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Standard Deviation Calculator

Standard Deviation
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Variance
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Mean (Average)
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Count (N)
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How to Use This Calculator

This tool quickly computes the standard deviation, variance, and mean for any dataset.

  1. Enter your data: Type or paste your numbers into the large input box. You can separate them using commas or spaces (e.g., 12, 14, 16 or 12 14 16).
  2. Choose the type: Select Sample if your numbers represent a small group taken from a larger population. Select Population if your numbers represent every single member of the group you are studying.
  3. Calculate: Click the button to instantly generate your statistical spread.

What is Standard Deviation?

In statistics, standard deviation is a measure of the amount of variation or dispersion within a set of values. A low standard deviation indicates that the values tend to be grouped very close to the mean (the average). A high standard deviation indicates that the values are spread out over a much wider range.

For example, if the average height of students in a class is 170cm, a low standard deviation means almost everyone is around 170cm. A high standard deviation means you have some very short students and some very tall students.

Sample vs. Population Standard Deviation

It is crucial to know whether you are working with a sample or a population, as the mathematical formulas are slightly different.

  • Population Standard Deviation (σ): Used when you have data for the entire population you are analyzing (e.g., the exact grades of every single student in a specific classroom). You divide by N.
  • Sample Standard Deviation (s): Used when your data represents only a fraction of a larger population (e.g., asking 100 people a survey question to predict how an entire country feels). You divide by N - 1. This is called Bessel's correction, and it provides a more unbiased estimate.

The Formulas

If you are calculating this by hand, you first find the Mean, subtract the Mean from every individual number, square those results, add them together to find the Variance, and then take the square root.

Population: σ = √ [ Σ ( xi - μ )2 / N ]

Sample: s = √ [ Σ ( xi - x̄ )2 / (N - 1) ]

Frequently Asked Questions (FAQ)

1. What is Variance?

Variance is simply the standard deviation squared. While standard deviation is expressed in the same units as the original data (like "centimeters" or "dollars"), variance is expressed in squared units. Standard deviation is much easier to interpret in real-world scenarios.

2. Can standard deviation be negative?

No. Because the formula requires squaring the differences from the mean (which removes all negative signs) and then taking the principal square root, standard deviation is always a positive number or exactly zero.

3. What does a standard deviation of 0 mean?

A standard deviation of exactly zero means there is absolutely no variation in your dataset. Every single number you entered is exactly the same (e.g., 5, 5, 5, 5, 5).