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Standard Deviation Calculator

Calculate Mean, Variance & Dispersion

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Comprehensive Guide to Standard Deviation

What is Standard Deviation?

Standard deviation is a fundamental statistical metric that measures the amount of variation or dispersion in a set of values. It quantifies how much the individual data points in a group differ from the average (mean) of the group.

  • Low Standard Deviation: Indicates that the data points tend to be very close to the mean, suggesting high consistency.
  • High Standard Deviation: Indicates that the data points are spread out over a wider range of values, suggesting high volatility or diversity.
Sample vs. Population Standard Deviation

Calculating standard deviation depends on whether you are analyzing an entire population or just a sample of it.

Population Standard Deviation (σ)
σ = √[ Σ(x - μ)² / N ] Used when you have data for every member of the group.
Sample Standard Deviation (s)
s = √[ Σ(x - x̄)² / (n - 1) ] Used when the data is a subset of a larger population.

*Note: Bessel's correction (n-1) is used in the sample formula to provide an unbiased estimate of the population variance.

How to Calculate Standard Deviation Step-by-Step
  1. Calculate the Mean (average) of the numbers.
  2. For each number, subtract the Mean and square the result (Difference squared).
  3. Calculate the Mean of those squared differences (this is the Variance).
  4. Take the square root of the Variance to get the Standard Deviation.
Frequently Asked Questions (FAQ)

There is no universal "good" value. It depends entirely on the context. In manufacturing, a very low SD (Six Sigma) is good. In investment, a higher SD might indicate higher potential returns but with higher risk.

No, standard deviation is always a non-negative number because it involves squaring values and taking a positive square root. A value of zero means all data points are identical.

Standard Deviation is simply the square root of the Variance. Variance is measured in squared units, while SD is measured in the same units as the original data, making it easier to interpret.
Disclaimer

This tool is intended for educational and general informational purposes only. While we strive for mathematical accuracy, we do not guarantee the results for critical scientific, medical, or financial decision-making. Always verify important calculations with professional-grade statistical software or a certified statistician.

Quick Summary
  • Mean: The average of all data points.
  • Variance: Average of squared differences from Mean.
  • Range: Difference between Max and Min values.
  • Count: Total number of data points entered.
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