Standard deviation is the single most important number in statistics. Our standard deviation calculator computes the SD of any dataset with a complete step-by-step breakdown β trusted by students, teachers, data scientists, and financial analysts worldwide.
What is Standard Deviation?
Standard deviation (SD) measures the average amount by which individual data points differ from the mean. A small SD means data is tightly clustered; a large SD means it's widely spread. It's the square root of variance and is expressed in the same units as the original data, making it directly interpretable.
Formula
Population SD (Ο) = β[Ξ£(x β ΞΌ)Β² / N]
Sample SD (s) = β[Ξ£(x β xΜ)Β² / (Nβ1)]
Example Calculation
Dataset: {10, 20, 30, 40, 50}
- Mean = 30
- DeviationsΒ²: 400, 100, 0, 100, 400 β Sum = 1000
- Population Variance = 1000/5 = 200
- Population SD = β200 β 14.14
How to Use This Calculator
- Enter your numbers separated by commas.
- Click Calculate to see the mean, variance, and both population and sample standard deviations.
Real Life Use Cases
- Finance: Stocks with high SD are riskier investments. Portfolio managers use SD as the primary risk metric.
- Manufacturing: Machine precision is measured by the SD of component dimensions.
- Medicine: Clinical trials use SD to determine if a drug's effect is consistent across patients.
- Education: Grading on a curve uses mean and SD to assign letter grades fairly.
FAQs
What is a good standard deviation value?
It depends on context. In investment, Nifty 50 has a daily SD of ~1%. A high-risk small-cap stock might show 3-4% daily SD. In test scores, an SD of 10-15 on a 100-point test is typical.
What is the 68-95-99.7 rule?
In a normal distribution, 68% of data falls within Β±1 SD, 95% within Β±2 SD, and 99.7% within Β±3 SD of the mean. This is called the empirical rule.
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