 20, 25
Count: 5 values
Statistical Results
Population Standard Deviation
7.0711
Sample Standard Deviation
7.9057
Mean (Average)
15.0000
Population Variance
50.0000
Sample Variance
62.5000
Statistics:
Count: 5
Sum: 75.00
Min: 5.00
Max: 25.00
About This Calculator
The Standard Deviation Calculator helps you calculate the standard deviation, variance, and mean of a set of numbers. Standard deviation measures how spread out the numbers are from the mean (average) value.
This calculator supports both population standard deviation (divide by n) and sample standard deviation (divide by n-1). Enter your data as comma-separated values, and the calculator will automatically compute all statistical measures.
Formula & Calculation Method
Standard Deviation Formulas:
Mean (μ):
μ = (Σx) / n
Population Variance (σ²):
σ² = Σ(x - μ)² / n
Population Standard Deviation (σ):
σ = √[Σ(x - μ)² / n]
Sample Variance (s²):
s² = Σ(x - x̄)² / (n - 1)
Sample Standard Deviation (s):
s = √[Σ(x - x̄)² / (n - 1)]
Where n is the number of data points, x is each data value, μ (or x̄) is the mean, and Σ represents summation. Use population standard deviation when you have data for the entire population, and sample standard deviation when you have a sample from a larger population.
How to Use This Calculator
- Enter Data: Input your numbers separated by commas (e.g., 5, 10, 15, 20, 25).
- View Results: See mean, variance, population standard deviation, and sample standard deviation.
- Choose Type: Use population standard deviation for entire population, sample for a sample.
- Interpret: Lower standard deviation means data is closer to the mean; higher means more spread out.
Frequently Asked Questions
What is the difference between population and sample standard deviation?
Population standard deviation (σ) divides by n and is used when you have data for the entire population. Sample standard deviation (s) divides by n-1 (Bessel's correction) and is used when you have a sample from a larger population. The sample version is slightly larger to account for estimation uncertainty.
What does standard deviation tell us?
Standard deviation measures the spread or dispersion of data around the mean. A low standard deviation means data points are close to the mean, while a high standard deviation means they are spread out over a wider range.
What is a good standard deviation?
There is no universal "good" standard deviation - it depends on the context and what you're measuring. In general, you want to compare standard deviation relative to the mean (coefficient of variation) or to other similar datasets.
How is standard deviation related to variance?
Standard deviation is the square root of variance. Variance measures the average squared deviation from the mean, while standard deviation measures the average deviation in the same units as the original data.
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Calculator by
Prof. Emily Watson, PhD
PhD in Applied MathematicsStatistics Certification