Standard Deviation Calculator
About Standard Deviation Calculator
Standard deviation (SD, denoted by σ for population and s for sample) is a statistic that measures the dispersion or spread of a dataset relative to its mean. It is calculated as the square root of the variance. A low standard deviation means values are close to the mean, while a high standard deviation indicates values are more spread out. Standard deviation is widely used in statistics, science, finance, quality control, and many other fields to quantify variability, risk, and reliability.
Why do we use standard deviation?
Standard deviation helps us understand how much the values in a dataset deviate from the mean. It is essential for comparing distributions, measuring risk (e.g., in finance), assessing quality (e.g., in manufacturing), and interpreting experimental results. It is also used to calculate confidence intervals and margins of error.
Step-by-Step Calculation Example
Let's calculate the sample standard deviation for the numbers: 30, 20, 28, 24, 11, 17.
| xi | xi - x̄ | (xi - x̄)² |
|---|---|---|
| 30 | 8.33 | 69.4 |
| 20 | -1.67 | 2.78 |
| 28 | 6.33 | 40 |
| 24 | 2.33 | 5.43 |
| 11 | -10.67 | 113.85 |
| 17 | -4.67 | 21.80 |
| Sum | 253.26 | |
- Calculate the mean (x̄): (30 + 20 + 28 + 24 + 11 + 17) / 6 = 21.67
- Subtract the mean from each value and square the result.
- Sum the squared differences: 253.26
- Divide by n-1 (sample): 253.26 / (6-1) = 50.65 (sample variance)
- Take the square root: √50.65 ≈ 7.12 (sample standard deviation)
Related Statistics
- Variance (σ² or s²): The average of the squared differences from the mean.
Variance = SD² - Deviation score (xi - μ or xi - x̄): The difference between each value and the mean.
- Sum of squares (Σ(xi - μ)²): The sum of squared deviations, a key step in variance and SD calculation.
- Mean (μ or x̄): The average value.
- Margin of error: An estimate of the uncertainty in the mean, often used in confidence intervals.
Common Tools & Related Calculators
- Variance Calculator
- Coefficient of Variation Calculator
- Covariance Calculator
- Correlation Coefficient Calculator
- Mean, Median, Mode Calculators
- Quartile, Z Score, P Value, Confidence Interval Calculators
References
- Standard deviation. Investopedia.
- Understanding and calculating standard deviation. Scribbr.
- Standard deviation: Calculating step by step. Khan Academy.