Standard Deviation Calculator
Calculate population and sample standard deviation with detailed statistics
σ = √[Σ(xᵢ - μ)² / N]
Population standard deviation - use when you have data for the entire population
s = √[Σ(xᵢ - x̄)² / (n-1)]
Sample standard deviation - use when you have a sample from a larger population
Minimum 2 numbers required
Quick Examples
Standard Deviation
=
Mean (μ/x̄)
Variance
Count (n)
Sum (Σx)
Min
Max
Range
Sum of Squares
Step-by-Step Breakdown
Step 1: Your Data
Step 2: Calculate Mean
μ = () / =
Step 3: Calculate Deviations
| x | x - μ | (x - μ)² |
|---|---|---|
Step 4: Calculate Variance
= Σ(x - μ)² / =
Step 5: Standard Deviation
= √ =
Note: Step-by-step breakdown is hidden for datasets larger than 10 values for readability.
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About Standard Deviation Calculator
What is Standard Deviation?
Standard deviation is a measure of how spread out numbers are from the mean (average). A low standard deviation indicates that data points are close to the mean, while a high standard deviation indicates more spread.
Population vs Sample Standard Deviation
Population Standard Deviation (σ)
Used when you have data for the entire population:
σ = √[Σ(xᵢ - μ)² / N]
Where:
- σ = population standard deviation
- μ = population mean
- N = total number of values
- xᵢ = each value in the dataset
Sample Standard Deviation (s)
Used when you have a sample from a larger population:
s = √[Σ(xᵢ - x̄)² / (n-1)]
Where:
- s = sample standard deviation
- x̄ = sample mean
- n = sample size
- Uses (n-1) for Bessel's correction
How to Calculate Standard Deviation
- Find the mean (average) of your data
- Subtract the mean from each value
- Square each difference
- Find the average of squared differences
- Take the square root
Example Calculation
Dataset: 2, 4, 4, 4, 5, 5, 7, 9
- Mean = (2+4+4+4+5+5+7+9) / 8 = 5
- Differences from mean: -3, -1, -1, -1, 0, 0, 2, 4
- Squared differences: 9, 1, 1, 1, 0, 0, 4, 16
- Sum of squares = 32
- Population variance = 32/8 = 4
- Population SD = √4 = 2
When to Use Each Type
| Scenario | Use |
|---|---|
| Survey of all employees | Population SD |
| Sample from a larger group | Sample SD |
| Scientific measurements | Depends on context |
| Quality control | Usually Sample SD |
Key Statistics Explained
- Mean: The average of all values
- Variance: The square of standard deviation
- Range: Difference between max and min values
- Sum of Squares: Used in many statistical calculations
Applications
- Finance - Measuring investment volatility
- Quality Control - Process variation analysis
- Research - Data variability assessment
- Education - Test score analysis
- Weather - Temperature variations
Tip: For most practical applications with sample data, use Sample Standard Deviation (s) which provides an unbiased estimate of the population parameter.