Formula & Calculator
Standard Deviation (Sample)
Measures how spread out a set of sample data values are from their mean, using n-1 in the denominator to correct for sample bias.
Interpretation
s = √( Σ(xᵢ − x̄)² / (n−1) ). Measures spread of sample data. Used in statistics, quality control, and finance. Essential for data analysis.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| s | Sample standard deviation | |
| xi | Each individual data value | |
| x_bar | Sample mean | |
| n | Number of data points in the sample |
What it means
The sample standard deviation (s) is a measure of the dispersion or spread of a set of data points relative to the sample mean. It is calculated by taking the square root of the variance. The denominator is n−1 (Bessel’s correction) to provide an unbiased estimate of the population standard deviation. This metric is widely used in statistical analysis to quantify uncertainty, in quality control (six sigma), in finance (volatility of returns), and in many scientific fields. A small s indicates data points are close to the mean; large s indicates wide spread. Understanding standard deviation is essential for interpreting confidence intervals, hypothesis testing, and regression analysis.
Worked example
Sample Standard Deviation – Two Examples
Real‑World| Parameter | Value |
|---|---|
| n | 5 |
| Sum sq dev | 10 |
| Parameter | Value |
|---|---|
| n | 5 |
| Sum sq dev | 250 |
Common mistakes
- Sample standard deviation: s = √[Σ(xᵢ − x̄)²/(n−1)] – denominator is n−1 (degrees of freedom).
- Population: For population σ, use N in denominator (not n−1).
- Mean: x̄ is the sample mean – compute it first.
- Units: The standard deviation has the same units as the data.
- Zero: If all values are equal, s = 0.
Applications
The sample standard deviation, s = √(Σ(xi − x̄)²/(n−1)), measures the spread of a dataset around its mean. It is essential in quality control, engineering tolerances, and statistical inference. Engineers use it to assess process capability, to set control limits, and to evaluate measurement uncertainty. In finance, it measures risk (volatility). In research, it quantifies variability. By understanding standard deviation, professionals can judge the consistency of processes, identify outliers, and make data‑driven decisions. This formula is a cornerstone of descriptive statistics and is used in virtually every field that handles numerical data.
- Quality control and process capability (Six Sigma)
- Measurement system analysis and uncertainty estimation
- Risk assessment in finance and investment
- Experimental data analysis and hypothesis testing
- Performance variation analysis in manufacturing
Frequently Asked Questions
The sample standard deviation (s) measures the spread of data around the sample mean. It is calculated as s = √( Σ(xᵢ – x̄)² / (n – 1) ), where n is the sample size, x̄ is the sample mean, and the sum is over all data points.
Dividing by n instead of n–1, which is the correction needed when working with a sample rather than a full population.
Population standard deviation uses n (number of data points) in the denominator; sample standard deviation uses n–1 to correct for bias.
This is Bessel's correction. It accounts for the loss of one degree of freedom when estimating the sample mean from the same data.
Variance is the square of the standard deviation: s² for sample, σ² for population.
It is zero, since all values are the same and there is no spread.
The same units as the original data (e.g., if data is in kg, s is in kg).
In a normal distribution, about 68% of data falls within ±1σ, 95% within ±2σ, and 99.7% within ±3σ.
- Measuring variability in data.
- Quality control (process capability).
- Risk assessment in finance.
Standard error (SE) is the standard deviation of the sample mean: SE = s/√n. It measures the precision of the mean estimate.