Standard Deviation Calculator
Enter a set of numbers to calculate the standard deviation, variance, mean, and other statistical measures. Choose between sample and population data.
Statistical Results
Enter your data and click "Calculate".
Understanding Standard Deviation
Standard Deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (the average) of the set, while a high standard deviation indicates that the values are spread out over a wider range.
Sample vs. Population
It's crucial to know whether your data represents a whole population or just a sample of one.
- Population: Includes every member of a group being studied. For example, the test scores of every student in a specific school. The population standard deviation is denoted by the Greek letter sigma (σ).
- Sample: A subset of a larger population. For example, the test scores of 50 students selected randomly from that same school. The sample standard deviation is denoted by the letter 's'.
The key difference is in the formula: the sample variance is divided by n-1 (Bessel's correction) to provide a better estimate of the population variance, while the population variance is divided by N.
Formulas Used by the Calculator
- Mean (μ or x̄): The average of the data set.
Formula: μ = (Σx) / N - Variance (σ2 or s2): The average of the squared differences from the Mean.
Population Variance: σ2 = Σ(xi - μ)2 / N
Sample Variance: s2 = Σ(xi - x̄)2 / (n-1) - Standard Deviation (σ or s): The square root of the Variance. It brings the measure back to the original unit of the data.
Formula: σ = √σ2 or s = √s2
You can find the square root of any number using our Root Calculator.
Why is Standard Deviation Important?
- Finance: In investing, standard deviation of an asset's price is a common measure of its volatility or risk.
- Manufacturing: Used in quality control to ensure products meet specifications. A low standard deviation means more consistency.
- Science: To understand the statistical significance of experimental results and the margin of error.
- Weather: To describe the range of typical temperatures for a location.
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