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# Indicators of variations

Calculation of variation - the coefficient of variation, dispersion, mean square deviation, etc.

User Maria asked me to make this calculator: /680/

The calculations in this calculator are not so hard, so here it is. Traditionally, the theory is below the calculator.

#### Population of analysis

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Items per page:

Digits after the decimal point: 2
Arithmetic average

Range of variability

Mean deviation

Dispersion

Mean square deviation

Oscillations coefficient (percentage)

Relative linear deviation (percentage)

Variation coefficient (percentages)

Variation - it is a difference of individual values any indication within the target population.

For example, we have a class of students - target population, and they have an annual rating of the Russian language. Somebody have an A, somebody have a B and so on. Set of these ratings throughout the class, along with their frequency ( i.e. the occurrence, for example, 10 persons have an "A", 7 persons have a - "B", 5 persons - "C") that is a variation on which you can calculate a lot of indicators.

That's what we will do.
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### Absolute indicators

1. Range of variability - the difference between the maximum and minimum of attribute value
$R=x_{max}-x_{min}$

2. Mean deviation - the arithmetic mean deviation of individual values from the mean
$\bar{l}=\frac{\sum{|x_i-\bar{x}|f_i}}{\sum{f_i}}$,
where $f_i$ - occurrence frequency of $x_i$.

If there are too many individual values, the data can be simplified for calculations by grouping i.e. combined into intervals.
Then $x_i$ have meaning of i-interval or have a mean observation on i-interval.

1. Dispersion - the average of the squared deviationsзначений of characteristic values of the average.
$\sigma^2=\frac{\sum{(x_i-\bar{x})^{2}f_i}}{\sum{f_i}}$

Dispersion can also be calculated the following way:
$\sigma^2=\bar{x^2}-\bar{x}^2$, where $\bar{x^2}=\frac{\sum{x^{2}f}}{\sum{f}}$

1. Mean square deviation - $\sigma$, root of dispersion

### Relative indicators

Absolute indicators are measured in the same magnitude as the indicator itself and show the absolute size of deviations, therefore they are inconvenient to use for comparison the variability of different population indicators. Therefore, relative indicators of variations are calculated additionally.

1. Oscillation coefficient - it characterizes the variability of extreme values of indicators around the arithmetic mean.
$K_o=\frac{R}{\bar{x}}$

2. Relative linear deviation или linear coefficient of variation - it describes the proportion of the average value out of arithmetical mean
$K_l=\frac{\bar{l}}{\bar{x}}$

3. Variation coefficient - It characterizes the degree of homogeneity of the population, the most frequently used indicator.
$V_\sigma=100\frac{\sigma}{\bar{x}}$

The population is considered to be homogenous at values less than 40%. For values greater than 40% indicate the large indicator oscillation and it's considered to be inhomogeneous.

PLANETCALC, Indicators of variations