Standard Deviation Calculator
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About Standard Deviation
Standard deviation is a measure of how spread out numbers are from their average value. A low standard deviation indicates that values tend to be close to the mean, while a high standard deviation indicates that values are spread out over a wider range.
Key Concepts:
| Term | Definition | Formula |
|---|---|---|
| Mean (Average) | Sum of all values divided by count | µ = (Sx)/N |
| Variance | Average of squared differences from mean | s² = S(x-µ)²/N (population) s² = S(x-x¯)²/(n-1) (sample) |
| Standard Deviation | Square root of variance | s = vs² (population) s = vs² (sample) |
Population vs Sample Standard Deviation:
- Population standard deviation (s): Use when your data includes all members of the population you're studying
- Sample standard deviation (s): Use when your data is only a sample from a larger population (uses n-1 in denominator)
Interpretation Guidelines:
- About 68% of values fall within 1 standard deviation of the mean
- About 95% of values fall within 2 standard deviations
- About 99.7% of values fall within 3 standard deviations
- These percentages assume a normal (bell-shaped) distribution
Common Applications:
- Quality control in manufacturing
- Risk assessment in finance
- Test scoring in education
- Scientific research data analysis
- Weather forecasting models