Standard Deviation Calculator with Steps
Calculate standard deviation with complete step-by-step explanations. Learn the statistical process, formulas, and data interpretation. Free calculator.
Quick Answer
Standard deviation measures data spread from the mean. For population: σ = √(Σ(x-μ)²/N). For sample: s = √(Σ(x-x̄)²/(n-1)). Lower values indicate data clustered near mean, higher values show greater spread. Essential for statistical analysis and quality control.
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Calculate With Full ToolWhat is Standard Deviation?
Standard deviation measures the amount of variation or dispersion of a set of values. A low standard deviation indicates that values tend to be close to the mean, while a high standard deviation indicates values are spread out over a wider range.
How Standard Deviation Works
The calculation involves finding the mean, then calculating the squared differences from the mean, averaging these squared differences, and finally taking the square root. This process gives us the standard deviation in the same units as the original data.
Population vs Sample Standard Deviation
Population standard deviation uses N in the denominator when you have complete data. Sample standard deviation uses n-1 (Bessel's correction) when working with a sample of the population, providing an unbiased estimate of the population parameter.
Standard Deviation Formulas
Population: σ = √(Σ(x-μ)²/N)
Sample: s = √(Σ(x-x̄)²/(n-1))
Population Formula: σ = √(Σ(x-μ)²/N)
Sample Formula: s = √(Σ(x-x̄)²/(n-1))
Where: μ = population mean, x̄ = sample mean, N = population size, n = sample size
Variance: σ² = Σ(x-μ)²/N (population) or s² = Σ(x-x̄)²/(n-1) (sample)
Step-by-Step Example
Example: Calculate standard deviation of [10, 15, 12, 18, 20]
Step 1: Calculate the mean: (10 + 15 + 12 + 18 + 20) ÷ 5 = 15
Step 2: Find differences from mean: [-5, 0, -3, 3, 5]
Step 3: Square the differences: [25, 0, 9, 9, 25]
Step 4: Sum squared differences: 25 + 0 + 9 + 9 + 25 = 68
Step 5: Divide by n-1: 68 ÷ 4 = 17 (variance)
Step 6: Take square root: √17 ≈ 4.12 (standard deviation)
This example shows the complete process of calculating sample standard deviation. The result of 4.12 indicates that, on average, data points deviate from the mean by about 4.12 units.
Who Should Use This Calculator?
Data Analysts
Analyze data variability and quality metrics
Researchers
Evaluate experimental data and statistical significance
Quality Control
Monitor process consistency and product variation
Students
Learn statistical concepts and complete assignments
Frequently Asked Questions
What's the difference between variance and standard deviation?
Variance is the average of squared deviations from the mean, while standard deviation is the square root of variance. Standard deviation is in the same units as the original data, making it more interpretable for practical applications.
When should I use sample vs population standard deviation?
Use population standard deviation when you have complete data for the entire population. Use sample standard deviation when working with a subset of data, as it provides an unbiased estimate of the population parameter.
What does a high standard deviation indicate?
A high standard deviation indicates that data points are spread out over a wider range from the mean. This suggests greater variability or inconsistency in the data. The interpretation depends on the context and scale of your data.
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