CL
Back to Calculators

Standard Deviation Calculator - Calculate Variance & Mean Online

Free online standard deviation calculator. Calculate population and sample standard deviation, variance, mean, and range. Fast, accurate statistical analysis.

8 min read

iAbout This Calculator

The standard deviation calculator is an essential statistical tool for analyzing data dispersion. Standard deviation measures how spread out numbers are from their average (mean). A low standard deviation indicates data points cluster close to the mean, while a high standard deviation shows data is spread over a wider range. This calculator handles both population and sample standard deviation, along with variance, mean, and range calculations.

?How to Use

  1. 1

    Enter your numbers in the input field, separated by commas or spaces. For example: 10, 20, 30, 40, 50 or 10 20 30 40 50.

  2. 2

    Select whether your data represents a population (all possible values) or a sample (subset of a population).

  3. 3

    Click Calculate to see the standard deviation, variance, mean, and other statistics.

  4. 4

    Review the step-by-step breakdown to understand how each value was calculated.

  5. 5

    Use the results for statistical analysis, quality control, or academic purposes.

fFormula

\sigma = \sqrt{\frac{\sum_{i=1}^{N}(x_i - \mu)^2}{N}}

Population standard deviation divides by N (total count). Sample standard deviation divides by n-1 (Bessel's correction) to provide an unbiased estimate.

sigma
Population standard deviation
s
Sample standard deviation
xi
Each data value
mu
Population mean
N
Population size
n
Sample size

Examples

Test Scores Analysis

Inputs: numbers: 85, 90, 78, 92, 88
Sample SD: 5.36

Mean = 86.6, Variance = 28.7, Standard Deviation = 5.36. Scores are relatively consistent.

Temperature Readings

Inputs: numbers: 72, 68, 75, 71, 69, 73
Population SD: 2.38

For all readings in a controlled environment, population SD shows low variation.

Stock Price Changes

Inputs: numbers: -2.5, 3.1, -1.2, 4.5, -0.8, 2.3
Sample SD: 2.64

Daily percentage changes show moderate volatility in stock price.

Manufacturing Quality Control

Inputs: numbers: 10.2, 10.1, 9.9, 10.0, 10.1, 9.8
Population SD: 0.13

Very low SD indicates consistent product dimensions meeting quality standards.

Use Cases

Academic Research

Analyze experimental data, survey results, and test scores. Determine data reliability and identify outliers in research datasets.

Quality Control

Monitor manufacturing processes and product consistency. Lower standard deviation indicates better quality control and more uniform products.

Financial Analysis

Measure investment volatility and risk. Higher standard deviation in returns indicates greater investment risk.

Sports Analytics

Analyze player performance consistency. Athletes with lower standard deviation in scores are more reliable performers.

Weather Analysis

Study temperature variations, rainfall patterns, and climate data. Helps in understanding weather predictability for a region.

Frequently Asked Questions

What is the difference between population and sample standard deviation?
Population standard deviation (sigma) is used when you have data for an entire population and divides by N. Sample standard deviation (s) is used for a subset of data and divides by n-1 (Bessel's correction) to provide an unbiased estimate of the population standard deviation.
Why is standard deviation important?
Standard deviation quantifies the amount of variation in a dataset. It helps identify outliers, compare different datasets, assess risk in finance, ensure quality in manufacturing, and understand data reliability in research.
What is variance and how is it related to standard deviation?
Variance is the average of squared differences from the mean. Standard deviation is the square root of variance. While variance measures spread in squared units, standard deviation is in the same units as the original data, making it easier to interpret.
What does a high standard deviation mean?
A high standard deviation indicates that data points are spread out over a wide range of values, showing high variability. In contrast, a low standard deviation means data points are clustered close to the mean.
Can standard deviation be negative?
No, standard deviation cannot be negative. It is calculated as the square root of variance (which is always non-negative). The minimum value is 0, which occurs when all data points are identical.

Conclusion

Standard deviation is a fundamental statistical measure that helps you understand data variability. Whether you are analyzing test scores, monitoring quality control, assessing investment risk, or conducting research, our standard deviation calculator provides accurate results with detailed step-by-step breakdowns. Understanding standard deviation empowers better data-driven decision making across all fields.