How Does the Standard Deviation Calculator Work?
Enter a set of numbers separated by commas. The calculator computes the population standard deviation (σ), variance (σ²), arithmetic mean and count. Standard deviation measures how spread out the data is from the mean - the higher the value, the more dispersed the data. A standard deviation of zero means all values are identical.
Formulas
| Measure |
Formula |
Description |
| Mean (x̄) |
Σxᵢ / n |
Sum of all values divided by count |
| Variance (σ²) |
Σ(xᵢ - x̄)² / n |
Average of squared deviations from mean |
| Std Dev (σ) |
√(σ²) |
Square root of variance |
| Sample Std Dev (s) |
√(Σ(xᵢ - x̄)² / (n-1)) |
Bessel's correction for sample data |
This calculator uses the population standard deviation formula (divides by n). For sample standard deviation (divides by n-1), the difference is significant only for small datasets (n < 30).
The 68-95-99.7 Rule (Empirical Rule)
For data that follows a normal (bell curve) distribution:
| Range |
% of data |
Meaning |
| Mean ± 1σ |
68.27% |
About 2 out of 3 data points |
| Mean ± 2σ |
95.45% |
Almost all data points |
| Mean ± 3σ |
99.73% |
Virtually all data (outliers beyond 3σ are rare) |
In quality control, the "Six Sigma" methodology targets 3.4 defects per million - that is how tight ±6σ is. In finance, events beyond 3σ are called "black swan" events.
Practical Examples
Example 1: Test scores: 85, 90, 78, 92, 88
Mean: 86.60, Std Dev: 4.8166, Variance: 23.20
Example 2: Heights in cm: 170, 165, 180, 175, 160
Mean: 170.00, Std Dev: 7.0711, Variance: 50.00
Example 3: Daily temperatures (°C): 22, 24, 21, 23, 25, 22, 24
Mean: 23.00, Std Dev: 1.2472 - low dispersion, stable weather
Example 4: Stock returns (%): 5, -3, 8, 2, -1, 6, 4
Mean: 3.00, Std Dev: 3.5590 - high volatility, risky investment
Example 5: Identical values: 5, 5, 5, 5, 5
Mean: 5.00, Std Dev: 0.0000 - zero dispersion, all values equal
Example 6: Manufacturing tolerances (mm): 10.02, 9.98, 10.01, 9.99, 10.00, 10.03, 9.97
Mean: 10.00, Std Dev: 0.0200 - very tight tolerance, good QC
Example 7: Salaries ($K): 45, 50, 48, 120, 52, 47, 49
Mean: 58.71, Std Dev: 24.47 - the 120K outlier skews both mean and σ
FAQ - Frequently Asked Questions
What is standard deviation?
Standard deviation measures how spread out numbers are from the mean. A low standard deviation means data points are close to the mean (consistent data), while a high standard deviation means data is spread over a wide range (variable data). It's expressed in the same units as the original data, unlike variance which is in squared units.
What is the difference between population and sample standard deviation?
Population standard deviation (σ) divides by n (total count) and is used when you have data for the entire population. Sample standard deviation (s) divides by n-1 (Bessel's correction) and is used when working with a sample from a larger population. For large datasets (n > 30), the difference becomes negligible. This calculator uses the population formula.
What is variance and how does it relate to standard deviation?
Variance is the average of the squared differences from the mean. It's the square of standard deviation: σ² = variance, σ = √variance. While variance gives a mathematical measure of spread, standard deviation is more intuitive because it's in the same units as the original data. If your data is in centimeters, σ is also in centimeters, but σ² is in cm².
When is standard deviation useful in practice?
Quality control (manufacturing tolerances - if σ is too high, products are inconsistent), finance (portfolio risk/volatility - higher σ = riskier asset), education (test score analysis - how much do students vary?), science (experimental measurement precision), weather forecasting (temperature variability), and sports analytics (consistency of performance).
What does a standard deviation of 0 mean?
A standard deviation of 0 means all values in the dataset are identical - there is zero variation. Every data point equals the mean. This is rare in real-world data but common in theoretical examples.
What is the 68-95-99.7 rule?
For normally distributed data: 68% of values fall within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. This is also called the empirical rule or three-sigma rule. A data point beyond 3σ is extremely unusual (only 0.3% chance) and is often flagged as an outlier.
How do outliers affect standard deviation?
Outliers dramatically increase standard deviation because deviations are squared. One extreme value can inflate σ significantly. For example, in {50, 50, 50, 50, 200}, the mean is 80 and σ is 60 - mostly because of the single outlier at 200. Consider using median absolute deviation (MAD) for datasets with suspected outliers.
What is the coefficient of variation (CV)?
CV = (σ / mean) x 100%. It expresses standard deviation as a percentage of the mean, allowing comparison between datasets with different scales. A CV of 5% means low relative variability regardless of whether the data is in grams or tons. Useful when comparing volatility of stocks at different price levels.
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