Standard Deviation Calculator - σ & Variance Std Dev Calculator
Enter a list of numbers to instantly calculate standard deviation, variance, mean, and more.
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Your Results
Population Std Dev (σ) 4.8990
Formula σ = √(Σ(xi − x̄)² / N)
Population Std Dev (σ) 4.8990
Sample Std Dev (s) 5.2372
Population Variance (σ²) 24.0000
Sample Variance (s²) 27.4286
Mean (x̄) 18
Sum 144
Count (N) 8
Min 10
Max 23
Range 13
Sorted Data (Ascending)
10, 12, 16, 16, 21, 23, 23, 23
Standard Deviation Calculator - Guide
What is Standard Deviation?
Standard deviation is a measure of how spread out numbers are from the mean (average). A low standard deviation means the data points are close to the mean, while a high standard deviation indicates the data is spread over a wide range of values.
Formulas
1. Population Standard Deviation (σ)
σ = √[ Σ(xi − μ)² / N ]
Used when data represents the entire population.
2. Sample Standard Deviation (s)
s = √[ Σ(xi − x̄)² / (N − 1) ]
Used when data is a sample from a larger population (Bessel's correction).
3. Variance
Variance = (Standard Deviation)²
Examples
- Data: 2, 4, 4, 4, 5, 5, 7, 9 → Mean = 5, Population σ = 2, Sample s ≈ 2.14
- Data: 10, 10, 10 → σ = 0 (no spread)
- Data: 1, 100 → σ = 49.5 (large spread)
When to Use Each
- Population σ: When your data includes every member of the group (e.g., all students in a class).
- Sample s: When your data is a subset of a larger group (e.g., survey respondents).
- Variance: Useful in probability theory and statistical inference; units are squared.
How This Calculator Works
- Enter data: Type or paste your numbers separated by commas or spaces.
- Instant results: Population and sample standard deviation, variance, mean, sum, count, min, max, and range update in real time.
- Sorted view: See your data sorted in ascending order below the results.
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