Calculate standard deviation instantly with step-by-step breakdown. Supports both sample (n-1) and population (n) standard deviation, variance, and mean calculations with complete formulas.
This standard deviation calculator computes both sample and population standard deviation with complete step-by-step calculations. Enter your dataset to instantly calculate the mean, variance, and standard deviation with detailed formulas and working shown for each step.
This tool is ideal for students, exams, statistics homework, and data analysis.
Use Sample when analyzing a subset of data (most common in research and exams).
| Symbol | Description |
|---|---|
| σ | Population standard deviation |
| s | Sample standard deviation |
| μ | Population mean |
| x̄ | Sample mean |
| ∑ | Summation operator (capital sigma) |
| x | Individual data value |
| i | Index of observation |
| n | Number of observations |
The sum is the total of all data values.
The size is the number of data points in the dataset.
The mean (average) is the sum of all values divided by the number of values.
For a Population
For a Sample
The sum of squares is the sum of squared differences from the mean.
For a Population
For a Sample
Standard deviation measures how spread out values are from the mean. It is the square root of variance.
For a Population
For a Sample
Variance is the squared form of standard deviation, representing the average squared deviation from the mean.
For a Population
For a Sample
Reviewed by: VIP Calculator Statistics Research Team
This Standard Deviation Calculator follows internationally accepted statistical formulas used in mathematics, data science, finance, and academic research.
Transparency Note: Results are intended for educational, analytical, and informational purposes. Minor rounding differences may occur depending on dataset size and precision.
Standard deviation measures the amount of variation in a set of values.
Sample Standard Deviation: s = √[Σ(x - x̄)² / (n - 1)]
Population Standard Deviation: σ = √[Σ(x - μ)² / n]
Where: x̄ or μ = mean, x = each value, n = number of values
Population standard deviation (σ) uses n in the denominator and is used when you have data for the entire population. Sample standard deviation (s) uses n-1 to correct for bias when analyzing a subset of data representing a larger population.
Use sample standard deviation (s) when working with a subset of data from a larger population—this is most common in research, surveys, and exams. Use population standard deviation (σ) only when you have complete data for every member of the entire population.
A high standard deviation indicates that data points are spread out over a wider range of values, showing more variability. Values vary significantly from the mean, meaning the data is less consistent or more dispersed.
Variance measures how far data points spread from the mean. It's the square of standard deviation and is crucial for understanding data dispersion. Variance is used in advanced statistical methods like ANOVA, regression analysis, and quality control to assess data variability.
Yes, the calculator accepts decimal numbers and provides precise calculations for any numerical dataset including integers, decimals, and negative values.