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Standard Deviation Calculator

Calculate sample or population standard deviation, variance, mean, range, and coefficient of variation from a dataset.

Standard Deviation Calculator

Population: Complete dataset. Sample: Part of a larger population.

Results
Dataset (8 values)
[2, 4, 4, 4, 5, 5, 7, 9]

Standard Deviation (σ)

2.0000

Population std dev

Variance (σ²)

4.0000

Population variance

Mean (μ)

5.0000

Range

7.0000

2.00 to 9.00

Interpretation

• Data spread: Moderate variability (40.0% coefficient of variation)
• About 68% of values fall within ± 1 standard deviation: 3.00 to 7.00
• About 95% of values fall within ± 2 standard deviations: 1.00 to 9.00

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What is Standard Deviation Calculator?

Standard deviation is a measure of how spread out numbers are from their average (mean). It tells you whether your data points are clustered close to the mean or scattered far apart. A low standard deviation means data points are close to the mean, while a high standard deviation indicates they are spread out over a wider range.

Population vs Sample Standard Deviation

  • Population Standard Deviation (σ): Used when you have data for the entire population. Divides by N.
  • Sample Standard Deviation (s): Used when you have a sample from a larger population. Divides by N-1 (Bessel's correction).
  • When to use which: Use population if you have complete data, sample if your data represents part of a larger group.

Understanding the Results

  • Variance: The average of squared differences from the mean
  • Standard Deviation: The square root of variance, in the same units as your data
  • Coefficient of Variation: Standard deviation divided by mean, useful for comparing variability
  • 68-95-99.7 Rule: In normal distributions, ~68% of data falls within 1σ, ~95% within 2σ, ~99.7% within 3σ

Real-World Applications

  • Quality Control: Manufacturing tolerances and defect rates
  • Finance: Investment risk assessment and portfolio volatility
  • Education: Test score analysis and grade distributions
  • Healthcare: Medical measurements and treatment effectiveness
  • Research: Experimental data analysis and statistical significance

Interpreting Standard Deviation

  • Lower standard deviation = more consistent, predictable data
  • Higher standard deviation = more variable, less predictable data
  • Compare standard deviations to understand relative variability
  • Consider the context and units when interpreting magnitude

Worked Standard Deviation Example

For the population dataset 2, 4, 4, 4, 5, 5, 7, 9, the mean is 5.

The squared differences sum to 32, so population variance is 32 / 8 = 4 and population standard deviation is √4 = 2.

If these values are a sample, divide by 7 instead; the sample standard deviation is approximately 2.138.

Method reference: NIST Engineering Statistics Handbook: measures of scale.



FAQ - Standard Deviation Calculator

Use population standard deviation when you have data for the complete population you're studying. Use sample standard deviation when your data is a subset representing a larger population. For example, if you survey all employees in a small company, use population. If you survey 100 people to represent a city, use sample.