Standard Deviation Calculator
Calculate standard deviation, variance, and other statistical measures for your dataset.
Enter Your Data
Provide numerical data points separated by commas
Enter numbers separated by commas. Example: 23, 45, 67, 89, 12, 34, 56, 78, 90, 11
Statistical Results
Based on your dataset analysis
Enter your data points and click "Calculate Standard Deviation" to see your results
About Standard Deviation
What is Standard Deviation?
Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.
How Standard Deviation is Calculated
Population Standard Deviation Formula:
σ = √[Σ(x - μ)²/N]
Where:
- • σ = standard deviation
- • Σ = sum of
- • x = each value in the dataset
- • μ = mean of the dataset
- • N = number of values in the dataset
Population vs Sample Standard Deviation
Population Standard Deviation
- • Used when you have data for the entire population
- • Denominator is N (total number of values)
- • Symbol: σ (sigma)
- • Formula: σ = √[Σ(x - μ)²/N]
Sample Standard Deviation
- • Used when you have a sample of the population
- • Denominator is N-1 (Bessel's correction)
- • Symbol: s
- • Formula: s = √[Σ(x - μ)²/(N-1)]
Interpreting Standard Deviation
- Normal Distribution: In a normal distribution, approximately 68% of values fall within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.
- Low Standard Deviation: Values are clustered closely around the mean, indicating consistency and low variability.
- High Standard Deviation: Values are spread out over a wider range, indicating high variability and diversity in the dataset.
- Zero Standard Deviation: All values in the dataset are identical.
Applications of Standard Deviation
- Finance: Measuring investment risk and volatility
- Quality Control: Monitoring manufacturing processes and product consistency
- Research: Analyzing experimental data and statistical significance
- Education: Grading on a curve and assessing student performance
- Weather: Analyzing temperature variations and climate patterns
- Sports: Evaluating player performance consistency
Tips for Using This Calculator
- Enter numbers separated by commas (e.g., 1, 2, 3, 4, 5)
- Ensure all values are numerical - non-numeric values will be ignored
- Use decimal points for precise values (e.g., 3.14, 2.718)
- The calculator works with both positive and negative numbers
- For large datasets, consider using statistical software for more advanced analysis
- Remember the difference between population and sample standard deviation when interpreting results