Calculate sample or population standard deviation, variance, mean, and range for any data set, with every step shown.
| # | Value (xᵢ) | Deviation (xᵢ − x̄) | Squared Dev (xᵢ − x̄)² |
|---|
Sum of all values divided by total observations count.
Sum of each observation's squared distance from mean.
Sample variance uses N − 1 (Bessel's correction).
Square root of variance to return to original units.
Use when your dataset represents a smaller sample taken from a larger population (e.g., 50 survey respondents out of 10,000 users).
Use when your dataset includes every single individual or measurement in the target group (e.g., exam marks for all 30 students in a class).
Estimates how far the sample mean is likely to be from the true population mean.
Measures relative variability to compare datasets measured in different units.
Indicates how many standard deviations a specific data point lies from the mean.
Standard deviation measures the dispersion of data points relative to their mean. A low standard deviation indicates values tend to cluster close to the mean, while a high standard deviation indicates wide variance.
In a normal bell-curve distribution, approximately 68.27% of all data observations fall within one standard deviation of the mean (μ ± 1σ).
Approximately 95.45% of all data points lie within two standard deviations of the mean (μ ± 2σ). Values outside this range are considered uncommon.
Approximately 99.73% of all data values fall within three standard deviations (μ ± 3σ). Data beyond 3σ is often flagged as statistical outliers.
| Metric Feature | Sample Standard Deviation (s) | Population Standard Deviation (σ) |
|---|---|---|
| When to Use | When data represents a sample drawn from a larger group. | When you have complete data for the entire target population. |
| Formula | s = √[ ∑(xᵢ − x̄)² / (N − 1) ] | σ = √[ ∑(xᵢ − μ)² / N ] |
| Divisor (Degrees of Freedom) | N − 1 (Bessel's Correction) | N (Full Count) |
| Purpose of Divisor | Corrects bias when estimating population variance from samples. | Exact calculation across every item in the dataset. |
| Common Examples | Survey responses from 500 voters out of 100,000. | Exam marks of all 30 students in a closed classroom test. |
In investing, standard deviation measures market volatility. Fund managers use SD to calculate the Sharpe Ratio—higher standard deviation indicates higher historical price swings and risk.
Factory production lines use standard deviation to monitor product dimensions and tolerances. Six Sigma methodology aims to keep manufacturing defect rates within 3.4 defects per million.
The statistical range formula measures total dispersion by calculating the numerical distance between the highest and lowest values in a distribution.
Variance is the average of the squared differences between each value and the mean: variance = (sum of (x - mean)^2) / divisor. Standard deviation is simply the square root of variance, expressed in the same units as the original data.
Population standard deviation divides by N because the data represents the entire group being studied. Sample standard deviation instead divides by N - 1, known as Bessel's correction, which compensates for the fact that a sample tends to underestimate the true variability of the full population; using the wrong divisor on small datasets can noticeably skew the result.
For the dataset 2, 4, 6, 8 the mean is 5, and the squared deviations are 9, 1, 1, and 9, which sum to 20. Dividing by N = 4 gives a population variance of 5 (SD approximately 2.24), while dividing by N - 1 = 3 gives a sample variance of 6.67 (SD approximately 2.58).
Use the Average Calculator when only the mean is needed without variability, and the Percentage Calculator when a spread needs to be expressed as a relative percentage rather than an absolute standard deviation.
Quick answers about how this tool works.
Variance (σ² or s²) is the average of squared differences from the mean. Standard deviation is simply the square root of variance.
Standard deviation measures the dispersion or spread of a dataset relative to its mean. A low SD means numbers are clustered close to the average; a high SD means they are widely spread.
Population SD calculates dispersion when you have the entire population data. Sample SD divides by (n-1) instead of n, adjusting for bias when analyzing a subset sample of a larger population.
In finance, standard deviation is a key metric for volatility and risk. It measures the historic price swings of an asset or portfolio.
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