How to know the variation series in statistics. An example of a typical task. The Kolmogorov-Smirnov criterion is now victorious
An example of a perfection of the control work of mathematical statistics
Head 1
Pochatkov data : students of the singing group, which consists of 30 osіb, folded іspit from the course "Informatics". The grades taken by the students are approved by the next series of numbers:
I. Storable variation series
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m x nak |
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II. Graphic presentation of statistical reports.
III. Numerical characteristics of the vibrator.
1. Arithmetic mean
2. Mean geometric
3. Fashion 
4. Median
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5. Vibration dispersion
7. Coefficient of variation
8. Asymmetry
9. Coefficient of asymmetry
10. Excess
11. Coefficient of excess
Manager 2
Pochatkov data : the students of the singing group wrote the final control work. The group consists of 30 items. Gather students bali to satisfy the next series of numbers
Solution
I. Shards of the sign gain a rich variety of meanings, then for the new one we will create an interval variation series. For which order, we set the value of the interval h. Speed up with the Sturgeer Formula
Let's create a scale of intervals. For the upper interval of the first interval, the value is taken, which is assigned to the formula:
The upper between offensive intervals is significant for the offensive recurrent formula:
also
Pobudov’s scale of intervals is over, so that the upper boundary of the chalk interval has become larger or more equal to the maximum value of the vibration 
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II. Graphic display of the interval variation series
III. Numerical characteristics of the vibrator
To determine the numerical characteristics of the selection, we add an additional table
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Suma: |
1. Arithmetic mean
2. Mean geometric
3. Fashion 
4. Median
10 11 12 12 13 13 13 13 14 14 14 14 15 15 15 |15 15 15 16 16 16 16 16 17 17 18 19 19 20 20
5. Vibration dispersion
6. Vibirkove standard resignation
7. Coefficient of variation
8. Asymmetry
9. Coefficient of asymmetry
10. Excess
11. Coefficient of excess
Manager 3
Umov : the price of the sub-scale of the ammeter is 0.1 A. The indication is rounded to the nearest integer sub-dilu. Know the immovability of what, if you see it, will break the pardon, which will exceed 0.02 A.
Solution.
The pardon is rounded and can be seen as a vipadical value X yak rozpodіlena equally in the interval between two sudіdnіmi tsilimi podіla. Shchilnіst equal rozpodіlu
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de 
- Dovzhina interval, in which way the value is possible X; position by interval 
This task has an interval, in which possible values are placed X, more 0.1, to that
Pardon should be moved to 0.02, so it will be placed in the interval (0.02; 0.08). Todi
Suggestion: R=0,6
Head office 4
Postal data: mathematical grading and standard recognition of normally distributed signs X 10 and 2. Know the significance of what is the result of testing X dial the value, put in the interval (12, 14).
Solution.
Speeding up with the formula
І theoretical frequencies 
For Х її the mathematical calculation of M(X) and the variance of D(X). Solution. We know the function of subdivision F(x) vipadkovy size... vibirka's pardon). Sklademo variational row Interval width stock: For skin value row cheer up, skilki...
Solution: alignment with the changes that are divided
SolutionFor chatting private solution heterogeneous equalization storeable I'm going to destroy the system... ; +47; +61; +10; -eight. Induce interval variational row. Give statistical estimates of the mean.
Solution: We will conduct a survey of the Lancian and basic absolute growth rates, growth rates, growth rates. Subtract the value from the table 1
SolutionObsyag of virobnitstva. Solution: Arithmetic mean of the interval variational row calculated as follows: for... stock: Δ w \u003d t * μ \u003d 2 * 0.0146 \u003d 0.02927 Significantly between ...
Solution. Oznaka
SolutionAbout the work experience of some folded vibrka. The average experience for the vibirka ... of the working day folded vibrka. The mean value of trivality... 1.16, significance level α = 0.05. Solution. Variation row given selections may look: 0.71 ...
Working basic program in biology for grades 10-11 Ukladach: Polikarpova S. V
Working programThe simplest breeding schemes” 5 L.R. " Solution elementary genetic tasks” 6 L.R. " Solution elementary genetic tasks” 7 L.R. "..., 110, 115, 112, 110. Store variational row, baptize variation curve, find the average value of the sign ...
As a result of mastering the given division, the student is responsible for: nobility
- showcases variations and their interrelationships;
- the main laws of the rozpodіlu sign;
- the essence of the criteria is good; remember
- rozrakhovuvat pokazniki variations and criteria fit;
- determine the characteristics of the rozpodіlu;
- evaluate the main numerical characteristics of the statistical rows of the subdivision;
Voloditi
- methods of statistical analysis of a series of roses;
- basics of variance analysis;
- by methods of re-verification of statistical series, it was determined according to the basic laws of analysis.
Indicators of variation
With a statistical analysis of the signs of different statistical marriages, it is of great interest to show variations of the signs of four statistical singles of the marriage, and to instill the character of the rozpodil of the singles by the sign. Variation - tse vіdmіnnostі іndivіdіdualnyh znacheny znacheny vіdnі suupnostі, scho vyvchaєtsya. The following variations may be of great practical importance. For equal variation, you can make vysnovkas about the boundaries of the variation signs, the uniformity of the marriage for the price of the sign, the typicality of the middle, the mutual connection of the chinniks, the initial variation. Indicators of variation are used to characterize that ordering of statistical aggregates.
The results of the grouping of the materials of the statistical guardianship, arranged in the visual statistical rows of the subdivided, are the ordering of the subdivisions of the single totality, which twists, on the group behind the group (variable) sign. Since the basis of grouping is taken as a sign, then such a series is called subdivided attributive(Rose for the profession, for the article, for the color, etc.). If a number of urges were given a number of urges for a kіlkіsnoy sign, then such a series is called variant(rozpodіl for growth, vaga, for rosem salary and etc.). Encourage a variational series - also put in order the number of numbers of numbers for the meanings of the signs, pick up the number of numbers of numbers for the numbers of signs (frequency), arrange the results to the table.
The replacement of the frequency of the variant can be zastosuvannya її setting to the solemn obligation of the guard, which is called the frequency (external frequency).
There are two types of variation series: discrete and interval. Discrete series- This is such a variational series, the basis for which was laid signs with a variable change (discrete signs). Until the rest, you can add the number of workers at the enterprise, the tariff range, the number of children at home. A discrete variational series represents a table that is composed of two graphs. In the first column, the specific meaning of the sign is indicated, while in the other - the number of single marriages with the first meaning of the sign. If there is a sign that I can change without interruption (the amount of income, the work experience, the number of fixed assets of the enterprise is too small, if at the singing boundaries they can take on any significance), then for the price the signs can be interval variation series. The table for the hour of awakening of the interval variation series also has two graphs. For the first one, the value of the characters in the interval is indicated - up to (options), for the other - the number of ones that are included in the interval (frequency). Frequency (frequency of repetition) – the number of repetitions of the selected option and the value of the sign. Intervals can be closed and open. Close the intervals of the fence along the sides, tobto. to wash between the bottom (“vіd”), and the top (“before”). Vіdkritі intervals mаyut yakus one boundary: either the upper one, or the lower one. If there are options for spreading for growing or falling, then the rows are called ranked.
For variation series, there are two types of frequency response options: accumulated frequency and accumulated frequency. The accumulated frequency is showing, for some of the warning signs, the value of the sign has taken on a value less than the specified value. The accumulated frequency is determined by the way of summing up the value of the frequency signs of the group with the usual frequencies of the forward groups. Accumulated part characterizes the pet vag of one guard, as it may mean signs to shift the upper boundary between the day group. In this rank, a part has been accumulated to show the pet vag an option for marriage, which may mean no more than this. Frequency, frequency, absolute and visual power, accumulated frequency and frequency are characteristics of the value of the variant.
Варіації ознаки статистичних одиниць сукупності, і навіть характер розподілу вивчаються з допомогою показників і показників варіаційного ряду, до яких ставляться середній рівень низки, середнє лінійне відхилення, середнє квадратичне відхилення, дисперсія, коефіцієнти осциляції, варіації, асиметрії, ексцесу та інших.
To characterize the center of the rose under the vicorist, average values are used. The middle is a typical statistical characteristic, in which we take into account how many manifestations of the typical rіven signs, like the limbs of the marriage, which are twisted. However, it is possible to increase the arithmetic averages with a different character of the subdivision, because the statistical characteristics of the variation series can be classified as structural averages - mode, median, as well as quantiles, so that the subdivision row is subdivided into equal parts (quarters, deciles, percents). ).
Fashion the same meaning signs, which are used in a number of cases, are more frequent, lower and other meanings. For discrete rows - ce options to achieve the highest frequency. In the interval variation series with the method of designation, it is necessary to designate the interval in front of it, in which case it is known, so the titles of the modal interval. In the variational series with equal intervals, the modal interval is assigned to the highest frequency, in the series with uneven intervals, the modal interval is subdivided into the largest thicket. Let's put the formula
de Mo - the meaning of modi; x Mo - the lower boundary of the modal interval; h- modal interval width; / Mo - modal interval frequency; / Mo j - frequency of the pre-modal interval; / Mo + 1 - the frequency of the post-modal interval, and for a number of irregular intervals in the given formula, the change of frequencies / Mo, / Mo, / Mo Rosum 0 _| , Rosum 0> Umo+"
Even though it is a single fashion, it is called unimodal; although there is more than one mode, it is called multimodal (polymodal, multimodal), in case of two modes it is bimodal. As a rule, the richness of modality indicates that it was filed, that it should be followed, that it does not comply with the law normal rozpodіlu. For homogeneous marriages, sound, characteristic single-peak roses. The richness of the perfection is also about the heterogeneity of the marriage, which is twisted. Appearance of two and more peaks to work with the necessary regrouping of data with the method of seeing the same groups.
In the interval variation series, the mode can be graphically determined using additional histograms. For each of the upper points of the largest column, draw two lines with histograms to the upper points of the two sums of columns, which are intertwined. Then, from the point їх the crossbar, lower the perpendicular to the entire abscissa. The meaning of the signs on the abscissa axis, which corresponds to the perpendicular, is the mode. In rich vipadkas, with the characteristics of the marriage, as the pokagalneniy pokaznik, the fashion prevails, and not the arithmetic mean.
Median - tse central significance signs, it may be the central member of the ranged row rozpodіlu. In discrete series, in order to know the value of the median, the serial number is indicated on the back. For which, with an unpaired number of ones, one is added to the sum of all frequencies, the number is divided by two. With a paired number of 1s, the row will have two medians of 1s, in which case the median will be shown as the average of the value of the two medians of 1s. In such a rank, the median in a discrete variation series is the value, as if dividing the series into two parts, in order to avenge the same number of options.
In the interval rows, after the designation of the ordinal number of the median, there is a medial interval for the accumulated frequencies (frequencies), and then, after the additional formula for the distribution of the median, the value of the median itself is displayed:
de Me is the value of the median; x Me - lower boundary of the median interval; h- median interval width; - Sum of frequencies to a number of subdivisions; /D - accumulated frequency of pre-median interval; / Me - the frequency of the median interval.
The median can be known graphically for additional cumulative. For which on the scale of accumulated frequencies (parts) cumulate points that correspond to the ordinal number of the median, a straight line is drawn, parallel to axis abscissa, to the line with cumulate. Far from the point of the crossbar, the designated straight line is lowered from the cumulative perpendicular to the entire abscissa. The value of the signs on the abscissa axis, which shows the ordinate (perpendicular), is the median.
The median is characterized by such powers.
- 1. Won to lie down in the quiet meaning of the signs, like roztashovani from both sides in it.
- 2. There is a power of minimality, which means that the sum of the absolute values of the sign in the median and the minimum value is equal to the value of the sign in the case of any other value.
- 3. When combining two divergences from the given medians, it is impossible to transfer the value of the median of the new divergence later.
The values of the power of the media are widely used in the design of mass service points - schools, polyclinics, gas stations, water pumps, etc. For example, if there is a place to call a polyclinic near a singing quarter, then you need to spread the dotsilnishche near such a point in a quarter, as if to add not to a dovzhina quarter, but to a large number of inhabitants.
Spivvіdnoshennia modi, median and arithmetic mean indicates the nature of the distribution of signs in marriage, which allows to evaluate the symmetry of the distribution. Yakscho x Me may be right-sided asymmetry in a row. With normal distribution X - Me - Mo.
K. Pearson, on the basis of the variation of different types of curves, determined that for moderately asymmetric roses, the similarity between the arithmetic mean, median and mode is just:
de Me is the value of the median; Mo is the meaning of modi; x arithm - arithmetic mean value.
To blame the need to remember the structure of the variation series of the report, then the values of signs similar to the median are calculated. Such meanings of the signs are divided by the unity of the rozpodil on equal numbers, they are called quantiles and gradients. Quantiles are subdivided into quartiles, deciles, and percentiles.
Quartiles divide the sukupnіst chotirma equal parts. The first quartile is calculated similarly to the median behind the formula for the calculation of the first quartile, indicating the first quarterly interval in advance:
de Qi – value of the first quartile; x Q^- lower boundary of the first quartile interval; h- Width of the first quarterly interval; /, - Frequency of the interval series;
Accumulated frequency in the interval before the first quarter interval; Jq (- Frequency of the first quartile interval.
The first quartile shows that 25% of the marriages are less than the її value, and 75% are more. The other quartile is closer to the median, that is. Q2 = me.
By analogy, pay for the third quartile, knowing in advance the third quarterly interval:
de - lower boundary of the third quartile interval; h- Width of the third quartile interval; /, - Frequency of the interval series; /X"- accumulated frequency in the interval, which is forwarded
G
third quarter interval; Jq is the frequency of the third quartile interval.
The third quartile shows that 75% of the marriages are less than the її value, and 25% are more.
The difference between the third and first quartiles is the interquartile interval:
de Aq is the value of the interquartile interval; Q 3 - value of the third apartment; Q - the value of the first quartile.
Deciles divide the stock into 10 equal parts. Decile - the meaning of the signs of the middle rose, which is given to ten parts of the number of marriages. By analogy with quartiles, the first decile shows that 10% of the totality is less than the 1st value, and 90% is greater, and the ninth decile shows that 90% of the 10% of the totality is less than the 10th value, and 10% is greater. Spivvіdshenie ninth and first deciles, tobto. decile coefficient, widely zastosovuєtsya shdo diferentіatsії income for the world spіvvіdnoshennya іvnіv іvnіv іn 10% vіbіlsh prosperous і 10% vіnlsh savvy naselenija. Percentiles divide the ranged order by 100 equal parts. Rozrahunok, the value of that zastosuvannya percentiles similar to deciles.
Quartiles, deciles and other structural characteristics can be graphically calculated by analogy with the median for additional cumulation.
To overcome the variation, the following indicators are used: the range of variation, average linear variation, average quadratic variation, dispersion. The rosemary of the range of variations as a whole is low. Tsey pokazynnik tsіkavyi vypadki, if it is important to know, kak amplitude kolivan sign:
de R- the meaning of the range of variations; x max - the maximum value of the sign; x tt - minimum value signs.
In case of rozrahunka rozmakh variation of the most important value of the number of members is not low, as the variation is related to the skin values of a member of the series. Some minor improvement in indications, yakі є middle, otrimani z vіdkhilenі іndivіdualnyh znachenі znachі vіd avіdnії ї: srednє іnіyne vіdhilennyа i srednє kvadhіlennі vіdhilennya. Mіzh іndivіdualnymi vіdhіlennymi vіd srednёї and kolyvannya kondividnye znachenі іsnuє sprаlіє zalezhnistі. What is the strongest colivannya, what is the greater absolute expansion of the mind in the middle.
The average of the linear deviations is the arithmetic mean of the absolute values of the deviations of the other options in the form of their average value.
Middle line care for non-grouped data
de / pr - the value of the average linear ventilation; x, - sign values; X - P - kіlkіst loneliness suupnostі.
Middle line clearance of a grouped row
de / vz - the value of the average linear ventilation; x - sign values; X - the average value of the signs for the enduring marital status; / - The number of single marriages in an okremіy group.
Vidhilen signs to this particular type are ignored, otherwise the sum of all expenses is equal to zero. The middle line of income is fallow in the form of grouping of data analyses, it is secured according to different formulas: for grouping and non-grouping of data. Середнє лінійне відхилення в силу його умовності окремо від інших показників варіації застосовується практично порівняно рідко (зокрема, для характеристики виконання договірних зобов'язань щодо рівномірності поставки; в аналізі обороту зовнішньої торгівлі, складу працюючих, ритмічності виробництва, якості продукції з урахуванням технологічних особливостей виробництва та etc.).
The mean quadratic viability characterizes, the mean values are characterized by individual signs, which grow in the mean value for the condition, and the signs that vibrate are expressed in loneliness. The core of the quadratic widhilen, being the main vicorist, is widely vicoristed in the qi -rations in a one -rod praise, the knowledge of the curvature of normal roseshoks, which is consequences of the organs of vibric. Mean quadratic variation, but not grounded, is calculated after the next algorithm: skin variation in the average value is squared, all squares are summed up, after which the sum of squares is divided by the number of members in the series and the square root is taken from private:
de a Iip is the value of the root-mean-square adjustment; Xj- meaning signs; X- The average value of signs for doslіdzhuvanoї suupnostі; P - kіlkіst loneliness suupnostі.
For grouped analysis of data, the average allowance for data is secured according to the known formula 
de - the value of the root-mean-square correction; Xj- meaning signs; X - the average value of the signs for the enduring marital status; fx- a lot of single marriages in an okremіy group.
Viraz under the root in both vipads is called dispersion. Thus, the variance is calculated as the mean square of the average value. For unimportant (simple) values, the variance sign is calculated as follows:
For the meanings of the signs
Use also a special way of spreading the dispersion: at a wild look 
for unimportant (simple) meaning signs 
for the meanings of the signs 
with the help of the method of looking at mental zero
de a 2 – dispersion value; x, - sign values; X - average sign, h- the size of the group interval, t 1 - wag (A =
The dispersion may be independent of the statistics and lie up to the most important indications of variation. She will die in loneliness, which will give signs to the square of loneliness that will win.
Dispersion can still be powerful.
- 1. The dispersion of a constant value is closer to zero.
- 2. Changing all the values of the sign by the same value L does not change the value of the dispersion. Tse means that the average square of the number can be calculated for the given values, and, according to the values of their number, it can be calculated.
- 3. Change of all sign values k times change the variance in k 2 times, and the mean quadratic deviation - y k razіv, tobto. all the values of the signs can be divided as a constant number (say, by the value of the interval in a row), calculate the mean quadratic deviation, and then multiply them by a constant number.
- 4. How to calculate the average square of the width in any size And at tієyu chi іnshoy іroy vіdіznyаієєі іd іn аn arithmetic mean, іn аvzhdі іѕ аlѕо thе average square оf іdhilen, counted іnіd аn arithmetic mean. The average square of the figure, if it is greater by a full value - by the square of the difference of the average value of the mentally taken value.
The variation of the alternative signs is poised in the presence or in the presence of the dosledzhuvana power of the single marriage. A large variation of the alternative signs is expressed by two values: the presence of a single person with sufficient power is indicated by a single (1), like a daily presence - by zero (0). A part of the loneliness, which can reach power, is denoted through R, and a part of the loneliness, which does not lead to power, - through G. In this way, the dispersion of alternative signs is a good way to get more often alone, which will lead to power (P), to one, this power will not lead (G). Nyabilsha Variaziya pepper from vipads, a number of pepper, pusk 50% of the ud with a pepper, May, and Insha part is 50%, not a maximum of the maximum value of the maximum significant value, the maximum bore 0.25 , i.e. P = 0.5, G= 1 - P \u003d 1 - 0.5 \u003d 0.5 and pro 2 \u003d 0.5 0.5 \u003d 0.25. The lower boundary of the indicator is equal to zero, which shows the situation, if the marriage has a daily variation. Practically zastosuvannya variance of alternative signs polugaє pobudovі prevіrchih Іnvalіv pіd іn the hour of carrying out the vibratory vigilance.
Chim less value variance and mean quadratic variation, while uniform consistency and more typical will be the average value. In practice, statistics are often blamed for the need to match variations of different signs. For example, let's take a closer look at the differences in the number of workers and their qualifications, the length of service and the increase in wages, co-working and income, the length of service and the productivity of work, too. For such performances, the indications of absolute coliving are a sign of inapplicability: it is impossible to compare the labor experience, expressed in rock, with the variation of wages, expressed in rubles. Для здійснення таких порівнянь, а також порівнянь коливання однієї й тієї ж ознаки в кількох сукупностях з різними середніми арифметичними використовуються показники варіації - коефіцієнт осциляції, лінійний коефіцієнт варіації та коефіцієнт варіації, які показують міру коливань крайніх значень навколо середньої.
Oscillation coefficient:
de V R - the value of the oscillation coefficient; R- The value of the range of variations; X -
Linear coefficient of variation.
de vj- the value of the linear coefficient of variation; I- the value of the average linear breath; X - the average value of the signs for the enduring marriage.
Coefficient of variation:
de Va- the value of the coefficient of variation; a - the value of the root mean square deviation; X - the average value of the signs for the enduring marriage.
Keephizynt Osciasi - Tsnotkovo Viashnya Rozoshakhu Variazi to the middle of the veneration, pusklіdzhuhn, and the layfitsyt vagiasi - cubic worship of the middle -naked valethery to the middle -sulfide of the premium, and the scorigal venerates, the scorchings in the veneles of the veneration, and the scorigal venerates in The coefficient of variation is the average value of the quadratic deviation to the average value of the sign. As an obvious value, expressed in hundreds, the coefficient of variation is fixed for the equal degree of variation of different signs. With an additional coefficient of variation, the homogeneity of statistical marriage is estimated. If the coefficient of variation is less than 33%, then the marriage is considered homogeneous, and the variation is weak. If the coefficient of variation is greater than 33%, then the marriage rate is considered to be heterogeneous, the variation is strong, and the average value is atypical and it is impossible to win as an indicator of the marriage rate. In addition, coeficient variations of vikoristovuyutsya for equalization of the same signs in different combinations. For example, with the method of assessing the variation in the experience of working as a practical worker in two businesses. The greater the value of the coefficient, the more variation marks the suttevish.
On the basis of the foreclosed quartiles, the possibility of foreclosing is also the indicator of the quarterly variation according to the formula
de Q 2 і
The interquartile range is assigned to the formula
Quarterly inspiration should be replaced by a range of variations, in order to overcome the shortcomings, related to the most extreme values:
For non-interval variation rows, the thickness of the rose is also expanded. It will vary as a rule, depending on the frequency of the frequency, or the frequency by the value of the interval. In the non-interval rows, the vicorist is absolute and the viability of the rozpodіlu. The absolute width of the rose is the same frequency that falls on one of the last interval. Vіdnosna bushy rozpodіlu - frequency, which falls on one dozhini interval.
All of the above is true for rozpodіl, the law of rozpodіl any good is described by the normal law of rozpodіlu or close to new.
Variations lavi: designation, vidi, main characteristics. Rozrahunka technique
modi, median, arithmetic mean of medical and statistical records
(Show on the mental butt).
Variation series - a series of numerical values of the additional sign, which are considered one kind of one for their size and rotting in the song order (in the vishide or recession order). The skin of the numerical value of the series is called the variant (V), and the numbers that show how often such a chi іnsha variant is used in the warehouse series are called the frequency (p).
Zagalne number vipadkіv poserezhen, in that number the variation series is added up, denoted by the letter n. Vіdmіnіst v znachennі doslіdzhuvanih sign is called variation. At the same time, the variability of the sign does not have a kіlkіsnoї world, the variation is called yakіsnoy, and a number of roses are attributive (for example, rose for the result of illness, the camp of health is thin).
As a sign, which is variant, if there is a kіlkіsne expression, such a variation is called kіlkіs, and a row of roses is called a variant.
Variation rows are divided into perevnі and without perevnі - according to the nature of the kіlkіsnoї signs, simple and important - according to the frequency of the variant.
In the simple variation series, the skin variant is streaked only once (p = 1), in the advanced one, one and the same variant is streaked once (p> 1). Apply such rows to be looked at further behind the text. Yakshcho kolkіsny sign may be uninterrupted character, tobto. between integer values and intermediate fractional values, the variational series is called uninterrupted.
For example: 10.0 - 11.9
14.0 - 15.9 etc.
As a kіlkіsny sign may have a perervny character, tobto. Okremі znachennya (options) vіdіznyayutsya one vіd one on the whole number i can't think of intermediate fractional values, the variational series is called interchangeable or discrete.
Victory data of the front butt about the pulse rate
in 21 students, there will be a variation series (Table 1).
Table 1
Rated by medical students for pulse rate (beats/hv)
Otzhe, induce a variational series - means numerical value (options) to systematize, order, tobto. roztashuvaty in the singing sequence (in the upper or lower order) with the highest frequencies. In the application of the variant of rasterization in the visual order and the expression in the view of the number of primary (discrete) numbers, the skin variant is sharpened by a sprig of times, tobto. we may take it to the right, let's change it with a discrete variational series.
As a rule, as a rule, the number of guards we see in the statistical totality does not exceed 30, then all the signs are sufficient, which are twisted, arranged in a variational series in an accruing manner, like in Table. 1, or the decay order.
With a large number of warnings (n> 30), the number of options can be even greater, in which case the interval or groupings of the variation series are formed, in order to simplify the further processing and to explain the nature of the distribution of the options in the group.
Indicate the number of group options in the range of 8 to 15.
Їх may not be less than 5, because Otherwise, it will be something rougher, more superficially enlarged, which creates a radical picture of variation and is also indicated by the accuracy of average values. When the number of group options is more than 20-25, the accuracy of calculating the average values is increased, but the peculiarities of the variation signs and the mathematical processing become more complicated.
When folding a grouped row, it is necessary to rotate,
− group variant due to roztashovuvatisya in singing order (for the highest or the lowest);
- intervals in groups of options may be the same;
− the value between the intervals is not due to change, because it will be unclear to which groups there are more than one option;
- it is necessary to ensure the particularity of the material, which is selected when intervening between intervals (for example, when vaccinating older people, an interval of 3-4 kg is acceptable, but for children during the first months of life, it is not guilty to exceed 100 g.)
Let's consider a grouping (interval) series that characterizes the data on the pulse rate (number of beats per beat) in 55 medical students before sleep: 64, 66, 60, 62,
64, 68, 70, 66, 70, 68, 62, 68, 70, 72, 60, 70, 74, 62, 70, 72, 72,
64, 70, 72, 76, 76, 68, 70, 58, 76, 74, 76, 76, 82, 76, 72, 76, 74,
79, 78, 74, 78, 74, 78, 74, 74, 78, 76, 78, 76, 80, 80, 80, 78, 78.
To encourage a grouped row, it is necessary:
1. Determine the value of the interval;
2. Designate the middle, the cob and the final group option of the variation series.
● The interval interval (i) is determined by the number of transferring groups (r), the number of such groups is set according to the number of guards (n) for a special table
The number of groups in the fallow area according to the number of guards:
In our case, for 55 students it is possible to put together 8 to 10 groups.
Expanding the interval (i) is determined by such a formula -
i = Vmax-Vmin/r
In the butt, the value of the interval is more 82- 58/8= 3.
How is the value of the interval є fractional number subtracting the result is rounded up to an integer number.
Distinguish the number of types of average values:
● arithmetic mean,
● geometric mean,
● middle harmony,
● root mean square,
● mid-progressive,
● median
In medical statistics, arithmetic mean values are most often covered.
The arithmetic mean value (M) is the most important value, as it signifies those typical ones that are common to all marriages. The main ways of rozrahunku M є: the arithmetic mean method and the method of moments (smart mindfulness).
The arithmetic mean method is used to calculate the arithmetic mean simple and the arithmetic mean known. Select the method of rozrahunku arithmetic mean value to fall in the form of a variational series. In the case of a simple variational series, in which the skin variant is used only once, the arithmetic mean is calculated simply for the formula:
de: M - Arithmetic mean value;
V - value of variable characters (options);
Σ - vkazuє dіyu - pіdsumovuvannya;
n is the number of guards.
Butt of a rozrahunka of the arithmetic mean. The frequency of diarrhea (the number of dyhal ruptures per whilin) in 9 people in 35 years: 20, 22, 19, 15, 16, 21, 17, 23, 18.
To determine the average level of the frequency of respiratory distress in people over 35 years, it is necessary:
1. Induce a variational series, expanding all variants in an increasing and decreasing order. We took away a simple variational series, because The value of the option is displayed only once.
M = ∑V/n = 171/9 = 19 dyhal per quill
Visnovok. The frequency of breathing among people is 35 years, and the average is 19 times less for whilin.
As soon as the meanings of the variant are repeated, there is not much to write in the line of the skin variant, it is sufficient to reconsider the variant (V), which are sharpened, and instruct to indicate the number of their repetition (p). such a variational series, in which the variants are rated for the number of frequencies that are similar to them, have a name - the ranks of the variational series, and the average value that is being developed is the arithmetic mean.
The arithmetic mean is determined by the following formula: M= ∑Vp/n
de n - the number of guards, which are worth the sum of frequencies - Σр.
Butt of the rozrahunka of the arithmetic mean.
Trivality of ineffectiveness (in days) in 35 ailments on hospital respiratory disease (GRZ), who were delighted with a long-term doctor for the I-st quarter of in-line fate, added up: 6, 7, 5, 3, 9, 8, 7, 5, 6, 4 , 9, 8, 7, 6, 6, 9, 6, 5, 10, 8, 7, 11, 13, 5, 6, 7, 12, 4, 3, 5, 2, 5, 6, 6, 7 days
The methodology for determining the average triviality of non-practice in ailments on the GRZ is advancing:
1. We will encourage the ranks of the variation series, because Okremi values of the option are repeated for a few times. For which one can accommodate all options in increasing or decreasing order with variable frequencies.
We have options for arranging for a growing order
2. Calculate the arithmetic mean for the formula: M = ∑Vp/n = 233/35 = 6.7 days
I got sick from the GRZ for the trivality of unproductiveness:
| Trivality of Unprecedentedness (V) | Number of sick people (p) | vp |
| ∑p = n = 35 | ∑Vp = 233 |
Visnovok. The incidence of incompetence in patients with hospitalized respiratory illness was 6.7 days on average.
Fashion (Mo) - options that are most commonly seen in the variation series. For the rozpodіlu, presented in the table, the mode of the variant, which is more expensive 10, will be more frequent for others - 6 times.
Razpodіl ailing for trivality perebuvannya on a drug bed (in days)
| V |
| p |
Sometimes it’s important to set the exact value of the mod, the shards in data can be used as a guardian, which is reduced “most often”.
Median (Me) is a non-parametric indicator that divides the variation series into two equal halves: on both sides of the median, the same number of options is expanded.
For example, for a rozpodіlu, designated in the table, the median is 10 because. on the offending side of the value of the size, it will be sorted out according to the 14th option, tobto. the number 10 takes the central position of the second row and the second median.
Vrakhovuyuchi, scho how many guards have a couple butt (n=34), the median can be assigned as follows:
Me = 2+3+4+5+6+5+4+3+2/2 = 34/2 = 17
Tse means that the middle of the row sits seventeen behind the rachunk of the option, which means the median, which is more expensive 10. For the rozpodіlu presented in the table, the arithmetic mean is more:
M = ∑Vp/n = 334/34 = 10.1
Also, for 34 guards from Table. 8 mi were taken away: Mo=10, Me=10, arithmetic mean (M) is 10.1. In our butt, all three pokazniki appeared equal or close one to one, although the stench is absolutely different.
The arithmetic mean is the effective sum of all inflows, the molded її take the fate of all the variants without a blame, the extreme, often atypical for the whole thing of the marriage.
Mode and median, on the basis of the arithmetic mean, do not lie in the magnitude of all individual values of signs that vary (the value of the extreme options and the degree of development in a row). The arithmetic mean characterizes the entire mass of the guard, the mode and the median - the main mass
The grouping method also allows variation(minlivist, colivannya) sign. With an apparently small number of singles, the variation of the variance wins over the improvement of the ranked low ones, which make the perfection of the marital. The row is called ranked, yakscho alone roztashovani for rostannyam (changed) signs.
Prote the ranking of the rows to give low visibility to the same, if you need a proportional characteristic of the variation. In addition, in rich situations, mothers are brought to the right with statistical aggregates that add up great number loneliness, yakі it is practical important to show at the sight of a particular row. At the link with the cym for the cob zagalny recognition of the statistical data and especially the reduction of the variation, the sign of the continued appearance and the process should be called by the group, and the results of the grouping should be drawn up as a group table.
As a group table, there are only two graphs - groups for seeing the sign (options) and the number of groups (frequencies and frequencies), they are called rose order.
A number of roses - the simplest variety of structural grouping for one sign, shown in a group table with two columns, in which there is a variant of the frequency of signs. In bagatioh vipadkah such a structural grouping, tobto. From the folding of the rows, the distribution of the statistical material begins.
Structural grouping in a sub-divided row can be transformed into a right structural grouping, as the group will be characterized not only by frequencies, but also by other statistical indicators. The headline recognition of a row of roses is a sign of variation. The theory of series was analyzed in detail by mathematical statistics.
Row rozpodіlu dilyat on attributive(grouped by attributive signs, for example, the population was divided by status, nationality, family camp) and variation(Grouping for kіlkіsnimi signs).
Variation series is a group table, so as to replace two graphs: grouping of individuals for one number of signs, that number of individuals in a skin group. The intervals of the variational series are confirmed by sounding equal and closed. Variations next to the advance of the grouping of the population of Russia for the size of average per capita penny incomes (Table 3.10).
Table 3.10
Rozpodіl number of the population of Russia for the value of average per capita income in 2004-2009 years.
|
Groups of the population by the size of average per capita penny incomes, rub./mіs. |
Number of population of the group, in % before the result |
|||||
|
8 000,1-10 000,0 |
||||||
|
10 000,1-15 000,0 |
||||||
|
15 000,1-25 000,0 |
||||||
|
Ponad 25,000.0 |
||||||
|
All population |
||||||
Variation rows are subdivided into discrete and interval intervals. Discrete variation rows combine the variants of discrete signs that change at narrow boundaries. With the butt of a discrete variational series, you can butt the Russian families for the number of new children.
interval variant rows combine options or without interruptions, or they change in wide ranges of discrete characters. Interval є variatsiyny series of rozpodіlu population of Russia for the value of average per capita penny incomes.
Discrete variational rows are not practical to stop very often. Time to time the folding is clumsy, but the warehouse of the group is marked by specific options, which really can be the final group signs.
The largest width of the interval variation series. With their folded wines seriously ill about the number of groups, as well as about the size of the intervals, which may be installed.
Principles of excellence in nutrition were discussed in the section on the methodology of encouragement statistical groupings(Div. paragraph 3.3).
Variation rows are a zasіb gortannya or compression of various information in a compact form, behind them you can add a clear judgement about the nature of the variation, vichity of the identity of the sign of the phenomena that should be added to the sequence. And more importantly, the value of the variation series is the one that counts the most important characteristics of the variation (div. Chapter 7).
A group of numbers that unite as a sign is called sukupnistyu.
As it was appointed more, the first statistical sports material is a group of disparate numbers, so as not to give the coach a notice about the day of the appearance or the process. Zavdannya polagaє in order to transform tsyu sukupnіst on the system that skoristatisya її pokazyki for otrimannya nebhіdnoї іnformatsії.
Folding of a variational series of sings and moldings of mathematical
Example 2. 34 sportsmen-skiers registered such an hour of heart rate renewal after passing the distance (in seconds):
81; 78: 84; 90; 78; 74; 84; 85; 81; 84: 79; 84; 74; 84; 84;
85; 81; 84; 78: 81; 74; 84; 81; 84; 85; 81; 78; 81; 81; 84;
As you can see, this group of numbers does not carry the same information.
To fold the variational row backwards, the operation is ranking - spreading numbers in order of growth and decay. For example, in order to increase the ranking of the order to the offensive;
78; 78; 78; 78; 78; 78;
81; 81; 81; 81; 81; 81; 81; 81; 81;
84; 84; 84; 84; 84; 84; 84; 84; 84; 84; 84;
In order to change the ranking, bring to such a group of numbers:
84; 84; 84; 84; 84; 84; 84; 84: 84: 84; 84;
81; 81; 81; 81; 8!; 81: 81; 81; 81;
78; 78; 78; 78; 78; 78;
After the ranking has been carried out, the form of recording a group of numbers becomes obviously irrational - one and the same numbers are repeated differently. The reason for this is the natural thought of rewriting the record in such a manner, to show how many times it is repeated. For example, vrakhovuyuchi ranking in order of growth:
Here the number is written, which indicates the hour of the athlete's pulse, the right-handed number of repetitions of this indication in this group of 34 athletes.
It is obvious to understand more about mathematical symbols and the examined group of vimiryuvan meaningfully whether it is a letter, for example x. Vrahovyuchi growing order of numbers in tsіy groupі: x 1 -74 s; x 2 - 78 s; x 3 - 81 s; x 4 - 84 s; x 5 - 85 s; x 6 - x n - 90 s, look at the number on the skin can be marked with the symbol X i.
Significantly, the number of repetitions looked at by the letter n. Todi:
n 1 = 4; n 2 = 6; n 3 = 9; n4 = 11; n 5 \u003d 3; n 6 \u003d n n \u003d 1, and the skin number of repetitions can be calculated as n i.
The total number of conducted vimiryuvans, as if crying out from the butt of the mind, є 34. Tse means that the sum of all n is worth 34. But it is symbolic:
Significantly tsyu sum one letter - n. The same output data for the butt can be recorded in this way (Table 1).
Let's take away a group of numbers and transform them into a series of chaotically scattered indications, taken by the coach on the cob of work.
Table 1
| x i | n i |
| n=34 |
Such a group is a singing system, the parameters that characterize the conduct of the experiment. The numbers that are the results of vimiryuvan (x i) are called options; n i - numbers їх repetition - are called frequencies; n - sum of all frequencies - є oath of marriage.
The entire system is called variation nearby. Some of the series are called empirical or statistical.
It is not important to remember that a possible number of occurrences in the variational series, if all frequencies are equal to ones n i == 1, then the skin is vimir in this group of numbers only once.
The omission of the variational series, like and be-like the other, can be shown graphically. To encourage the graph of the selected series, it is necessary to think about the scale on the horizontal and vertical axes.
In this task, on the horizontal axis, it is necessary to add the value of the hour of the pulse (x 1) in such a way that the unit of time, taken sufficiently, is equal to the value of one second. Please note that the values are almost equal to 70 seconds, mentally stepping into the space between the two axes 0.
On the vertical axis, we show the values of the frequencies of our series (n i), taking the scale: one and the other frequency units.
Having prepared in such a rank, think for a schedule, we proceed to work with a variation next to it.
The first pair of numbers x1 = 74, n1 = 4 is plotted on the graph as follows: on the x axis; known x 1 =74 and we add the perpendicular from the center of the point, on the axis n we know n 1 \u003d 4 і we draw from it a horizontal line to the crossbar with the previously established perpendicular. Offset lines-vertical and horizontal lines with additional lines and that are applied to the little ones with a dotted line. Point їх crossline є by the scale of this graph of spіvvіdnoshennia Х 1 =74 і n 1 =4.
So all other points of the graph are applied by themselves. Then we get the stench in straight lines. Schob graph maw closures looked, the extreme points are connected with the borders with the susceptible points of the horizontal axis.
Otriman's figure is a graph of our variational series (Fig. 1).
Fully understood, the skin variation series seems to be its own schedule.
Rice. 1. Graphic manifestation of the variation series.
On fig. 1 is visible:
1) athletes made up the largest group from the most obstezhenyh, the hour of renewal of the pulse in them was 84 s;
2) for bagats, the hour is 81 s;
3) the smallest group was composed by athletes with a small hour of renewal of the pulse - 74 with more - 90 s.
In this way, after vikonavshi series of trials, next rank subtract numbers and add a variation series, which is a simple mathematical system. For accuracy, the variational series can be illustrated with a graph.
Guidance is more variational series is called more discrete order - such that the skin has a variant of expressions in one number.
Let's add a few more examples of folding variation rows.
example 3. 12 shooters, who were right lying down for 10 shots, showed the following results (in eyepieces):
94; 91; 96; 94; 94; 92; 91; 92; 91; 95; 94; 94.
For the establishment of a variational series, it is necessary to rank the given numbers;
94; 94; 94; 94; 94;
After ranking, we add up a variation series (Table 3).
Enthusiasm...