Statistics Tool
Standard Deviation Calculator
Calculate Variance and Standard Deviation (Population or Sample) with detailed steps.
Separated by commas or spaces.
Σ The Formula
σ = √(Σ(x - μ)² / N) for population | s = √(Σ(x - μ)² / (N-1)) for sample
Real World Examples
Test Scores
Dataset: 85, 90, 78, 92, 88 → Mean = 86.6, σ = 4.93
Quality Control
Sample: 10.1, 10.3, 9.9, 10.2 → Mean = 10.125, s = 0.171
Consistent Data
Dataset: 50, 50, 50, 50 → σ = 0 (no variation)
High Variance
Dataset: 10, 50, 90 → Mean = 50, σ = 32.66 (spread out)
# About This Calculator
Standard Deviation is a fundamental statistical measure that quantifies the amount of variation or dispersion in a set of data values. A low standard deviation indicates that the data points tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.
It is calculated as the square root of variance. In finance, standard deviation is a standard proxy for risk or volatility. In manufacturing, it's used to measure process consistency. In education, it helps analyze test score distributions.
Population vs. Sample
- Population Standard Deviation ($\sigma$): Used when you have data for the entire population. Formula divides by $N$.
- Sample Standard Deviation ($s$): Used when you have a sample of data from a larger population. Formula divides by $N-1$ (Bessel's correction) to provide an unbiased estimate of the population standard deviation.
How To Use
- Enter your data points separated by commas or spaces.
- Select **Population** if you have all data, or **Sample** if it's a subset.
- Click **Calculate Deviation**.
- Review the Mean, Variance, Std Deviation, and step-by-step table.
Frequently Asked Questions
Why divide by N-1 for Sample Standard Deviation?+
Dividing by N-1 (Bessel's correction) corrects the bias in the estimation of the population variance. It makes the sample variance a better estimator of the true population variance, especially for small sample sizes.
What does standard deviation tell you?+
It tells you how spread out the data is. For example, if two classes have the same average test score of 75, but Class A has SD=2 and Class B has SD=15, Class A's scores were very consistent (mostly 73-77), while Class B's were all over the place (some 50s, some 100s).
How is it related to the Bell Curve?+
In a normal distribution (Bell Curve), approximately 68% of data falls within 1 standard deviation of the mean, 95% within 2 SDs, and 99.7% within 3 SDs. This is known as the Empirical Rule.
Can standard deviation be negative?+
No. Standard deviation is a measure of distance (spread) and is calculated as the square root of a sum of squares, so it must always be non-negative. It can be zero if all numbers in the dataset are identical.
Difference between Variance and Standard Deviation?+
Variance is the average of squared differences from the mean (units squared). Standard deviation is the square root of variance (same units as original data). Standard deviation is generally easier to interpret because it's on the same scale as the data.
Is Standard Deviation Calculator free to use?+
Yes, Standard Deviation Calculator on Matheric is completely free to use. We believe in accessible education and utility for everyone.