Standard Deviation Calculator - σ Calculator Online

What is Standard Deviation?

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a dataset. It tells us how spread out the data points are from the mean (average) value. A low standard deviation indicates that data points are close to the mean, while a high standard deviation indicates that data points are spread out over a wider range of values.

σ = √(Σ(x-μ)²/N) for population

Definition of Standard Deviation

Standard deviation is the square root of the variance. It measures the average distance between each data point and the mean. The standard deviation is expressed in the same units as the original data, making it easier to interpret than variance.

Types of Standard Deviation

Population Standard Deviation (σ): Used when you have data for the entire population
Sample Standard Deviation (s): Used when you have a sample of the population
Key Difference: Sample standard deviation uses n-1 in the denominator (Bessel's correction)
Sample SD is always larger: This accounts for the uncertainty in estimating the population parameter

Frequently Asked Questions

What's the difference between population and sample standard deviation?

Population standard deviation (σ) is used when you have data for the entire population and divides by N. Sample standard deviation (s) is used when you have a sample of the population and divides by n-1 (Bessel's correction). The sample standard deviation is always larger than the population standard deviation to account for the uncertainty in estimating the population parameter from a sample.

When should I use population vs sample standard deviation?

Use population standard deviation when you have data for the entire population you're studying. Use sample standard deviation when you have a sample of the population and want to make inferences about the population. In most real-world situations, you're working with samples, so sample standard deviation is more commonly used.

Can standard deviation be negative?

No, standard deviation cannot be negative. It's the square root of variance, and since variance is the average of squared deviations (which are always non-negative), the square root is also always non-negative. A standard deviation of zero means all data points are identical.

How do I interpret a high vs low standard deviation?

A low standard deviation indicates that data points are close to the mean (less spread), suggesting consistency or precision. A high standard deviation indicates that data points are spread out from the mean (more spread), suggesting greater variability or uncertainty. The interpretation depends on context - sometimes high variability is good (e.g., diverse performance), sometimes it's bad (e.g., inconsistent quality).