Statistics · Data Analysis
Standard Deviation
Calculator
Calculate population and sample standard deviation, variance, mean, median, mode, quartiles, IQR, and more — with step-by-step solutions and histogram visualization.
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Pop SD (σ)
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Sample SD (s)
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Mean (x̄)
Enter Your Data
Separate values with commas, spaces, or new lines. Example: 10, 20, 30, 40, 50
Quick Datasets
Population Std Dev
σ (sigma) — divide by N
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Variance σ² = —
Sample Std Dev
s — divide by (N−1)
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Variance s² = —
Mean (x̄)
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Median
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N (count)
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Range
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Normal
Outlier (>2σ from mean)
📋 Step-by-Step: Population Standard Deviation
🔢 Sorted Data & Deviations Table
Understanding Standard Deviation
Standard deviation (σ or s) measures how spread out data is from the mean. A low SD means data points are clustered near the mean; a high SD means data is spread widely. It's the most commonly used measure of variability in statistics.
Population vs Sample Standard Deviation
Population SD (σ): σ = √[ Σ(xᵢ − μ)² / N ]
Use when you have ALL data points (the entire population).
Sample SD (s): s = √[ Σ(xᵢ − x̄)² / (N−1) ]
Use when your data is a SAMPLE from a larger population.
Dividing by (N−1) is Bessel's correction — it removes bias.
Variance = SD²
Coefficient of Variation (CV) = (SD / Mean) × 100%
When should I use population vs sample standard deviation?
Use population SD (σ) when you have data for every member of the group you're studying — e.g., grades of all students in a class, heights of all players on a team. Use sample SD (s) when your data is a subset taken from a larger population — e.g., a survey of 100 voters to estimate all voters, lab measurements from a batch. Sample SD uses N−1 (Bessel's correction) to give an unbiased estimate of the population SD.
What does a high vs low standard deviation mean?
A low SD means data points are close to the mean — values are consistent. For example, a class where all students score between 78–82% has a low SD. A high SD means data is spread widely — values vary a lot. A class with scores from 40–100% has a high SD. In finance, high SD means high volatility (risk). In manufacturing, low SD means consistent product quality. A SD of 0 means all values are identical.
What is the 68-95-99.7 rule?
For a normal distribution: approximately 68% of data falls within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. This is called the empirical rule or 68-95-99.7 rule. For example, if test scores have mean=75 and SD=8: about 68% score between 67–83, about 95% score between 59–91, and about 99.7% score between 51–99. Values beyond 2σ from the mean are often flagged as outliers.
What is the coefficient of variation?
The Coefficient of Variation (CV) = (SD / Mean) × 100%. It expresses SD as a percentage of the mean, making it possible to compare variability between datasets with different units or scales. For example, comparing the variability of salaries ($) vs. ages (years): CV normalizes for the different scales. A higher CV means more relative variability. Generally, CV below 15% is considered low, 15–35% moderate, and above 35% high variability.