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Standard Deviation Calculator

Calculate standard deviation, variance, mean, median, range, and other descriptive statistics for sample and population datasets.

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What is the Standard Deviation Calculator?

A Standard Deviation Calculator helps you measure how spread out numbers are within a dataset. It is one of the most commonly used statistical tools for analyzing variability, consistency, and dispersion.

Whether you're analyzing exam scores, financial returns, scientific measurements, survey results, or business metrics, standard deviation provides insight into how closely values cluster around the average.

This calculator supports both sample standard deviation and population standard deviation calculations. Simply enter your numbers separated by commas, spaces, or new lines, and instantly calculate:

  • Standard deviation
  • Variance
  • Mean (average)
  • Median
  • Minimum and maximum values
  • Range
  • Coefficient of variation (CV)
  • Dataset size

The tool performs all calculations directly in your browser, ensuring fast results and complete privacy.

How it works

Standard deviation measures the average distance of data points from the mean.

Step 1: Calculate the Mean

Add all values together and divide by the number of values.

Formula:

Mean (x̄) = Σx / n

Example dataset:

10, 15, 18, 22, 30

Mean:

(10 + 15 + 18 + 22 + 30) ÷ 5 = 19

Step 2: Find Deviations from the Mean

Subtract the mean from each value.

10 - 19 = -9
15 - 19 = -4
18 - 19 = -1
22 - 19 = 3
30 - 19 = 11

Step 3: Square Each Deviation

Squaring removes negative values and emphasizes larger differences.

81, 16, 1, 9, 121

Step 4: Calculate Variance

For a population:

Variance = Σ(x - μ)² / N

For a sample:

Variance = Σ(x - x̄)² / (n - 1)

The sample formula uses n − 1, known as Bessel's correction, to provide a more accurate estimate of the population variance.

Step 5: Calculate Standard Deviation

Take the square root of the variance.

Standard Deviation = √Variance

The resulting value indicates how much variation exists within the dataset.

Interpreting Standard Deviation

Low Standard Deviation

A low standard deviation means data points are clustered closely around the mean.

Example:

48, 49, 50, 51, 52

High Standard Deviation

A high standard deviation means values are spread farther from the mean.

Example:

10, 25, 50, 75, 90

Sample vs Population Standard Deviation

Use Sample Standard Deviation When

  • Analyzing a subset of a larger population
  • Conducting surveys
  • Working with research samples
  • Estimating population characteristics

Use Population Standard Deviation When

  • You have the complete dataset
  • Analyzing all employees, customers, or records
  • Evaluating full populations

Additional Statistics Provided

This calculator also computes:

  • Mean
  • Median
  • Variance
  • Minimum value
  • Maximum value
  • Range
  • Coefficient of variation (CV)
  • Number of observations

These statistics provide a more complete understanding of your dataset's distribution and variability.

Examples

Sample Standard Deviation

Calculate the sample standard deviation for a small dataset.

10, 15, 18, 22, 30
 
Mean: 19, Sample Standard Deviation: 7.84

Population Standard Deviation

Calculate the population standard deviation for an entire dataset.

2, 4, 4, 4, 5, 5, 7, 9
 
Mean: 5, Population Standard Deviation: 2

Class Test Scores

Analyze student scores using descriptive statistics.

72, 75, 81, 85, 88, 90, 94
 
Mean, Median, Variance, Standard Deviation, and Range

Frequently asked questions

What is standard deviation?

Standard deviation is a statistical measure that shows how much data points vary from the mean. A smaller standard deviation indicates that values are closer to the average, while a larger standard deviation indicates greater spread within the dataset.

What is the difference between sample and population standard deviation?

Population standard deviation is used when you have every value in the population. Sample standard deviation is used when analyzing only a subset of a larger population and uses the n − 1 adjustment to reduce estimation bias.

How do I calculate standard deviation?

To calculate standard deviation, first find the mean, then calculate each value's deviation from the mean, square those deviations, compute the variance, and finally take the square root of the variance.

Why is standard deviation important?

Standard deviation helps measure variability, risk, consistency, and reliability. It is widely used in statistics, finance, education, science, engineering, and business analysis.

What does a standard deviation of zero mean?

A standard deviation of zero means every value in the dataset is identical. Since there is no variation, all values are exactly equal to the mean.

Can standard deviation be negative?

No. Standard deviation can never be negative because it is the square root of variance, and variance is always zero or positive.

When should I use sample standard deviation?

Use sample standard deviation when your data represents only part of a larger population. This is common in surveys, experiments, market research, and statistical sampling.

When should I use population standard deviation?

Use population standard deviation when you have data for every member of the population being studied and do not need to estimate values from a sample.

What is variance?

Variance measures the average squared distance between data points and the mean. Standard deviation is simply the square root of variance, making it easier to interpret because it uses the original unit of measurement.

What is a good standard deviation value?

There is no universally good or bad standard deviation. Whether a value is considered high or low depends on the dataset, industry, and context being analyzed.

Can I calculate standard deviation for decimals and negative numbers?

Yes. Standard deviation calculations work with positive numbers, negative numbers, decimal values, and mixed datasets.

Is this Standard Deviation Calculator accurate?

Yes. The calculator uses standard statistical formulas for both sample and population standard deviation and performs all calculations directly in your browser.

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