Math & Statistics
Standard Deviation
Calculate mean, variance, and standard deviation for a dataset.
Standard deviation measures the spread of data around the mean. A small SD means data is clustered; a large SD means it is spread out. It is fundamental to statistics, finance, quality control, and scientific research.
Standard Deviation
2.6693
- mean
- 14.6250
- variance
- 7.1250
- count
- 8 values
- range
- 8.0000
Formula and method
σ = √[Σ(xi - μ)² / (n-1)] for sample; √[Σ(xi - μ)² / n] for population
Sample SD (n-1 denominator, Bessel's correction) is used when the data is a sample of a larger population. Population SD (n denominator) when you have the complete dataset.
Worked examples
Test scores
- Data Points (comma-separated):
- 72, 85, 91, 68, 79, 88, 74, 83
- Calculation Type:
- Sample (÷ n-1, for data samples)
Standard Deviation: 8.1416
Stock daily returns
- Data Points (comma-separated):
- 1.2, -0.8, 2.1, -1.5, 0.9, -0.3, 1.8
- Calculation Type:
- Sample (÷ n-1, for data samples)
Standard Deviation: 1.3680
Assumptions and limits
- Sample SD uses Bessel's correction (n-1)
- Data is numeric and properly separated
This tool provides an educational estimate. Rules, rates and personal circumstances can change the answer. Review the assumptions before relying on a result. Report a correction.
Common mistakes to avoid
- Using population SD on sample data — this underestimates the true variability of the population.
- Forgetting to square the deviations before averaging — squaring ensures negatives don't cancel positives.
- Confusing standard deviation with standard error — SE = SD / √n and applies to sampling distributions.
Frequently asked questions
When should I use sample vs population SD?
Use sample SD (n-1) when your data is a subset of a larger group (a survey sample, test scores from one class, etc). Use population SD (n) only when you have data for the entire population.
What is the 68-95-99.7 rule?
In a normal distribution: 68% of data falls within ±1 SD, 95% within ±2 SD, and 99.7% within ±3 SD. This is useful for identifying outliers and understanding data spread.
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