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📊 Variance Calculator
Calculate variance, standard deviation, and measures of variability
📏 Enter Your Data
Choose "Sample" for a subset or "Population" for complete data
Enter numbers separated by commas, spaces, or new lines
📊 Results
Sample Variance (s²)
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Std Dev: --
Mean
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Count (n)
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Range
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Coefficient of Variation
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Data Distribution
Calculation Steps
📚 Understanding Variance
What is Variance?
Variance is a statistical measure that quantifies the spread or dispersion of a set of data points around their mean. A higher variance indicates that data points are more spread out, while a lower variance means they are closer to the mean.
Population vs Sample Variance
Population Variance (σ²): Used when you have data for the entire population. The formula divides by n (the total number of values).
σ² = Σ(xᵢ - μ)² / n
Sample Variance (s²): Used when you have data from a sample of a larger population. The formula divides by (n-1) to provide an unbiased estimate. This adjustment is called Bessel's correction.
s² = Σ(xᵢ - x̄)² / (n - 1)
Standard Deviation
Standard deviation is the square root of variance. It's expressed in the same units as the original data, making it more interpretable than variance. A low standard deviation indicates data points are close to the mean, while a high standard deviation indicates greater spread.
σ or s = √variance
Coefficient of Variation (CV)
The coefficient of variation expresses the standard deviation as a percentage of the mean. It's useful for comparing the variability of datasets with different units or scales.
CV = (Standard Deviation / Mean) × 100%
When to Use Each Type
- Use Population Variance: When analyzing complete census data, all test scores in a class, or any complete dataset
- Use Sample Variance: When working with survey data, quality control samples, or any subset of a larger population
- General Rule: If you're unsure, use sample variance as it provides a more conservative estimate
Practical Applications
- Finance: Measuring investment risk and portfolio volatility
- Quality Control: Monitoring manufacturing process consistency
- Research: Analyzing experimental data and survey results
- Weather: Studying temperature variations and climate patterns
- Education: Evaluating test score distributions
- Healthcare: Analyzing patient data and treatment outcomes
Frequently Asked Questions
What's the difference between variance and standard deviation?
Variance is the average of squared deviations from the mean, while standard deviation is the square root of variance. Standard deviation is in the same units as the original data, making it easier to interpret. For example, if measuring heights in inches, variance is in square inches, but standard deviation is in inches.
Why do we divide by (n-1) for sample variance?
Dividing by (n-1) instead of n is called Bessel's correction. It provides an unbiased estimate of the population variance from a sample. Since we use the sample mean (which is itself an estimate), we lose one degree of freedom, so we divide by (n-1) to compensate.
Can variance be negative?
No, variance cannot be negative. Since it's calculated by squaring the deviations from the mean, all values are positive or zero. A variance of zero means all data points are identical. The minimum possible variance is 0, and there's no maximum limit.
What is a good variance value?
There's no universal "good" variance value—it depends on your data and context. Low variance indicates consistency (good for manufacturing), while high variance might indicate diversity (sometimes desirable in portfolios). Use the coefficient of variation to compare variability across different datasets or scales.
How do outliers affect variance?
Outliers significantly increase variance because deviations are squared in the calculation. A single extreme value can dramatically inflate the variance. If outliers are errors or anomalies, consider removing them. If they're legitimate data points, they represent real variability in your dataset.
What's the relationship between variance and normal distribution?
In a normal distribution, about 68% of data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three. This relationship helps interpret variance: higher variance means data is more spread out across the distribution curve.
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