Standard Deviation Calculator
Calculate the sample and population standard deviation, variance, mean, and sum of squares for any dataset. Instantly analyze data dispersion and identify outliers.
Statistical Analysis Tool
Calculate SD, variance, and mean for any dataset
Paste or type your dataset. Minimum 2 values required for sample SD.
Use ‘Sample’ for surveys/subsets. Use ‘Population’ for complete datasets.
Statistical Results
Standard deviation, variance, and summary
Enter your dataset and click Analyze Data to calculate the standard deviation and other statistical metrics.
The Empirical Rule
For a normal distribution (bell curve), the empirical rule dictates how data spreads around the mean.
| Range | Percentage of Data | Interpretation |
|---|---|---|
| Within 1 SD (μ ± 1σ) | ~68.27% | Typical variation |
| Within 2 SD (μ ± 2σ) | ~95.45% | High confidence range |
| Within 3 SD (μ ± 3σ) | ~99.73% | Near certainty |
| Beyond 3 SD | ~0.27% | Statistical outliers |
Standard Deviation FAQ
Everything you need to know about calculating and interpreting standard deviation.
Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. A low standard deviation indicates that the data points tend to be very close to the mean, while a high standard deviation indicates that the data points are spread out over a wide range.
Population standard deviation (σ) is used when your data includes every single member of the group you are studying. It divides the sum of squared differences by N. Sample standard deviation (s) is used when your data is just a subset (sample) of a larger population. It divides by n – 1 (Bessel’s correction) to provide an unbiased estimate of the population’s variance.
To calculate standard deviation: Step 1: Find the mean (average) of the dataset. Step 2: Subtract the mean from each data point and square the result. Step 3: Sum all the squared differences. Step 4: Divide by N (for population) or n-1 (for sample) to find the variance. Step 5: Take the square root of the variance to get the standard deviation.
The empirical rule applies to normal distributions (bell curves). It states that approximately 68% of data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations. It is a quick way to understand data spread and identify outliers.
Variance is the average of the squared differences from the mean. It is the square of the standard deviation. While variance gives a raw measure of spread in squared units, standard deviation is more intuitive because it is expressed in the same units as the original data.
No, standard deviation can never be negative. Because it is calculated by squaring the differences from the mean (which makes them all positive) and then taking the square root, the result is always zero or a positive number. A standard deviation of zero means all values in the dataset are exactly identical.
The coefficient of variation (CV) is the ratio of the standard deviation to the mean, often expressed as a percentage. It is useful for comparing the relative variability of datasets with different units or vastly different means. Formula: CV = (Standard Deviation / Mean) × 100.
Standard deviation is crucial for understanding data reliability, calculating margins of error in surveys, assessing risk in finance, and performing hypothesis testing. It tells you how much you can trust the mean as a representative value and helps identify outliers or unusual data points.