Standard Deviation Calculator
Calculate standard deviation, variance, mean, and other statistical measures from your dataset. Choose between sample (n-1) or population (n) calculations.
Statistical Measures
Interpretation
A low SD means data points are close to the mean; high SD means they're spread out.
What is Standard Deviation?
Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a dataset. It tells you how spread out the numbers are from their average (mean). A low standard deviation indicates that data points tend to be close to the mean, while a high standard deviation shows that data points are spread across a wider range of values.
For example, test scores of 85, 87, 86, 88, 85 have a low standard deviation (SD = 1.22) because all values are close together. Test scores of 60, 85, 95, 75, 100 have a higher standard deviation (SD = 15.62) because the values are more spread out. Standard deviation is widely used in finance to measure investment risk, in quality control to monitor manufacturing processes, and in research to analyze experimental data.
How to Calculate Standard Deviation
Calculating standard deviation involves five steps. First, find the mean by adding all numbers and dividing by the count. Second, subtract the mean from each number to get deviations. Third, square each deviation to eliminate negative values. Fourth, sum all squared deviations and divide by n (population) or n-1 (sample) to get variance. Fifth, take the square root of variance to get standard deviation.
For the dataset [10, 12, 14, 16, 18]: Mean = 14. Deviations = [-4, -2, 0, 2, 4]. Squared deviations = [16, 4, 0, 4, 16]. Sum of squared deviations = 40. For sample SD: Variance = 40/(5-1) = 10. Standard deviation = √10 = 3.16. The formula ensures that larger deviations have more impact on the result by squaring them.
Sample vs Population Standard Deviation
| Type | Symbol | Divisor | When to Use |
|---|---|---|---|
| Sample | s | n - 1 | When analyzing a subset of data |
| Population | σ (sigma) | n | When you have all possible data |
Use sample standard deviation (s) with n-1 divisor when working with a sample from a larger population. This is called Bessel's correction and provides an unbiased estimate. Use population standard deviation (σ) with n divisor only when you have data for the entire population. For most real-world scenarios, you're working with samples, so use the sample formula.
Standard Deviation Formulas
Sample Standard Deviation: s = √[Σ(xi - x̄)² / (n-1)]
Where s = sample SD, xi = each data point, x̄ = sample mean, n = number of data points
Population Standard Deviation: σ = √[Σ(xi - μ)² / n]
Where σ = population SD, xi = each data point, μ = population mean, n = total count
Standard Deviation vs Variance
Variance is the average of squared deviations from the mean, while standard deviation is the square root of variance. Both measure spread, but standard deviation has the same units as the original data, making it easier to interpret. For example, if measuring heights in centimeters, variance is in cm², but standard deviation is in cm.
Variance amplifies outliers because of squaring, making it useful for mathematical calculations. Standard deviation is preferred for reporting and interpretation because it's in the original scale. In finance, variance is used in portfolio theory, while standard deviation (volatility) is reported to investors. For statistical analysis, calculate variance as an intermediate step, then take its square root to get the more interpretable standard deviation.
Common Uses of Standard Deviation
Finance and Investing: According to investment analysts, standard deviation measures volatility and risk. A stock with 15% SD is riskier than one with 5% SD. Portfolio managers use SD to construct diversified portfolios. For investment calculations, see our investment calculator.
Quality Control: Manufacturing uses SD to monitor process consistency. Six Sigma methodology aims for less than 3.4 defects per million, requiring tight control (low SD) of product specifications. Control charts track whether processes stay within acceptable SD limits.
Academic Testing: Educational institutions use SD to compare test score distributions. A class with mean score 75 and SD 5 is more consistent than one with mean 75 and SD 20. Standardized tests like SAT report scores with SD to show relative performance.
Interpreting Standard Deviation with Normal Distribution
In a normal (bell curve) distribution, approximately 68% of data falls within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD. This is called the empirical rule or 68-95-99.7 rule. For example, if IQ scores have mean 100 and SD 15, then 68% of people score between 85-115, 95% between 70-130, and 99.7% between 55-145.
Coefficient of Variation
The coefficient of variation (CV) expresses SD as a percentage of the mean: CV = (SD / Mean) × 100%. It's useful for comparing variability between datasets with different units or scales. A dataset with mean 100 and SD 10 has CV = 10%, while one with mean 1000 and SD 100 also has CV = 10%, showing equal relative variability despite different absolute SD values.
Limitations and Considerations
Standard deviation assumes data is roughly normally distributed and can be misleading with skewed data or outliers. For example, the dataset [1, 2, 3, 4, 100] has a high SD (43.2) dominated by the outlier 100, which doesn't reflect the clustering of the first four values. In such cases, consider using median absolute deviation or interquartile range instead. For statistical comparisons, try our z-score calculator.
Last Updated: January 2026 | This calculator uses standard statistical formulas. Results are accurate for properly formatted numeric data. For complex statistical analysis or research purposes, consult a statistician.