Standard Deviation Calculator

Calculate standard deviation and variance that measure data dispersion. Essential tool for statistical analysis, quality control, and data science.

Statistical Results

Count — Sum — Mean — Median — Standard Deviation (Sample) — Variance (Sample) — Minimum — Maximum — Range — Mode —

GUIDE

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What is Standard Deviation?

Standard deviation measures how far data points are from the mean. Small standard deviation means data is clustered around the mean; large means data is spread out. Used for analyzing test scores, quality control, stock volatility, etc.

Relationship Between Variance and Standard Deviation

Variance is the average of squared differences from the mean. Standard deviation is the square root of variance, using the same unit as original data for easier interpretation. For example, if height variance is 100cm², standard deviation is 10cm.

Population vs Sample Standard Deviation

Population standard deviation covers all data, dividing by n. Sample standard deviation estimates population from partial data, dividing by n-1 (Bessel's correction). Most real-world situations use sample standard deviation.

68-95-99.7 Rule (Empirical Rule)

In normal distribution, about 68% of data falls within ±1 standard deviation from mean, 95% within ±2, and 99.7% within ±3. This rule is used in quality control (Six Sigma), confidence intervals, outlier detection, etc.

Coefficient of Variation (CV) for Relative Comparison

Coefficient of Variation is standard deviation divided by mean, useful for comparing dispersion of data with different units. CV = (Std Dev/Mean) × 100%. For example, directly compare variability of height and weight.

Real-Life Applications of Standard Deviation

Measuring volatility (risk) in stock investing, verifying product quality consistency in manufacturing, analyzing student grade distribution in education, predicting temperature fluctuations in meteorology, evaluating clinical trial results in medicine.

Frequently asked questions

How should I enter my data?

Enter numbers separated by commas, for example: 10, 20, 30, 40, 50. Decimals and negative numbers are also accepted.

Should I use sample or population standard deviation?

Use population standard deviation if your data represents the entire group, and sample standard deviation (divided by n-1) if your data is only a subset used to estimate a larger population.

What does a standard deviation of zero mean?

It means every value in your dataset is identical. With no variation at all, both variance and standard deviation are always zero.

What if there is no single mode or multiple modes?

If several values share the highest frequency, one or more may be shown as modes; if every value appears equally often, there is no meaningful mode.

How do outliers affect the result?

Because variance squares each difference from the mean, a single extreme outlier can sharply increase both variance and standard deviation, while the median stays much less affected.