Calculations.one
Statistics • Research • Finance • School

Standard Deviation Calculator

Enter any data set to instantly calculate the mean, variance, and standard deviation — for a full population or a sample — along with range, coefficient of variation, and a complete step-by-step table.

Pop & Sample
Both Supported
Step-by-Step
Full Data Table
100%
Free & Accurate

Why Use This Calculator?

  • ✓ Supports both population and sample standard deviation.
  • ✓ Shows every deviation and squared deviation in a clear data table.
  • ✓ Plots your data points with the mean and ±1 SD band on a live chart.

On This Page

Standard Deviation Calculator Empirical Rule Reference Formulas Worked Example Best Practices FAQs Related Calculators

Standard Deviation Calculator

Separate values with commas, spaces, or new lines.

Use Sample mode when your data is a subset of a larger population (most common). Use Population mode only when your data set represents the entire population.

Data Chart

Results

Count (n)
8
Mean
18
Standard Deviation
5.3182
Variance
28.2857
Coefficient of Variation
29.55%
Min
10
Max
23
Range
13
Sum of Squared Deviations
198

Step-by-Step Breakdown

Value (x) Deviation (x − mean) Squared Deviation

Empirical Rule (68–95–99.7 Rule)

For data that follows a normal distribution, this rule estimates what percentage of values fall within a given number of standard deviations from the mean.

Range Approximate % of Data
Mean ± 1 standard deviation68%
Mean ± 2 standard deviations95%
Mean ± 3 standard deviations99.7%

Standard Deviation Formulas

Mean

x̄ = (Σx) ÷ n

Population Variance

σ² = Σ(x − μ)² ÷ N

Sample Variance

s² = Σ(x − x̄)² ÷ (n − 1)

Standard Deviation

σ or s = √variance

Coefficient of Variation

CV = (Standard Deviation ÷ Mean) × 100%

Worked Example

Data set: 10, 12, 23, 23, 16, 23, 21, 16

Mean = (10+12+23+23+16+23+21+16) ÷ 8 = 18
Sum of squared deviations = 198
Sample variance = 198 ÷ 7 = 28.29
Sample standard deviation = √28.29 = 5.32

Where Standard Deviation Is Used

📊 Statistics & Research

Measure how spread out survey results, experiment data, or study samples are.

💰 Finance & Investing

Quantify volatility and risk in stock returns, portfolios, and market indices.

🏭 Quality Control

Monitor manufacturing consistency and detect process variation early.

🎓 Education & Grading

Understand how spread out test scores are around the class average.

⚽ Sports Analytics

Compare player or team performance consistency across a season.

🧪 Science & Engineering

Assess measurement precision and repeatability in experiments.

Best Practices & Common Mistakes

✅ Best Practices

  • Use sample standard deviation (n − 1) unless you truly have the entire population.
  • Check for outliers, since they can heavily skew the standard deviation.
  • Report the mean alongside the standard deviation for context.
  • Use the coefficient of variation to compare spread across different units or scales.
  • Keep enough decimal precision for scientific or financial work.

❌ Common Mistakes

  • Dividing by n instead of n − 1 for sample data (or vice versa).
  • Forgetting to square the deviations before averaging them.
  • Confusing variance with standard deviation.
  • Ignoring outliers that distort the spread of the data.
  • Assuming a normal distribution when the data is heavily skewed.

Frequently Asked Questions

Population standard deviation divides by N (the total number of data points) and is used when your data covers an entire population. Sample standard deviation divides by n − 1 and is used when your data is a sample drawn from a larger population.
This is called Bessel's correction. It corrects the bias that occurs when estimating a population's variance from a smaller sample, producing a more accurate, slightly larger estimate.
A low standard deviation means data points are clustered closely around the mean. A high standard deviation means the data is more spread out and variable.
No. Standard deviation is always zero or positive, since it comes from squared deviations and a square root, both of which cannot produce a negative result.
Standard deviation is the square root of variance. Variance is expressed in squared units, while standard deviation is expressed in the same units as the original data, making it easier to interpret.
The coefficient of variation expresses standard deviation as a percentage of the mean, making it useful for comparing variability between data sets with different units or very different averages.
Yes. It uses the standard statistical formulas for mean, variance, and standard deviation, and shows the full step-by-step breakdown for verification.

Need More Math Tools?

Explore hundreds of free online calculators for math, finance, health, construction, education, and everyday calculations.

Browse Math Calculators All Calculators