Standard Deviation Calculator
Paste your data set and this calculator returns every key descriptive statistic: both standard deviations (population and sample), mean, median, mode, min, max, range, variance, sum, coefficient of variation, sum of squares, and margin of error at your chosen confidence level. A step-by-step panel shows exactly how each number is derived.
Formula
Worked example
For 2, 4, 4, 4, 5, 5, 7, 9: N = 8, sum = 40, mean = 5, SS = 32, population variance = 4, population σ = 2, sample s = 2.1381, range = 7, CV = 40.00%.
Population vs sample standard deviation: which one to use
Use population standard deviation (σ) when your data set contains every member of the group you are describing, such as all exam scores in a single class you are analyzing as a closed group. Use sample standard deviation (s) when your data is a subset drawn from a larger population. Dividing by N-1 instead of N applies Bessel's correction, which removes the downward bias that results from estimating the true population spread from a sample. In practice, most statistical analyses, surveys, experiments, and quality-control sampling call for the sample formula. Choosing the wrong formula gives a spread estimate that is systematically too small (using N on a sample) or unnecessarily inflated.
What each statistic tells you
Mean is the arithmetic average and is the center the standard deviation radiates from. Median is the middle value when the data is sorted; when the mean and median differ substantially, the distribution is skewed. Mode is the most frequent value; a data set can have multiple modes or none. Range is simply the gap from the smallest to the largest value and captures extreme spread in a single number. Sum of squares (SS) is the total squared deviation from the mean and is the building block for both variance and standard deviation. Coefficient of variation (CV) expresses standard deviation as a percentage of the mean, making spread comparable across data sets measured on different scales. Margin of error uses the z* critical value for your chosen confidence level to state the interval within which the true mean is likely to fall, assuming a large population.
How to enter your data
Paste or type your numbers into the data set field separated by commas, spaces, semicolons, or line breaks. The calculator accepts any real numbers including negatives and decimals, and it updates all statistics instantly. Select whether you want to see population values, sample values, or both from the data type dropdown. Choose the confidence level for the margin of error calculation (90%, 95%, or 99%). The step-by-step panel below the results traces every arithmetic operation with your actual numbers substituted in.
The empirical rule and what standard deviation means in practice
For a normally distributed data set, approximately 68% of values fall within one standard deviation of the mean, about 95% fall within two standard deviations, and roughly 99.7% fall within three. This is called the empirical rule or the 68-95-99.7 rule. If your data is roughly bell-shaped, these percentages let you quickly characterize how unusual any particular value is: a value more than two standard deviations from the mean occurs in only about 5% of a normal distribution. For heavily skewed or multimodal data, the empirical rule does not apply directly, but standard deviation still measures spread and is used throughout regression, hypothesis testing, and confidence intervals.
Coefficient of variation and comparing spread across data sets
Standard deviation is measured in the same units as the original data, so comparing the spread of two data sets measured in different units (say, heights in centimetres and weights in kilograms) is not directly meaningful. The coefficient of variation (CV = σ / |mean|) solves this by expressing spread as a proportion of the mean. A CV below 15% is generally considered low variability; 15-30% is moderate; above 30% is high. CV is widely used in quality control, laboratory science, and finance to compare consistency across different measurements or assets.
Limitations and edge cases
A data set with a single value produces a mean but leaves the sample standard deviation undefined, since dividing by N-1 requires at least two observations. Standard deviation assumes that the mean is a meaningful center; for heavily skewed data or data with extreme outliers, it can overstate or misrepresent typical variability. The calculator treats all values as exact and does not propagate measurement uncertainty. For very large data sets entered by hand, a single transcription error has an outsized effect on variance, so double-check your input. The margin of error shown here is an approximation using a z* critical value and is most accurate when the sample is large (N > 30) and the population is much larger than the sample.
Empirical rule: normal distribution coverage by standard deviations
| Interval | Coverage | Interpretation |
|---|---|---|
| mean +/- 1 SD | 68.27% | Typical values; about 2 in 3 observations |
| mean +/- 2 SD | 95.45% | Nearly all data; outliers start beyond here |
| mean +/- 3 SD | 99.73% | Extreme values; about 1 in 370 observations |
| mean +/- 1.96 SD | 95.00% | Standard 95% confidence interval boundary |
| mean +/- 2.576 SD | 99.00% | Standard 99% confidence interval boundary |
Applies to data that is approximately normally distributed.
Frequently asked questions
What is the difference between population and sample standard deviation?
Population standard deviation (σ) divides the sum of squared deviations by N and is correct when your data set is the entire population. Sample standard deviation (s) divides by N-1 (Bessel's correction) to produce an unbiased estimate of the true population spread when your data is a sample. Using N instead of N-1 on a sample consistently underestimates the spread of the full population.
What is the coefficient of variation and when should I use it?
The coefficient of variation (CV) is standard deviation divided by the absolute mean, usually expressed as a percentage. It lets you compare relative spread between data sets measured on different scales or in different units. For example, a CV of 10% means the standard deviation is 10% of the mean. Use CV when you need a unitless measure of consistency, such as comparing the variability of stock returns against crop yields, or evaluating measurement precision in a lab.
How is the margin of error calculated here?
The margin of error is computed as z* times s divided by the square root of N, where z* is the critical value for the chosen confidence level (1.645 for 90%, 1.96 for 95%, 2.576 for 99%). This formula assumes a large population and an approximately normal distribution of sample means. It tells you the range within which the true population mean is likely to fall at the chosen confidence level.
Why does standard deviation use squared deviations instead of absolute differences?
Squaring the deviations ensures that negative and positive differences do not cancel each other out, and it penalizes larger deviations more heavily than small ones. The squared form is also mathematically tractable, fitting naturally into algebra and calculus and connecting directly to regression, hypothesis testing, and other statistical methods. The square root at the end brings the result back to the original unit of measurement.
What does a high or low standard deviation tell you?
A low standard deviation means most values cluster tightly around the mean, indicating consistent or uniform data. A high standard deviation means values are widely spread, indicating greater variability. Neither is inherently good or bad; the appropriate level of spread depends entirely on the context, such as manufacturing tolerances, test score distributions, or financial return data. Use the coefficient of variation to compare spread across different data sets.
What is the sum of squares (SS)?
Sum of squares is the total of each value's squared deviation from the mean: SS = sum of (xi - mean)^2 for all i. It is the numerator in the variance formula and is an intermediate value the calculator shows in the steps panel. Dividing SS by N gives population variance; dividing by N-1 gives sample variance. SS is also used in ANOVA and regression analysis.