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The Variance Calculator computes both population variance ($$\sigma^2$$) and sample variance ($$s^2$$) for a dataset of five values, along with the sum of squares, mean, and standard deviations. Variance is a fundamental measure of dispersion that quantifies how far data values spread from the mean.
Variance is defined as the average of the squared deviations from the mean:
$$\sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2 = \frac{SS}{N}$$
For five values:
$$\sigma^2 = \frac{(x_1-\bar{x})^2 + (x_2-\bar{x})^2 + (x_3-\bar{x})^2 + (x_4-\bar{x})^2 + (x_5-\bar{x})^2}{5}$$
$$s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2 = \frac{SS}{n-1}$$
The $$n - 1$$ denominator (Bessel's correction) produces an unbiased estimate of the population variance when working with sample data.
The sum of squares is the numerator shared by both variance formulas:
$$SS = \sum_{i=1}^{n}(x_i - \bar{x})^2$$
This quantity appears throughout statistics — in ANOVA (between-group and within-group SS), regression (SSR, SSE, SST), and the coefficient of determination $$R^2$$.
An equivalent formula that avoids computing deviations individually:
$$\sigma^2 = \frac{\sum x_i^2}{N} - \left(\frac{\sum x_i}{N}\right)^2 = E[X^2] - (E[X])^2$$
This is known as the raw score formula or the "mean of squares minus square of the mean."
In portfolio theory, variance measures investment risk — the Markowitz model minimizes portfolio variance for a target return. In ANOVA, variance is decomposed into between-group and within-group components to test for significant differences. In physics, variance characterizes energy fluctuations in thermodynamic systems. In signal processing, variance measures signal power and noise levels.
Enter five values. The calculator computes the mean, then calculates each squared deviation from the mean. The sum of these squared deviations (SS) is divided by 5 for population variance or by 4 for sample variance. Standard deviations are the square roots of the respective variances.
Population variance is appropriate when your 5 values represent the entire dataset of interest. Sample variance is appropriate when the 5 values are drawn from a larger population. A variance near zero indicates data clustered around the mean; a large variance indicates widely dispersed data. The sum of squares (SS) shows the total squared deviation before averaging.
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Results
Mean = 4, SS = 32.8. Population variance (6.56) is 80% of sample variance (8.2). The data is moderately dispersed around the mean.
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Results
Variance is very small (σ² = 2) relative to the mean (100), indicating highly consistent data with CV ≈ 1.4%.
Variance measures how spread out numbers are from their average. It calculates the average of the squared differences between each value and the mean. A variance of 0 means all numbers are identical; a large variance means numbers are widely scattered.
Squaring has several advantages: (1) it ensures all deviations are positive, (2) it penalizes larger deviations more heavily, (3) it produces a smooth, differentiable function (useful in optimization), and (4) squared deviations have convenient mathematical properties like additivity for independent variables.
The sum of squares $$SS = \sum(x_i - \bar{x})^2$$ is the total squared deviation from the mean. It appears in many statistical procedures: ANOVA decomposes total SS into between-group and within-group SS; regression uses SS to compute $$R^2$$. It is the raw measure of total variability before dividing by $$N$$ or $$n-1$$.
Because variance involves squaring the deviations, the units are also squared (e.g., if data is in kg, variance is in kg²). This is why standard deviation (the square root of variance) is often preferred — it returns to the original units, making interpretation more intuitive.
Use population variance ($$\sigma^2$$, divide by $$N$$) when your data includes every member of the group of interest. Use sample variance ($$s^2$$, divide by $$n-1$$) when your data is a subset of a larger population and you want to estimate the population's variability.
Standard deviation is the square root of variance: $$\sigma = \sqrt{\sigma^2}$$ and $$s = \sqrt{s^2}$$. They contain the same information but standard deviation is in the original data units, making it easier to interpret alongside the mean.
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