How to Use
- Enter data
Input numbers separated by commas or one number per line.
- Select type
Choose population or sample standard deviation.
- View results
Click Calculate to see the standard deviation, variance, and mean.
What is standard deviation?
Standard deviation is a measure of dispersion that summarizes, in a single number, how far data values lie from their mean. Two data sets can share the same mean yet behave very differently in terms of stability and risk if their standard deviations differ.
Why it matters
Standard deviation reveals the variability that the mean alone cannot show. A small value means the data cluster tightly around the mean and are highly predictable, while a large value means they are widely spread and carry more uncertainty.
Where it is used
- Finance: the standard deviation of prices or returns measures investment risk (volatility).
- Quality control: methods such as Six Sigma manage process spread to reduce defect rates.
- Testing and research: it gauges the uniformity of score distributions and the error range of experimental measurements.
For a normal distribution, the so-called 68-95-99.7 rule makes interpretation intuitive: about 68% of the data fall within the mean ±1σ and about 95% within ±2σ.
Formula
The population standard deviation is σ = √(Σ(xᵢ − μ)² / N), and the sample standard deviation is s = √(Σ(xᵢ − x̄)² / (N − 1)). Here xᵢ is each data point, μ (or x̄) is the mean, and N is the number of data points.
Step-by-step example
For the data {2, 4, 4, 4, 5, 5, 7, 9}:
- 1. Mean μ = (2+4+4+4+5+5+7+9) / 8 = 40 / 8 = 5
- 2. Sum of squared deviations Σ(xᵢ−μ)² = 9+1+1+1+0+0+4+16 = 32
- 3. Population variance = 32 / 8 = 4 → σ = √4 = 2
- 4. Sample variance = 32 / (8−1) = 4.5714 → s ≈ 2.138
Because the sample divides by N−1, its standard deviation is always slightly larger than the population value.