Standard Deviation Calculator
Calculation Results:
Standard Deviation:
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Variance:
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Mean (Average):
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Count (N):
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Sum:
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Sum of Squares:
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Please enter at least two numbers to calculate the standard deviation.
Understanding Standard Deviation
Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of values. A low standard deviation indicates that the data points tend to be close to the mean (the expected value), while a high standard deviation indicates that the data points are spread out over a wider range.
Sample vs. Population
When calculating standard deviation, choosing between "Sample" and "Population" is critical:
- Population Standard Deviation (σ): Used when you have data for every member of the group you are studying (e.g., test scores for every student in a single small classroom).
- Sample Standard Deviation (s): Used when your data represents a random subset of a larger group (e.g., surveying 100 voters to estimate the behavior of an entire city). It uses Bessel's correction (n-1) to provide an unbiased estimate.
The Calculation Process
- Calculate the mean (average) of all numbers in the data set.
- Subtract the mean from each individual number and square the result (this gives you the squared deviations).
- Calculate the average of those squared deviations. (For samples, divide by n-1; for populations, divide by n). This result is the Variance.
- Take the square root of the Variance to find the Standard Deviation.
Calculation Example
Data Set: 2, 4, 4, 4, 5, 5, 7, 9
- Count (n): 8
- Sum: 40
- Mean (μ): 5
- Variance (σ²): 4
- Standard Deviation (σ): 2