Correlation | Linear Regression | Variance and Standard Deviation | Normal Distribution

Calculator Variance and Standard Deviation

Calculates the empiric variance and standard deviation from a data set of associated values. The variance is a measure for the dispersion, how far spread or how close together the values are. The higher variance and standard deviation are, the stronger is the spread. As the variance works with squares, more often the standard deviation is used, which is the square root of the variance.
Example calculates with the population (in millions) of some European countries.

Values

Round to   0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15   decimal places.

Result

Number of values:
Mean value:
Variance:
Standard deviation:
Coefficient of variation:

The formulas are:

x_i: values, n: number of values, Σ: sum i=1 to n
μ: mean, σ²: variance, σ: standard deviation, CV: coefficient of variation

μ = Σ(x_i) / n
σ² = Σ(x_i-μ)² / n
σ = √σ²
CV = σ / μ

Variance and standard deviation can never be negative. If they are zero, then all measured values ​​have exactly the same level. Otherwise, these figures are only meaningful in connection with the mean. The larger the mean, the larger the standard deviation will be if the distribution otherwise remains the same. The ratio of standard deviation and mean is the coefficient of variation. This value is dimensionless and meaningful on its own, but you have to make sure that the mean is not zero or close to zero. The larger the coefficient of variation, the wider the spread of the values. To make it clearer, you can also see it as a percentage value; a coefficient of variation of 0.5 then corresponds to a deviation of 50 percent.
Taking these statistical values ​​into account is often essential for a scientifically sound interpretation of empirical data. Outside of science, for the sake of simplicity, often only the mean is given, which is accompanied by a loss of information and can lead to incorrect conclusions.
A common form of distribution is the normal distribution, in which mean values ​​occur very frequently and extremes occur less frequently the more extreme they are. If the values ​​are arranged so that the mean is 0 and the standard deviation is 1, then it is a standard normal distribution.

Last updated on 06/24/2025. Author: Jürgen Kummer

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