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Approximation error
Name  Formula  Legend 
Absolute error  Show source$\Delta x=\left x x_0\right$ 

Relative error  Show source$\delta x_{rel.}=\left\frac{ x x_0}{ x_0}\right$ 

Average value (mean)
Name  Formula  Legend 
Arithmetic mean (average), expected value  Show source$\overline{x} = \frac{x_1 + x_2 + \cdots + x_n}{n}$ 

Geometric mean (average)  Show source$\overline{x}_{geom.} = \sqrt[n]{x_1 \cdot x_2 \cdot \dotso \cdot x_n}$ 

Weighted mean (average)  Show source$\overline{x}_{weight.} = \frac{w_1 x_1 + w_2 x_2 + \cdots + w_3 x_n}{w_1 + w_2 + \cdots + w_3}$ 

Harmonic mean (average)  Show source$\overline{x}_{harm.} = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}}$ 

Root mean square (RMS)  Show source$RMS(x) = \sqrt{\frac{x_1^2 + x_2^2 + \cdots + x_n^2}{n}}$ 

Weighted root mean square  Show source$RMS_{weight.}(x) = \sqrt{\frac{w_1 x_1^2 + w_2 x_2^2 + \cdots + w_3 x_n^2}{w_1 + w_2 + \cdots + w_3}}$ 

Generalized mean, power mean, Hölder mean  Show source$\mu_k = \sqrt[n]{\frac{x_1^p + x_2^p + \cdots + x_n^p}{n}}$ 

Generalized weighted mean  Show source$\mu_k = \sqrt[n]{\frac{w_1 x_1^p + w_2 x_2^p + \cdots + w_3 x_n^p}{w_1 + w_2 + \cdots + w_3}}$ 

Statistical dispersion
Name  Formula  Legend 
Standard deviation  Show source$\sigma = \sqrt{\frac{(x_1  \overline{x})^2 + (x_2  \overline{x})^2 + \cdots + (x_n  \overline{x})^2}{n}}$ 

Variance  Show source$\sigma^2 = \frac{(x_1  \overline{x})^2 + (x_2  \overline{x})^2 + \cdots + (x_n  \overline{x})^2}{n}$ 

Some facts
 Statistics delivers tools for processing big amout of data.
 The subject of statistical surveys are not individual events (e.g. one coin toss), but the properties of the whole data set. In this sense, the data set (e.g. people population in selected region) is a new entity with its own new properties that can be studied independently.
 Example applications of statistical methods are:
 analysis of macroeconomic data from a given country or region (e.g. to make optimal political decisions, at least in theory...),
 psychometric tests used by psychologists (e.g. to classify an unknown patient and quickly adjust the way of contact with him),
 analysis of medical data on the frequency of selected diseases (e.g. to assess the risk of complications after a surgery),
 statistical thermodynamics, an attempt to predict the properties of the macroscopic system (e.g. gas vessels) based on the microscopic system (e.g. single atoms),
 etc.
 analysis of macroeconomic data from a given country or region (e.g. to make optimal political decisions, at least in theory...),
 One of the most commonly used statistical concepts is an average value (sometimes called expected value). There are many different types of averages that differ in the definition and area where they are used. Example definitions of averages from n numbers with values of $x_1$, $x_2$, $x_3$, ... are:
 arithmetic mean  sum of numbers divided by their count, the most frequently used type of mean,
$\overline{x} = \dfrac{x_1 + x_2 + \cdots + x_n}{n}$  geometric mean  nth root from the product of numbers, where n is the numbers count in the set,
$\overline{x}_{geom.} = \sqrt[n]{x_1 \cdot x_2 \cdot \dotso \cdot x_n}$  root mean square (RMS)  used where sign of averaged numbers doesn't matter e.g. when calculating the accoustic power,
$RMS(x) = \sqrt{\dfrac{x_1^2 + x_2^2 + \cdots + x_n^2}{n}}$  harmonic mean  where we take into account the inverse numbers:
$\overline{x}_{harm.} = \dfrac{n}{\dfrac{1}{x_1} + \dfrac{1}{x_2} + \cdots + \dfrac{1}{x_n}}$
 arithmetic mean  sum of numbers divided by their count, the most frequently used type of mean,
 In the general case, different types of averages give different result for the same data set. The only case when all types of averages will give the same result is a situation when all numbers are equal. The relation between various types of averages have been systematically examined by a French mathematician Augustin Louis Cauchy. For this reason, the serie of averages in increasing values order is sometimes called the Cauchy inequality.
 Sometimes it happens that selected elements are more important than other ones e.g. when the current data is considered the most important and the past as less important. Then, the socalled weighted means are used. The elements, which have the highest weight assigned, have the biggest influence on the whole average and vice versa. Most types of means have their weighted equivalents, e.g. weighted arithmetic mean takes the form:
$\overline{x}_w = \dfrac{w_1 x_1 + w_2 x_2 + \cdots + w_n x_n}{w_1 + w_2 + \cdots + w_n}$where:
 $\overline{x}_w$  weighted arithmetic mean from numbers $x_1$, $x_2$, $x_3$, ..., etc.,
 $x_n$  nth number in the data set,
 $w_n$  the weight assigned to the nth number, zero weight means that number has no impact on mean, bigger weight means that a number is more important,
 $n$  count of numbers in the data set.
 $\overline{x}_w$  weighted arithmetic mean from numbers $x_1$, $x_2$, $x_3$, ..., etc.,
 Measuring variation is commonly used statistical concept. There are various types of variation measures. Simply speaking, if values are more diverse and deviate from the average, the more variation we assign to data set. In particular, if all values in the set are identical, then we will say that there is no variability at all. One of the most popular measures of variation is variance:
$\sigma^2 = \frac{(x_1  \overline{x})^2 + (x_2  \overline{x})^2 + \cdots + (x_n  \overline{x})^2}{n}$where:
 $\sigma^2$  variance,
 $\overline{x}$  arithmetic mean (the sum of a collection of numbers divided by the count of numbers in the collection),
 $x_1$  the first number in the collection,
 $x_2$  the second number in the collection,
 $x_n$  the nth number in the collection,
 n  count of numbers in the collection.
 Other popular measure of variation is standard deviation defined as below:
$\sigma = \sqrt{\frac{(x_1  \overline{x})^2 + (x_2  \overline{x})^2 + \cdots + (x_n  \overline{x})^2}{n}}$where:
 $\sigma$  standard deviation,
 $\overline{x}$  arithmetic mean (the sum of a collection of numbers divided by the count of numbers in the collection),
 $x_1$  the first number in the collection,
 $x_2$  the second number in the collection,
 $x_n$  the nth number in the collection,
 n  count of numbers in the collection.
 We can calculate standard deviation from variance and vice versa. So these are two different measures of the same quantity, which we can use depending on the needs:
$\text{variance} = \text{Var} = \sigma^2$$\sigma = \sqrt{\text{variance}} = \sqrt{\text{Var}}$  In the case of measurement data or data spread over time, the variation measures can be equated with uncertainty or treated as risk factor. For example, the purchase of shares of companies, for which the standard deviation is high, is more risky than other ones. In practice, this means that the share price of such a company is more susceptible to fluctuations. Colloquially, we will say that the share price of such a company is less predictable.
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