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Standard Deviation Calculator

Calculate mean, variance, and standard deviation

Standard Deviation Calculator

Numbers (comma or newline separated)

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About This Calculator

A standard deviation calculator measures the amount of variation or dispersion in a dataset, telling you how spread out the values are from the mean. The formula for population standard deviation is σ = √[Σ(xi − μ)² / N], and for a sample it is s = √[Σ(xi − x̄)² / (N−1)], where the N−1 denominator (Bessel's correction) provides an unbiased estimate for samples. The calculator also computes the mean, variance (standard deviation squared), range, and count. For example, test scores of {85, 90, 78, 92, 88} have a mean of 86.6 and a sample standard deviation of about 5.5, meaning most scores fall within 5.5 points of the average. A low standard deviation indicates data points cluster tightly around the mean, while a high value means they are widely scattered. This tool is used in academic research, quality control (Six Sigma), financial risk analysis (stock volatility), test score interpretation, scientific experiments, and any field that needs to quantify variability. Understanding standard deviation is essential for interpreting confidence intervals, z-scores, and normal distributions. Enter your numbers separated by commas or spaces to get instant results.

How to Use

1
Enter your data
Input your numbers separated by commas or line breaks.
2
Select mode
Choose between population or sample standard deviation.
3
View statistics
See the mean, median, variance, standard deviation, and other summary statistics.

Frequently Asked Questions

Q. What is the difference between population and sample standard deviation?

Population standard deviation (σ) divides by N and is used when you have data for every member of the group. Sample standard deviation (s) divides by N−1 and is used when your data is a subset of the larger population. The N−1 correction (Bessel's correction) prevents underestimation of the true variability.

Q. What does a high standard deviation mean?

A high standard deviation means the data points are widely spread from the mean. In test scores, a high SD indicates a wide range of performance. In finance, a high SD for stock returns means high volatility and risk. Whether a SD is "high" depends on the context and units — compare it to the mean or to expected values in your field.

Q. How is standard deviation related to the normal distribution?

In a normal distribution, about 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three — known as the 68-95-99.7 rule. This makes standard deviation the key measure for understanding how unusual a particular value is and for constructing confidence intervals.

Q. When should I use variance versus standard deviation?

Standard deviation is more intuitive because it is in the same units as your data (e.g., dollars, points). Variance (SD squared) is in squared units, which is less interpretable but is mathematically important in statistical formulas, ANOVA, regression analysis, and portfolio theory. Most people report standard deviation for communication purposes.

Disclaimer: Results are for informational purposes only and do not constitute professional advice. Always consult qualified professionals for important decisions.

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