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Standard Deviation Calculator
Enter comma-separated values to compute standard deviation, variance, confidence interval margins, and a frequency table.
Standard Deviation, σ: 4.898979485566
Count, N: 8
Sum, Σx: 144
Mean, μ: 18
Variance, σ²: 24
Steps
σ² = Σ(xi - μ)^2 / N
= ((10 - 18)^2 + (12 - 18)^2 + (23 - 18)^2 + (23 - 18)^2 + (16 - 18)^2 + (23 - 18)^2 + (21 - 18)^2 + (16 - 18)^2) / 8
= 192 / 8
= 24
σ = √24
= 4.898979485566
Margin of Error (Confidence Interval)
Standard error of mean, σx = σ / √N = 1.732050807569
| Confidence Level | Margin of Error | Error Bar |
|---|---|---|
| 68.3%, 1σx | 18 ± 1.732050807569 (±9.62%) | |
| 90%, 1.645σx | 18 ± 2.849223578451 (±15.83%) | |
| 95%, 1.96σx | 18 ± 3.394819582835 (±18.86%) | |
| 99%, 2.576σx | 18 ± 4.461762880297 (±24.79%) | |
| 99.9%, 3.291σx | 18 ± 5.700179207709 (±31.67%) | |
| 99.99%, 3.891σx | 18 ± 6.739409692251 (±37.44%) | |
| 99.999%, 4.417σx | 18 ± 7.650468417032 (±42.5%) | |
| 99.9999%, 4.892σx | 18 ± 8.473192550627 (±47.07%) |
Frequency Table
| Value | Frequency |
|---|---|
| 10 | 1 (12.5%) |
| 12 | 1 (12.5%) |
| 16 | 2 (25%) |
| 21 | 1 (12.5%) |
| 23 | 3 (37.5%) |
Standard deviation calculator guide
A standard deviation calculator measures how spread out values are around their mean. It is one of the most important descriptive statistics in data analysis, quality control, finance, engineering, and research reporting.
This tool supports both population and sample modes. Population standard deviation divides by N, while sample standard deviation divides by N-1, which is the common unbiased estimator correction used for samples.
Along with standard deviation, the calculator shows count, sum, mean, and variance so you can inspect each ingredient of the result.
The steps section reveals the variance and standard deviation flow using the entered dataset. This helps validate calculations and improves statistical intuition.
The margin of error table uses common confidence levels and z-score multipliers to show interval half-width around the mean based on standard error.
A frequency table is also included to summarize repeated values and their relative proportions in the dataset.
How to use
- Enter a comma-separated list of numeric values.
- Choose Population or Sample mode.
- Set precision digits and click Calculate.
- Read standard deviation, confidence margins, and frequency table.
Core formulas
Population variance: σ² = Σ(xi-μ)² / N. Population standard deviation: σ = √σ². Sample variance: s² = Σ(xi-x̄)² / (N-1). Standard error: SE = σ/√N (or s/√N in sample context). Margin at confidence level uses z × SE.
Notes and limitations
- Input parsing accepts comma-separated decimals and signed numbers.
- Sample mode requires at least two values.
- Confidence margins here use normal z multipliers for quick approximation.
- Outliers can strongly affect mean and standard deviation.
FAQ
Population or sample mode?
Use population for full datasets, sample for subsets estimating a larger population.
Why N-1 in sample variance?
It corrects bias in variance estimation from a sample.
What does margin of error mean here?
It is z times the standard error around the mean for each confidence level.
Why is frequency table useful?
It quickly shows repeated values and data concentration.