Statistics Β· Data Analysis
Standard Deviation
Calculator
Calculate population and sample standard deviation, variance, mean, median, mode, quartiles, IQR, and more β with step-by-step solutions and histogram visualization.
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Pop SD (Ο)
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Sample SD (s)
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Mean (xΜ)
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Separate values with commas, spaces, or new lines. Example: 10, 20, 30, 40, 50
Quick Datasets
Population Std Dev
Ο (sigma) β divide by N
β
Variance ΟΒ² = β
Sample Std Dev
s β divide by (Nβ1)
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Variance sΒ² = β
Mean (xΜ)
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Median
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N (count)
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Range
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Understanding Standard Deviation
Standard deviation (Ο 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 (Ο): Ο = β[ Ξ£(xα΅’ β ΞΌ)Β² / N ]
Use when you have ALL data points (the entire population).
Sample SD (s): s = β[ Ξ£(xα΅’ β xΜ)Β² / (Nβ1) ]
Use when your data is a SAMPLE from a larger population.
Dividing by (Nβ1) is Bessel's correction β it removes bias.
Variance = SDΒ²
Coefficient of Variation (CV) = (SD / Mean) Γ 100%
When should I use population vs sample standard deviation?
Use population SD (Ο) when you have data for every member of the group you're studying β 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 β e.g., a survey of 100 voters to estimate all voters, lab measurements from a batch. Sample SD uses Nβ1 (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 β values are consistent. For example, a class where all students score between 78β82% has a low SD. A high SD means data is spread widely β values vary a lot. A class with scores from 40β100% 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 1Ο of the mean, 95% within 2Ο, and 99.7% within 3Ο. 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 67β83, about 95% score between 59β91, and about 99.7% score between 51β99. Values beyond 2Ο from the mean are often flagged as outliers.
What is the coefficient of variation?
The Coefficient of Variation (CV) = (SD / Mean) Γ 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, 15β35% moderate, and above 35% high variability.