Variation series and statistical analysis of the selections. Pobudov of the interval variation series of uninterrupted calculus data. See statistical groupings

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 inter-variation signs, uniformity of marriage for the center sign, typicality of the middle, interrelation of chinniks, 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+"

If it's a single fashion, then it's a rip-off vipadkovy size 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 value of signs in a number of roses, which is given 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 more, and the ninth decile shows that 90% of the 10% of the totality is less than the 10th value, and 10% is more. Spivvіdshenie ninth and first deciles, tobto. decile coefficient, widely zastosovuєtsya shdo diferentіatsії income for the world spіvvіdnoshnja income equal to 10% of the most secure and 10% of the least prosperous population. 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 calculated graphically 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 pokaznik to become of interest to the vipads, if it is important to know how the amplitude of the sign is:

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 indicators, yakі є average, otrimani z vіdkhilenі іndivіdualnyh znachenі znachі vіd avіdnії ї: srednє іnіyne vіdhilennyа i srednє kvadhilenі 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 deviation 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 average lineage in the fallow in the form of grouping of data analysis is covered by 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.).

Average quadratic viability characterizes, on average in the average viability there are individual signs that grow in the average 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 middle 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 series), 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 one (1), and the presence of a day - by zero (0). A part of loneliness, which can reach power, is denoted through P, and a part of loneliness, which does not lead to power, - through G. In this manner, the dispersion of alternative signs is a sign of more often being alone, which will lead to this power (P), to a part of being alone, which cannot be given power (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.

The smaller the value of the variance of the root-mean-square rate, the more uniform the consistency, and the more typical the average value will be. In practice, statistics are often blamed for the need to match variations of different characters. 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 difference between the mean quadratic deviation to the mean 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 things are not valid for the rozpodіlu, the law of the rozpodіlu of some good is described by the normal law of the rozpodіlu or close to the new.

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, the mother is brought to the right with statistical combinations, which are made up of a large number of individuals, which is important to show in 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's appearances will be characterized not only by frequencies, but 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.

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 raspodіlu populyany Rosії by the value of the average per capita penny income.

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.

The principles of severance of this nutrition were discussed in the section on the methodology of inducing 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).

Variations name a row of roses, prompted for a kіlkіsny sign. The meaning of the kіlkіsnyh signs in the almighty loneliness of the marriage of the impermanent, the greater the lesser the difference between oneself.

Variation- kolyvannya, change in the size of the signs in the same marriage. Okremі numeral znachennya signs, sho strіchayutsya in sukupnostі, sho twisted, naming options value. The insufficiency of the average value for the overall characteristics of the marriage makes it necessary to supplement the average values ​​with indications, which allow us to evaluate the typicality of these average values, vimiryuvannya kolyvannya signs that show up.

The presence of the variation is framed by the influx of a large number of officials into the formation of the equal sign. Tsі chinniki dіyut with different strength and at different lines. For the description of the world of insincerity, a sign of vicorist showcases variations.

The task of the statistical registration of the variation:

• 1) the development of the character and the degree of variation is a sign of okremih loneliness of marriage;
• 2) designation of the role of other officials of this group in variations of quiet and other signs of marriage.

At statistics, there are special methods and follow-up variations, which are based on the best system of indicators, h to help those who win the variation.

The following variations may be important. Vymіryuvannya variatsіy nebhіdnі pіd іn hour wіbіrkovogo pоsterezhennja, correlаtіyіyomі variance аnіzі just. Ermolaev O.Yu. Mathematical statistics for psychologists: Pdruchnik [Text] / O.Yu. Ermolaev. - M .: View of Flint of the Moscow Psychological and Social Institute, 2012. - 335p.

For equal variations, you can make vysnovkas about the uniformity of the marriage, about the stability of the other sign and typicality of the middle. On the basis of rozroblyayutsya indicators of the accuracy of the connection between the signs, indicators of the assessment of the accuracy of the vibrating alert.

Distinguish the variation in the space and the variation in the hour.

Under the variation in the space, the meaning of the signs in the same totality, which represent the surroundings of the territory, is understood. Depending on the variation at the hour, it is possible to change the value of the signs at different periods of the hour.

For the elimination of variations in the lavas, it is necessary to carry out the expansion of all variants of the value of the sign in the increasing order. This process is called low ranking.

The simplest signs of variation are minimum and maximum- the least the most significant signs of marriage. The number of repetitions of the four options is the value of the sign called the frequency of repetition (fi). Frequencies can be manually replaced by frequencies - wi. A part is a prominent indicator of frequency, which can be expressed in parts of a single or in a few hundreds and allows you to set a variation in a row with a different number of guards. Expressed by the formula:

de Xmax, Xmin - the maximum and minimum values ​​of signs in marriage; n is the number of groups.

To overcome the variation, the signs are zastosovuyutsya different absolute and visual indications. To absolute indications of variation, there are ranges of variation, average linear variation, dispersion, average quadratic variation. Before the last indications, the oscillation coefficient, the linear deviation coefficient, the variation coefficient are added.

An example of the significance of a variational series

Manager. For the price of choice:

• a) Know the variation series;
• b) induce the function of subdivision;

No. = 42. Selection elements:

1 5 1 8 1 3 9 4 7 3 7 8 7 3 2 3 5 3 8 3 5 2 8 3 7 9 5 8 8 1 2 2 5 1 6 1 7 6 7 7 6 2

Solution.

• a) prompting a ranged variational series:
  • 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 5 5 5 5 5 6 6 6 7 7 7 7 7 7 7 8 8 8 8 8 8 9 9
• b) prompting a discrete variational series.

We calculate the number of groups in the variational series, using the Sturgess formula:

The number of groups is acceptable, equal to 7.

Knowing the number of groups, we analyze the value of the interval:

For transparency purposes, the number of groups is assumed to be equal 8 stock interval 1.

Rice. one Obsyag sale of goods by the store for a song promo hour

When processing great arrays of information, which is especially important for the hour of the daily scientific research, before the last day, it is necessary to seriously organize the correct grouping of the data. If given a discrete character, then the problems, as we had, are not blamed - it is necessary to simply improve the frequency of the skin signs. How do you finish the mark uninterrupted character (which may be more wide is practical), then the choice of the optimal number of intervals for grouping signs is good for trivial tasks.

For grouping without interruption of fluctuations, the entire variational range of signs is divided into sprats of intervals before.

Grouped by interval (uninterrupted) variation near name the range for the values ​​of the signs of the interval (), which are indicated at once with the most important frequencies () of the number of warnings that were spent in the i-th interval, or with the best frequencies ():

Histogramі cumulate (ogiva), already reportedly reviewed by us with a miraculous way of visualizing data, which allows us to take into account the first revelation about the structure of data. Such graphs (Fig. 1.15) will be the same for continuous data, as well as for discrete ones, only because of the fact that continuous data often fill the area of ​​their possible values, accepting whether they are significant.

Rice. 1.15.

Tom stovptsі on the histogram and cumulative guilt stick together, do not mothers of children, where they do not waste the meanings of the signs in the boundaries of the usable(that's why the histogram and cumulate are not due to the mother "dirok" on the abscissa axis, in which they do not use the value of the change, which is twisted, as in Fig. 1.16). The height of the stompchik is a warning to the frequency-number that was consumed in the given interval, or a warning to the visible frequency-particle. intervals not guilty to peretinatisya and may, as a rule, have the same width.

Rice. 1.16.

Histogram and polygon є approximations of the curve width of motion (differential function) f(x) theoretical rozpodіlu, which is considered in the course of the theory of imovirnosti. To this reason, it may be so important in the first statistical analysis of the calculus of uninterrupted data - at a glance, it is possible to make a visnovka about a hypothetical law of distribution.

Cumulate - curve of accumulative frequencies (parts) of the interval variation series. The graph of the integral function is plotted with a cumulative F(x), which is also considered in the course of the theory of imovirnosti.

In general, the histograms and cumulative concepts are related to the same with uninterrupted data and their interval variation series, so that their graphs are empirical estimates of the function of the density of fluctuations and the function of rozpodіlu obviously.

Pobudov of the interval variation series is started from the designated number of intervals k. The first task, perhaps, is the most complicated, important and ambiguous for the nutritional follower.

The number of intervals is not guilty, but too small, so that when the histogram comes out too smooth ( oversmoothed), consuming all the peculiarities of the lowness of the weekends - in fig. 1.17 you can think, like the data itself, with what prompted the graphs of fig. 1.15 to use histograms with a smaller number of intervals (line chart).

At the same time, the number of intervals is not guilty of being too great - otherwise we cannot estimate the span of the distribution of data, which scroll along the numerical axis: the histogram of the viide is not smoothed (undersmoothed) from non-replaceable intervals, uneven (div. Fig. 1.17, right graph).

Rice. 1.17.

How can you determine the best number of intervals?

Sche 1926 p. Herbert Sturges (Herbert Sturges) zaproponuvav formula for calculating the number of intervals, where it is necessary to break the unknown value of the last sign. This formula of truth has become overpopular - the majority of statistical assistants will propagate themselves, for the abbreviations they will win and impersonal statistical packages. Naskіlki tse it is true and in the usual moods - є even more serious nutrition.

So, what is the basis of the Sturges formula?

Look at bіnomny rozpodіl }