Standard Deviation Calculator - Sample & Population SD
Calculate sample and population standard deviation, variance, mean, coefficient of variation, and more from any dataset — enter numbers and get results instantly.
Paste your numbers separated by commas, spaces, or new lines. The calculator computes both sample and population SD simultaneously alongside six other descriptive statistics.
Standard Deviation Calculator - Sample & Population SD
Calculate sample and population standard deviation, variance, mean, coefficient of variation, and more from any dataset — enter numbers and get results instantly.
Separate numbers with commas, spaces, or new lines
About the Standard Deviation Calculator
Standard deviation is the most widely used measure of statistical dispersion. It answers the question: on average, how far are individual data points from the mean of the dataset? A small standard deviation means the values are tightly clustered; a large standard deviation means they are widely scattered. Understanding this spread is essential in science, engineering, finance, education, medicine, and virtually every other field that works with numerical data.
There are two versions of standard deviation depending on whether you are working with a complete population or a sample drawn from it. The population standard deviation σ uses n in the denominator and is appropriate when your dataset contains every member of the group you are studying. The sample standard deviation s uses n−1 (Bessel's correction) in the denominator to correct for the bias introduced when estimating the population spread from a subset of observations. In practice, unless you are truly measuring an entire population, you should use the sample formula. Both are computed here simultaneously so you can use whichever is appropriate for your situation.
This calculator also outputs variance (the square of standard deviation), which is used directly in statistical tests such as F-tests, ANOVA, and regression diagnostics. The mean, sum, minimum, maximum, and range give a complete picture of central tendency and spread. The coefficient of variation (CV = s / |x̄| × 100%) expresses the standard deviation as a percentage of the mean, which is especially useful when comparing the spread of datasets measured in different units or at different scales — for example, comparing the variability of stock prices that range from $10 to $10 000.
Common applications include quality control (monitoring whether a manufacturing process stays within tolerance), grading (understanding score distributions in a classroom), finance (measuring the volatility of an investment), clinical research (checking the consistency of measurements across subjects), and data science (detecting outliers, normalizing features, and assessing model residuals). Whenever you need to know not just what the typical value is, but how reliable that typical value is, standard deviation is the right tool.
A practical tip: if your coefficient of variation is less than 15–20%, the data is reasonably homogeneous and the mean is a reliable summary. Above 30–40%, the spread is large relative to the mean, which may signal outliers, multi-modal distributions, or the need for a logarithmic transformation before further analysis.
Standard deviation examples
Four real-world datasets that demonstrate the calculator across different domains.
| Dataset | Sample SD | Context |
|---|---|---|
| 85, 92, 78, 88, 90 | s ≈ 5.4589 | Student test scores for a class of 5. Mean = 86.6, Population SD ≈ 4.8826. |
| 150.25, 152.50, 149.75, 153.00, 151.50 | s ≈ 1.3987 | Weekly closing stock prices. Low SD indicates stable price over the period. |
| 502, 499, 505, 498, 501, 503 | s ≈ 2.5820 | Manufacturing batch weights (grams). CV ≈ 0.5% reflects tight production tolerance. |
| 250000, 275000, 260000, 280000, 265000 | s ≈ 11937 | House prices in a neighborhood. SD of $11 937 shows moderate price spread. |
How to use the standard deviation calculator
- Enter your numbers in the Data Set field, separated by commas, spaces, or new lines.
- Click Calculate. The results panel shows all eleven statistics simultaneously.
- Use Sample SD when your data is a sample drawn from a larger population. Use Population SD when your data is the entire population.
- Check the Coefficient of Variation to compare the relative spread against the mean, especially when comparing datasets in different units.
- Click Reset to clear the field, or use the example buttons to load a pre-built dataset and explore the output.
Standard deviation FAQ
When should I use sample SD versus population SD?
Use sample SD (s, Bessel's correction with n−1) when your data is a sample from a larger population and you are estimating the true population spread. Use population SD (σ, with n) only when your dataset contains every single member of the population you are analyzing. In most research and business contexts, sample SD is the correct choice.
What does a high standard deviation mean?
A high standard deviation means the data points are spread widely around the mean — there is high variability or dispersion. In finance this means high volatility. In manufacturing it means inconsistent output. In education it means scores are spread across a wide range. Whether 'high' is problematic depends entirely on context and what level of variation is acceptable for your application.
What is the coefficient of variation (CV)?
The coefficient of variation expresses the standard deviation as a percentage of the mean: CV = (s / |x̄|) × 100%. It is a dimensionless ratio, making it useful for comparing variability across datasets measured in different units or at very different scales. A CV of 5% indicates the standard deviation is 5% of the mean — tight clustering. A CV of 80% indicates the data is highly scattered relative to its average value.
Is standard deviation affected by outliers?
Yes. Because the formula squares each deviation from the mean, extreme outliers contribute disproportionately to the standard deviation. A single very large or very small value can inflate SD substantially. When outliers are present, consider reporting the median and interquartile range alongside the mean and SD to give a fuller picture of the distribution.
Can I calculate SD for negative numbers?
Yes. The standard deviation calculation works correctly for negative numbers, zero, and any mix of positive and negative values. Only the coefficient of variation becomes undefined or misleading when the mean is zero or close to zero, because division by a very small mean produces an arbitrarily large percentage.