—
—
19
—
%
5
—
—
19
—
%
5
The Standard Deviation Calculator measures how spread out the values in a dataset are from the mean. Standard deviation is the most widely used measure of statistical dispersion and is fundamental to virtually every area of statistics, from hypothesis testing to quality control.
A low standard deviation means the data points cluster tightly around the mean, while a high standard deviation indicates the data is spread over a wide range. This calculator supports both population and sample standard deviation, along with variance and the coefficient of variation.
Standard deviation is the square root of variance. The formulas differ depending on whether you have a complete population or a sample:
Population standard deviation:
$$\sigma = \sqrt{\frac{\sum_{i=1}^{N}(x_i - \mu)^2}{N}}$$
Sample standard deviation:
$$s = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n - 1}}$$
The sample version divides by \(n-1\) instead of \(n\) to correct for bias (Bessel's correction). This is because a sample tends to underestimate the true population variance.
Step-by-step process:
The coefficient of variation (CV) is also computed as \(\text{CV} = (s / |\bar{x}|) \times 100\%\), which expresses the standard deviation as a percentage of the mean, allowing comparison of variability between datasets with different units or scales.
Standard deviation is expressed in the same units as the original data, making it directly interpretable:
When comparing variability between datasets measured in different units, use the coefficient of variation (CV) instead of the raw standard deviation.
Inputs
Results
Sample SD of 6.20 means values typically deviate about 6.2 units from the mean of 19. The CV of 32.66% indicates moderate relative variability.
Inputs
Results
Population SD of 1.87 for these 8 values (the entire population). Values stay within about 2 units of the mean on average.
Use population standard deviation (divide by N) when your dataset includes every member of the group you are studying. Use sample standard deviation (divide by N-1) when your data is a subset drawn from a larger population, which is the more common scenario in practice.
This is called Bessel's correction. A sample tends to underestimate the true population variance because the sample mean is closer to the sample data points than the true population mean would be. Dividing by N-1 corrects this bias, producing an unbiased estimator of the population variance.
Variance is the square of standard deviation: \(\sigma^2 = \text{variance}\). Standard deviation is the square root of variance: \(\sigma = \sqrt{\text{variance}}\). Standard deviation is preferred for interpretation because it is in the same units as the data.
The CV expresses standard deviation as a percentage of the mean. It allows comparing variability between datasets with different scales or units. A CV below 15% typically indicates low variability, 15-30% moderate variability, and above 30% high variability.
No. Standard deviation is always zero or positive because it is the square root of variance, which itself is a sum of squared values (always non-negative). A standard deviation of zero means all values are identical.
In a normal distribution, approximately 68.27% of data falls within one SD of the mean, 95.45% within two SDs, and 99.73% within three SDs. This is the empirical rule (68-95-99.7 rule) and is the foundation for confidence intervals and hypothesis testing.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
