Use our Standard Deviation Calculator to quickly compute population or sample standard deviation, along with mean and variance, from any list of numbers. Paste your data, choose the dataset type, set your preferred precision, and get accurate results in seconds.
What is Standard Deviation?
Standard deviation is a measure of how spread out a set of numbers is. A low standard deviation indicates the data points tend to be close to the mean (average), while a high standard deviation shows that values are more dispersed. It is widely used in statistics, finance, quality control, education, and research to quantify variability and risk.
In simple terms, standard deviation helps answer the question: how far, on average, are the data points from the mean? By converting variance into the same units as your original data, standard deviation offers a more interpretable measure of dispersion.
Population vs. Sample Standard Deviation
There are two common ways to calculate standard deviation, and the choice depends on whether your data represent the entire population or just a sample from a larger population.
- Population standard deviation: Use when your dataset includes every member of the group of interest. It divides the sum of squared deviations by n.
- Sample standard deviation: Use when your dataset is a subset of a larger group. It divides by n ? 1 (Bessel's correction) to produce an unbiased estimator of the population variance.
How to Use the Standard Deviation Calculator
- Paste or type your values into the Data values box. You can separate numbers with commas, spaces, tabs, or new lines.
- Select Population or Sample depending on your scenario.
- Choose the number of decimal places for rounding.
- Click Calculate to get the mean, variance, and standard deviation.
Formulas Used
Mean (average): sum of values divided by the number of values.
Variance (population): average of the squared deviations from the mean, i.e., sum of squared deviations divided by n.
Variance (sample): sum of squared deviations divided by n ? 1.
Standard deviation: the square root of the variance.
Why Standard Deviation Matters
- Risk and volatility: In finance, higher standard deviation often means higher volatility.
- Quality control: Manufacturers monitor deviation to maintain consistent product quality.
- Education and testing: Standard deviation reveals how test scores spread around the average.
- Research and science: It indicates the reliability and consistency of experimental results.
Tips for Clean, Accurate Results
- Ensure all entries are numeric. The calculator ignores non-numeric tokens.
- Decide correctly between population and sample. If in doubt, you likely have a sample.
- Use an appropriate number of decimal places: 2–4 is common for most applications.
- Check for outliers: unusual values can substantially increase the standard deviation.
Example
Suppose your dataset is: 8, 9, 10, 12, 15. The mean is 10.8. The squared deviations are computed relative to 10.8 and then averaged. If treated as a population, variance divides by 5; as a sample, it divides by 4, resulting in a slightly larger variance and standard deviation for the sample case.
Common Questions
- Do I need to sort my data? No. The calculation is independent of order.
- Can I include negative numbers? Yes. Standard deviation works with any real numbers.
- What if I only have one value? Population standard deviation is zero; sample standard deviation is undefined.
Use this Standard Deviation Calculator whenever you need a fast, reliable way to assess variability—whether you are analyzing experiments, evaluating investments, or summarizing performance data.