Standard Deviation & Variance Calculator
Compute sample and population standard deviation, variance, mean, sum of squares, and step-by-step statistical derivations.
Data Input & Parameters
Statistical Results
Mode: sampleSquare root of variance in original units
Sum of Squares ÷ 6
Calculation Step-by-Step
Understanding Standard Deviation and Variance
In statistics, standard deviation and variance are fundamental measures of dispersion that quantify the spread of values in a dataset relative to their central mean. While the mean provides the central tendency, dispersion metrics reveal how tightly clustered or widely scattered individual observations are.
Sample Standard Deviation Formula
Applied when analyzing a representative subset of a larger population. Uses Bessel's correction ($N - 1$) to ensure an unbiased estimator.
Population Standard Deviation Formula
Applied when complete measurements are recorded for every single member of an entire target population ($N$).
Why Use Bessel's Correction ($N - 1$)?
When calculating sample variance, using $N$ as the denominator tends to systematically underestimate population variance because sample points are naturally drawn closer to the sample mean than to the true, unknown population mean. Replacing $N$ with $N - 1$ adjusts for this degrees-of-freedom constraint, producing an unbiased statistical estimate.
The Empirical Rule (68-95-99.7 Rule) in Normal Distributions
For standard bell-shaped (normal) distributions, standard deviation provides immediate probability boundaries for data spread:
Within ±1 Standard Deviation
Approximately 68% of all data points fall within $\mu \pm 1\sigma$.
Within ±2 Standard Deviations
Approximately 95% of observations fall within $\mu \pm 2\sigma$.
Within ±3 Standard Deviations
Over 99.7% of all sample data falls within $\mu \pm 3\sigma$.
Step-by-Step Calculation Breakdown Example
To demonstrate the underlying algorithm, consider the dataset: [4, 8, 6, 5, 12]. Mean (x̄) = 35 / 5 = 7.0.
| Data Point ($x_i$) | Mean (x̄) | Deviation (x_i - x̄) | Squared Deviation ((x_i - x̄)^2) |
|---|---|---|---|
| 4 | 7.0 | -3.0 | 9.0 |
| 8 | 7.0 | +1.0 | 1.0 |
| 6 | 7.0 | -1.0 | 1.0 |
| 5 | 7.0 | -2.0 | 4.0 |
| 12 | 7.0 | +5.0 | 25.0 |
| Sum = 35.0 | - | Sum = 0.0 | SS = 40.0 |
How to Use This Calculator
Select Dataset Type
Toggle between Sample ($N-1$) and Population ($N$) mode depending on whether your dataset represents a subset or a complete population.
Input Numerical Data
Paste or type values into the text area separated by commas, spaces, or line breaks. Or click one of the quick sample presets.
Review Results & Steps
Instantly inspect the standard deviation, variance, mean, sum of squares, coefficient of variation, and complete step-by-step derivation.
Export Analysis
Copy the plain-text summary report to your clipboard or export the detailed calculation matrix directly to a CSV spreadsheet.
Frequently Asked Questions (FAQ)
What is the difference between Sample and Population Standard Deviation?
Sample standard deviation ($s$) estimates variability in a larger population using a representative subset, dividing the sum of squared differences by ($N - 1$) Bessel's correction. Population standard deviation ($\sigma$) is used when you have complete data for every member of the entire group, dividing by $N$.
Why do we divide by N - 1 for sample standard deviation?
Dividing by $N - 1$ is known as Bessel's correction. It corrects the bias in the estimation of the population variance, compensating for the fact that sample values tend to be closer to the sample mean than to the true population mean.
What is the relationship between Variance and Standard Deviation?
Variance measures the average squared deviation from the mean, expressed in squared units. Standard deviation is simply the square root of variance, returning the dispersion metric back into the original units of measurement.
What does a low vs high Standard Deviation indicate?
A low standard deviation indicates that data points cluster tightly around the mean, representing high consistency. A high standard deviation means data points are spread widely across a broad range of values.
What is the Coefficient of Variation (CV)?
The Coefficient of Variation (CV) is the ratio of standard deviation to the mean expressed as a percentage (CV = (s / x̄) x 100). It measures relative variability, allowing you to compare dispersion between datasets with different units or scales.
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