Mathematical tables: typical statistics related formulas
Table shows common formulas used during statistical data processing such as various types of means (averages), standard deviation or variance.

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# Approximation error#

 Name Formula Legend Absolute error Show source$\Delta x=\left|x-x_0\right|$ $\Delta x$ - absolute error,x - measured value (value, which was measured, calculated or approximated),$x_0$ - reference values. Relative error Show source$\delta x_{rel.}=\left|\frac{x-x_0}{x_0}\right|$ $\delta x_{rel.}$ - relative error,x - measured value (value, which was measured, calculated or approximated),$x_0$ - reference values.

# Average value (mean)#

 Name Formula Legend Arithmetic mean (average), expected value Show source$\overline{x} = \frac{x_1 + x_2 + \cdots + x_n}{n}$ $\overline{x}$ - 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 n-th number in the collection,n - count of numbers in the collection. Geometric mean (average) Show source$\overline{x}_{geom.} = \sqrt[n]{x_1 \cdot x_2 \cdot \dotso \cdot x_n}$ $\overline{x}_{geom.}$ - geometric mean,$x_1$ - the first number in the collection,$x_2$ - the second number in the collection,$x_n$ - the n-th number in the collection,n - count of numbers in the collection. Weighted mean (average) Show source$\overline{x}_{weight.} = \frac{w_1 x_1 + w_2 x_2 + \cdots + w_3 x_n}{w_1 + w_2 + \cdots + w_3}$ $\overline{x}_{weight.}$ - weighted mean,$x_1$ - the first number in the collection,$x_2$ - the second number in the collection,$x_n$ - the n-th number in the collection,$w_1$ - weight assigned to the first number in the collection,$w_2$ - weight assigned to the second number in the collection,$w_3$ - weight assigned to the n-th number in the collection. Harmonic mean (average) Show source$\overline{x}_{harm.} = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}}$ $\overline{x}_{harm.}$ - harmonic average,$x_1$ - the first number in the collection,$x_2$ - the second number in the collection,$x_n$ - the n-th number in the collection,n - count of numbers in the collection. Root mean square (RMS) Show source$RMS(x) = \sqrt{\frac{x_1^2 + x_2^2 + \cdots + x_n^2}{n}}$ $RMS(x)$ - root mean square,$x_1$ - the first number in the collection,$x_2$ - the second number in the collection,$x_n$ - the n-th number in the collection,n - count of numbers in the collection. Weighted root mean square Show source$RMS_{weight.}(x) = \sqrt{\frac{w_1 x_1^2 + w_2 x_2^2 + \cdots + w_3 x_n^2}{w_1 + w_2 + \cdots + w_3}}$ $RMS_{weight.}(x)$ - root mean square weighted,$x_1$ - the first number in the collection,$x_2$ - the second number in the collection,$x_n$ - the n-th number in the collection,$w_1$ - weight assigned to the first number in the collection,$w_2$ - weight assigned to the second number in the collection,$w_3$ - weight assigned to the n-th number in the collection. Generalized mean, power mean, Hölder mean Show source$\mu_k = \sqrt[n]{\frac{x_1^p + x_2^p + \cdots + x_n^p}{n}}$ $\mu_k$ - generalized mean with with exponent p,$x_1$ - the first number in the collection,$x_2$ - the second number in the collection,$x_n$ - the n-th number in the collection,n - count of numbers in the collection,p - the exponent of the generalized mean. Generalized weighted mean Show source$\mu_k = \sqrt[n]{\frac{w_1 x_1^p + w_2 x_2^p + \cdots + w_3 x_n^p}{w_1 + w_2 + \cdots + w_3}}$ $\mu_k$ - weighted generalized mean with with exponent p,$x_1$ - the first number in the collection,$x_2$ - the second number in the collection,$x_n$ - the n-th number in the collection,$w_1$ - weight assigned to the first number in the collection,$w_2$ - weight assigned to the second number in the collection,$w_3$ - weight assigned to the n-th number in the collection,n - count of numbers in the collection,p - the exponent of the generalized mean.

# Statistical dispersion#

 Name Formula Legend Standard deviation Show source$\sigma = \sqrt{\frac{(x_1 - \overline{x})^2 + (x_2 - \overline{x})^2 + \cdots + (x_n - \overline{x})^2}{n}}$ $\sigma$ - 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 n-th number in the collection,n - count of numbers in the collection. Variance Show source$\sigma^2 = \frac{(x_1 - \overline{x})^2 + (x_2 - \overline{x})^2 + \cdots + (x_n - \overline{x})^2}{n}$ $\sigma^2$ - 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 n-th number in the collection,n - count of numbers in the collection.

# 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.
• 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 - n-th 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 acoustic 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:
$\overline{x}_{harm.} = \dfrac{n}{\dfrac{1}{x_1} + \dfrac{1}{x_2} + \cdots + \dfrac{1}{x_n}}$
• 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 so-called 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.,
• $x_n$ - n-th number in the data set,
• $w_n$ - the weight assigned to the n-th 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.
• 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 n-th 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 n-th 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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