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 how concentrated or spread out data points are around their arithmetic mean:
In a normal bell curve, 68.27% of all observations fall within one standard deviation of the mean (μ ± 1σ).
Approximately 95.45% of all data points lie within two standard deviations (μ ± 2σ). Observations outside are relatively uncommon.
Approximately 99.73% of data falls within three standard deviations (μ ± 3σ). Data beyond 3σ indicates statistical outliers.
| Feature | Sample SD (s) | Population SD (σ) |
|---|---|---|
| When to Use | When data is a representative sample of a larger group. | When you possess data for every item in the population. |
| Formula | s = √[ ∑(xᵢ − x̄)² ÷ (N − 1) ] | σ = √[ ∑(xᵢ − μ)² ÷ N ] |
| Divisor | N − 1 (Bessel's Correction) | N (Total Count) |
| Purpose of Divisor | Corrects downward bias when estimating population spread. | Direct exact calculation across the entire closed set. |
| Example Scenario | Survey data from 500 customers out of 50,000 users. | Quarterly exam scores of all 30 students in a classroom. |
Standard deviation measures investment volatility. Fund managers use SD to calculate Sharpe ratios—higher SD indicates wider price fluctuations and higher risk.
Factory assembly lines use standard deviation to control engineering tolerances. Six Sigma targets keeping production defect rates within 3.4 defects per million.
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.
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