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 measure of the amount of variation or dispersion in a dataset. It quantifies how spread out the values are from the mean (average). A low standard deviation means values cluster tightly around the mean; a high standard deviation means values are spread across a wider range.

This statistic is crucial in quality control (manufacturing tolerances), finance (investment risk/volatility), education (grade distributions), science (experimental error), and data analysis (outlier detection). Understanding standard deviation helps you assess consistency, reliability, and variability in any dataset.

There are two types: Population standard deviation (σ) when you have data for the entire group, and Sample standard deviation (s) when you have data from a subset. Sample SD uses N-1 (Bessel's correction) instead of N to provide an unbiased estimate of population variance.

This calculator computes both variance (average of squared differences) and standard deviation (square root of variance), showing detailed step-by-step calculations. It automatically handles both population and sample modes, making it perfect for statistics homework, research analysis, or quality control applications.

How To Use

  1. Enter data points (e.g., "10, 12, 23, 23, 16, 23, 21, 16").
  2. Select Population or Sample mode.
  3. Click Calculate.

Frequently Asked Questions

Difference between Population and Sample SD?+

Population SD (σ) divides by N when you have ALL the data. Sample SD (s) divides by N-1 (Bessel's correction) when estimating from a sample. Use sample SD for most real-world scenarios where you don't have complete data.

What does a high standard deviation mean?+

High SD means data points are spread far from the mean - high variability. For example, test scores of 50, 70, 90 have higher SD than 68, 70, 72. In investing, high SD means high volatility/risk. In manufacturing, it means inconsistent quality.

How is standard deviation different from variance?+

Variance is the average of squared differences from the mean. Standard deviation is the square root of variance. SD is in the same units as your data (easier to interpret), while variance is in squared units. Both measure spread.

What's a 'good' standard deviation?+

It depends on context! For test scores (0-100), SD of 10-15 is typical. For manufacturing tolerances, lower is better. In finance, compare SD to mean (coefficient of variation). Generally, lower SD means more consistency/predictability.

What is the 68-95-99.7 rule?+

For normal distributions: ~68% of data falls within 1 SD of the mean, ~95% within 2 SD, ~99.7% within 3 SD. This helps identify outliers and understand data distribution. Values beyond 3 SD are considered unusual.

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.

About

Standard Deviation is a measure of the amount of variation or dispersion in a dataset. It quantifies how spread out the values are from the mean (average). A low standard deviation means values cluster tightly around the mean; a high standard deviation means values are spread across a wider range.

This statistic is crucial in quality control (manufacturing tolerances), finance (investment risk/volatility), education (grade distributions), science (experimental error), and data analysis (outlier detection). Understanding standard deviation helps you assess consistency, reliability, and variability in any dataset.

There are two types: Population standard deviation (σ) when you have data for the entire group, and Sample standard deviation (s) when you have data from a subset. Sample SD uses N-1 (Bessel's correction) instead of N to provide an unbiased estimate of population variance.

This calculator computes both variance (average of squared differences) and standard deviation (square root of variance), showing detailed step-by-step calculations. It automatically handles both population and sample modes, making it perfect for statistics homework, research analysis, or quality control applications.

Related Tools

Mean, Median, Mode

Average Calculator

Box Plot Maker

Histogram Maker