The mean height allows us to say that the Swedes are, on average, taller than Italians, but does not reveal that many Italians are taller than many Swedes.
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We will look at the following mean: Let us then examine the fixed mean, those averages that take into account all data, regardless of their order. By varying, even slightly, even one of the data, they vary continuously and without jumps. The fixed average may only be used for numeric data. In statistics, they are usually distinguished two types of mean: I mean calculation or stationary , which satisfies a condition of invariance and is calculated taking into account all values of the distribution; II mean of position or loose , calculated taking into account only of some values.
Of course, the choice of the type of media to be used depends on the problem that is being examined. We study four types of medium calculation arithmetic, geometric, quadratic, and harmonic and two types of medium position median, mode or normal value. Given n values X1, X2, The mean properties are: I internalities; II barycentric interpolation; III equivariance compared to linear transformations; IV associative; V minimization of the sum of the quadratic deviations.
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The arithmetic mean, by far the best known and most used of the mean, is the most reliable value in the following two cases: I when performing different measurements of the same magnitude; II when measuring the typical value in a homogeneous population. In the first case, when measuring a physical quantity with a tool several times you do not always get the same result. This is due to several factors: Precisely for this reason, when you want to know the precise measurement of a magnitude, this can be shown by performing different measurements. If the differences between the obtained measures are due to accidental errors, the arithmetic average of the measurements is the most reliable value of the measure of the greatness.
In the second case, when reproducing metal pieces with a mold these should all have the same weight. But if you weigh the pieces produced, the weights will be different, as a result of measurement errors, as mentioned in the previous point, as to production errors the metallic material is not perfectly homogeneous, the various pieces have never form identical, the operation of the mold is influenced by environmental factors that vary over time, etc. It gives the typical weight that each piece should have according to the ideal model derived from the mold the arithmetic average of the weights obtained can be shown.
However, it may be greatly affected by extreme values in the case in which the distribution is not symmetric. Often, instead of the simple arithmetic average, using the weighted average: If the values are all positive and not zero you can calculate the geometric mean.
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It defines the geometric mean of the values x1, x2, …, xn, the G number that replaced the xi values bring no changes to their product:. Obviously, you cannot calculate the geometric mean if one of the values is zero because the product would be zero for any value taken from others. Furthermore, xi cannot be negative.
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For the calculation of the geometric average using formulas obtained from the previous two definitions using logarithms in any base that turn them into an arithmetical average, respectively, simple or weighted. Then, the logarithm of the geometric average simple or weighted is the arithmetic average simple or weighted of the logarithms of the statistical variable values.
It uses the geometric average when it makes sense to multiply together the statistical data. You must calculate the geometric average, and not the arithmetic average, for example, to determine the average rate of increase or decrease in prices, or the growth rate of a population. It uses the geometric average when the data varies in geometric progression.
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Even the geometric average simple or weighted , enjoys certain properties including. By multiplying or dividing all xi values for a same amount h, greater than zero, the geometric average is multiplied or divided for that quantity:. The reciprocal of the geometric average is equal to the geometric average of the reciprocal of the values:. If we assume function as the sum of the quadratics of the values, indicating with Q the quadratic mean, we have, for the usual definition.
If the values have different frequencies yi you have:. The quadratic mean simple or weighted is equal to the root mean square of the arithmetic mean simple or weighted of the quadratics of the data values. Among the considered medium, the quadratic mean is the one that has higher value and is the most influenced by very small or very large values of the distribution; the root mean square is therefore used to highlight the existence of values that differ a lot from the central values.
It also uses the quadratic mean when it has an interest to calculate an average value of available surface. The harmonic average is the value which when replaced leaves unchanged the sum of the reciprocals, in other words:.
If the values have different frequencies yi, with analogous procedure it goes on to the formula:. The harmonic mean, simple or weighted, is equal to the reciprocal of the arithmetic mean, simple or weighted, of the reciprocals.
The harmonic mean is applied when it makes sense to calculate the reciprocal of the data. The harmonic mean can also be applied also to discover the mean speed as a harmonic mean speed, since the reciprocal of a speed represents the time required to cover a unity of space. Among the four averages calculation examined, there is the following relationship:. The equal sign is necessary only in the case where the data are all equal among themselves and therefore equal to any average. The arithmetic, quadratic and harmonic mean, are special cases of the general formula:.
The median is an average of position and represents the central value of the distribution when the data are sorted. Sorted values, if the number of terms n is odd, the median is just the central value; if n is even, it is taken as the median semi-sum of the two central values. The above procedure applies to the series. For frequency distributions with discrete values, the data are generally already ordained; it is then necessary to calculate the cumulative absolute frequencies, which are obtained by associating to each value from the sum of the respective frequency with all those which precede it, and determine which value corresponds:.
If the data is grouped into classes, the median class is determined by using the absolute cumulative frequency. To obtain the exact median value, linear interpolation between the two extreme values of the class in which the median falls is applied, assuming that the frequencies are distributed at regular intervals in the class. For the approximate calculation in this case it is useful to make use of the polygonal plots of cumulative relative frequencies.
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The median is not influenced by the distribution of the extreme values, so even if the extreme classes, in the case of continuous distribution, are open, there is no need to close them. In addition, if the distribution is highly asymmetrical, the median value is more appropriate than the arithmetic mean to express a synthetic value of the distribution. A feature of the median property is as follows: Next to the median the first and third quartile are considered. Percentiles are a family of indicators similar to the median.
They are thus called because a percentile bisects the normal population so as to leave a certain amount of terms to its left and the remaining amount to its right. A synthetic value can be calculated in various ways 5. Some mean values satisfy a condition of invariance of a global value, namely: You use the arithmetic mean to determine a value that expresses a concept of equitable distribution when, for example, you want to determine an average of the costs, consumption, income, temperature.
It also applies the arithmetic mean, for the properties of its waste, to determine the precise value of a series of measures, provided that the measurement errors are accidental and not systematic in practice due to the instruments ; the arithmetic mean also applies if your data follow one another in arithmetic progression. The geometric mean is used to determine the average rate of increase or decrease of a phenomenon, the average interest rate of more rate in compound interest, or to determine an average exchange rates in the money.
The geometric average is used even when the data are followed in geometric progression. The quadratic mean is applied when you have to eliminate the influence of the signs and when you have to highlight the existence in the distribution of very large or very small values. The harmonic mean is applied when you want to know the average value using the reciprocal values of another character, such as the purchasing power of the currency.
The mode or the normal value of a frequency distribution is important when it is necessary to know the value that has is most probable to show up. The median value is the central value of the distribution and is independent of strong differences between the data. You cannot give a general rule for choosing the type of media, but you have to calculate more than an average value and choose the most adequate for the resolving the problem at hand.
The medium which is used most frequently in practice are the arithmetic mean, the median and, in the case of frequency distributions, the modal value. In many the coefficient software. In statistics having to do with a large number of data, it is convenient to consider the frequencies of the experimental units 7: If we are dealing with quantitative variables on a continuous scale, before calculating the frequencies it is convenient to split the range of measures in a number of frequency classes.
By dividing the absolute frequency by the total number of statistical units we get the so-called relative frequencies. The advantage with respect to the absolute frequencies consist in the comparison of frequency distributions based on different numbers of statistical units. I the relative frequency of a mode, the frequency of the uniform mode to the total number of frequencies; II the relative intensity of a mode, the intensity of the divided by the total of the intensity mode.
The characteristics are as follows: Those related to coincide with the empirical distribution function at the end of each interval. The absolute cumulative frequency Fi is the sum of the absolute frequency relative to Xi mode and absolute frequencies that precede it, then:. The relative cumulative frequency Ri is the sum of the relative frequency corresponding to Xi mode and its associated frequencies that precede it, then:. The graphs in tapes and columns are the first form of graphical representation and are often used to represent mutable or statistical variables.
The data tables are represented by drawing generic rectangles width and length proportional to the frequency or the intensity of the mode. The rectangles can be arranged horizontally graph tapes or vertically in columns graphics. They use the graphs in circular sectors in order to better highlight the subdivision of the phenomenon between the various modes that compose it. The frequency or the total intensity of the phenomenon is represented by the area around the circle, while the areas of each sector represent the frequencies or intensities of the individual modes:.
Histograms are made up of as many rectangles as there are classes. If all classes have the same amplitude, the rectangles have equal bases and the heights are proportional to the frequencies of the classes. In the presence of classes with different amplitude it is necessary that the heights of the rectangles be proportional to the frequency density, i.
There are two types of histograms: