Variability rate

Variability rates (dispersion/scatter/spread rates) indicate how close or distant the values ​​of the quantity are to each other in the individual elements of the set; they evaluate the dispersion of the values ​​of a statistical set around some mean value.

Examples of measures of variability are: range, interquartile range, standard deviation, variance, coefficient of variation, median absolute deviation, etc.

Range
Range (or span) je nejjednodušší mírou variability. Počítá se jako rozdíl největší a nejmenší hodnoty souboru:


 * $$ R = x_{max} - x_{min}.$$

The disadvantage of this value is that it depends on extreme values, so it can give very misleading information about the given phenomenon.

Interquartile range
Interquartile range (IQR) represents the difference between the third and first quartiles (that is, between the 75th and 25th percentiles). It therefore represents the range of values ​​that has the middle 50% of the values ​​of the variable:


 * $$ \mathrm{IQR} = q_{3} - q_{1}$$

Variance
Variance (s2) is a quite often used measure of variability. It is equal to the average square of the deviation of the value from the statistical set from the arithmetic mean. The greater the variance, the more the data deviates from the mean. The disadvantage is that the variance does not have the same physical dimension as the features of a statistical ensemble. The variance is calculated as:


 * $$s^2_n = \frac 1n \sum_{i=1}^n \left(y_i - \overline{y} \right)^ 2 = \frac{1}{n} \sum_{i=1}^{n}y_i^2 - \overline{y}^2$$

(the second formula is more suitable for manual calculation)

The above formulas for calculating variance are only used in descriptive statistics to determine the variability of a known set of statistics. However, if we infer the variance of an unknown random variable from a (known) sample set, it is necessary to use the formula for the so-called sample variance for calculation:


 * $$s^2=\frac{1}{n-1}\sum_{i=1}^n(x_i-\overline{x})^2=\frac{1}{n-1}\sum_{i=1}^n x_i^2-\frac{n}{n-1}\cdot\overline{x}^2.$$

Standard deviation
Standard deviation (s or SD) is the most commonly used measure of ensemble variability. Its great advantage over variance is that it has the same physical dimension as the mean. Typically, the sample standard deviation is used as the square root of the sample variance:


 * $$s = \sqrt{s^2}$$

Standard deviation and sample standard deviation are often marked as &sigma;n and &sigma;n-1 in the statistical functions used in the calculators.

Coeffficient of variation
Coefficient of variation represents a relative degree of variability. It is used to compare variability between data sets with different means. It is calculated as the quotient of the standard deviation and the mean:


 * $$k=\frac{s}{\bar{x}}$$.

Median absolute deviation
Median absolute deviation (MAD) represents a degree of variability that is only slightly affected by extreme values. It is defined by a formula:


 * $$\text{MAD}=\text{median}(|x_i-\tilde{x}|).$$

It's almost never used in medical statistics.

Related articles

 * Average
 * Distribution function