Standard Deviation Calculator
Calculate mean, population standard deviation (σ), and sample standard deviation (s) from a list of numbers.
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Numbers
Result
Count (N):
Mean (x̄):
Sum:
Population Variance (σ²):
Population Standard Deviation (σ):
Sample Variance (s²):
Sample Standard Deviation (s):
About Standard Deviation Calculator
This standard deviation calculator computes both population (σ) and sample (s) standard deviation from a list of numbers. It also calculates mean, sum, count, and variance for complete statistical analysis.
The calculator accepts numbers separated by commas or spaces. Results include up to 10 decimal places for precision in scientific and engineering applications.
Standard Deviation Formulas
Population Standard Deviation:
σ = √(Σ(xi - μ)² / N) Where: σ = population standard deviation xi = each value in the dataset μ = population mean (Σxi / N) N = total number of values
Sample Standard Deviation:
s = √(Σ(xi - x̄)² / (N - 1)) Where: s = sample standard deviation x̄ = sample mean (Σxi / N) N = number of values in sample (N-1) = Bessel's correction for unbiased estimation
Step-by-Step Calculation Example
For the dataset: 2, 4, 4, 4, 5, 5, 7, 9
Step 1: Calculate the mean Sum = 2+4+4+4+5+5+7+9 = 40 Mean = 40/8 = 5 Step 2: Calculate squared differences from mean (2-5)² = 9 (4-5)² = 1 (4-5)² = 1 (4-5)² = 1 (5-5)² = 0 (5-5)² = 0 (7-5)² = 4 (9-5)² = 16 Sum of squares = 32 Step 3: Calculate variance Population variance = 32/8 = 4 Sample variance = 32/7 = 4.571 Step 4: Calculate standard deviation Population σ = √4 = 2 Sample s = √4.571 = 2.138
When to Use Population vs Sample Formula
| Situation | Formula | Example |
|---|---|---|
| Complete census data | Population (N) | All employees in a company |
| Quality control (all items) | Population (N) | Every unit in a production batch |
| Survey sample | Sample (N-1) | 1000 voters polled from millions |
| Scientific experiment | Sample (N-1) | Test subjects representing larger population |
| A/B test results | Sample (N-1) | Users in test variant group |
Interpreting Standard Deviation
Standard deviation indicates data spread and is used in many applications:
- Quality Control: Low SD means consistent product quality; high SD signals process variation
- Finance: SD of returns measures investment risk and volatility
- Research: SD shows variability in experimental measurements
- Education: SD of test scores indicates how varied student performance is
- Weather: SD of temperatures shows climate stability vs variability
Empirical Rule (68-95-99.7 Rule)
For normally distributed data:
μ - 3σ | μ - 2σ | μ - σ | μ | μ + σ | μ + 2σ | μ + 3σ
0.13% 2.14% 13.59% | 34.13% 13.59% 2.14% 0.13%
|---------|---------|---------|---------|---------|---------|
68.27% of data within ±1σ
95.45% of data within ±2σ
99.73% of data within ±3σ
Coefficient of Variation
The coefficient of variation (CV) expresses standard deviation as a percentage of the mean:
CV = (σ / μ) × 100% Use CV to compare variability between datasets with different means or units. CV < 10%: Low variation CV 10-30%: Moderate variation CV > 30%: High variation
Common Applications
- Statistical Analysis: Foundation for hypothesis testing, confidence intervals, and regression
- Risk Assessment: Portfolio volatility, insurance risk modeling
- Manufacturing: Six Sigma quality control, process capability analysis
- Data Science: Feature scaling, outlier detection, normality testing
- Research: Error bars in scientific graphs, measurement precision
Frequently Asked Questions
- What is the difference between population and sample standard deviation?
- Population standard deviation (σ) uses N in the denominator and is used when you have data for the entire population. Sample standard deviation (s) uses N-1 (Bessel's correction) and is used when working with a sample to estimate population parameters. The N-1 correction accounts for the bias in estimating population variance from a sample.
- What is the standard deviation formula?
- Population: σ = √(Σ(xi - μ)² / N). Sample: s = √(Σ(xi - x̄)² / (N-1)). Where xi are individual values, μ is population mean, x̄ is sample mean, and N is the count of values. The squared differences from mean are summed, divided by N or N-1, then square rooted.
- When should I use N vs N-1 in standard deviation?
- Use N (population formula) when you have complete data for the entire group you're studying. Use N-1 (sample formula) when your data is a subset used to estimate population parameters. Using N-1 gives an unbiased estimator of population variance.
- What does standard deviation tell you?
- Standard deviation measures how spread out values are from the mean. Low SD means values cluster near the mean; high SD means values are more spread out. In normal distributions: ~68% of values fall within ±1 SD, ~95% within ±2 SD, ~99.7% within ±3 SD of the mean.
- What is variance and how does it relate to standard deviation?
- Variance is the average of squared differences from the mean (σ² or s²). Standard deviation is the square root of variance. Variance is in squared units (e.g., meters²), while standard deviation is in the original units (meters), making SD more interpretable.
- Can standard deviation be negative?
- No. Standard deviation is always non-negative because it's the square root of variance, which is a sum of squared values. A standard deviation of zero means all values in the dataset are identical. The minimum possible value is 0.