The square of standard ventilation. Estimation of variance, root-mean-square deviation. Vibirka can buti

Appointment

Root-mean-square ventilation ( English Standard Deviation, SD) є pokaznik, kaky vikoristovuєtsya in teorії ymovіrnostі and mathematical statistics for estimating the level of rozsіyuvannya vipadkoї value shodo її mathematical ochіkuvannya. In the investment standard, the profitability of the valuable papers of the portfolio is awarded according to the method of assessing the world risk. As for the steps of investing in the stock of valuable paper as well as of the estimated stock (mathematical grading of the stock), then there is more risk of investment, and navpaki.

The mean square deviation is indicated by the Greek letter σ (sigma), and the standard deviation by the Latin letter S or Std (X), de X is a vipad value.

Formula

Reference RMS value

Although the exact difference between the discrete epidemiological magnitude, and, apparently, the significance of the skin result, and can be assessed by the severity of the skin result, the formula for the mean-square respiration rate will look like an offensive rank.

De X i - the value of the vipadkovy value X for the i-th result; M(X) is a mathematical evaluation of the vertical value of X; p i - Immovability of the i-th result; N - the number of possible traces.

With any mathematical estimation of the vipadical value, it is settled according to the following formula:

Standard discharge of general marriage

In fact, the replacement of the exact rozpodіl vipadkovoї value sound is available only a selection of data. In which direction, the value of the root-mean-square respiration is assessed, in which case it is called the standard respiration (S). As an assessment is based on all the general aggregate of data, it is necessary to use such a formula.

De X i - i-th value of vipadkovoї value of X; X - arithmetic mean of the general population; N is the obligation of the general marriage.

Standard selection

As it is not the whole general collection of data that is victorious, but a selection from it, then the formula for the standard recovery is based on an unbiased estimate of dispersion.

De X i - i-th value of vipadkovoї value of X; X - arithmetic mean of the selections; N - obsyag vibirki.

Apply the rosemary

butt 1

The portfolio manager can evaluate the risks of investing in the shares of two companies A and B. When looking at 5 scenarios for the development of subdivisions, information about which of them is presented in the table.

Oskіlki we know more precisely raspodil dermal buttocks of the shares, we can razrahuvat the right value of the mean square of the dermal buttocks of them.

Krok 1. Razrahuyemo mathematical ochіkuvannya profitability of skin shares.

M(A) = -5%×0.02+6%×0.25+15%×0.40+24%×0.30+34%×0.03 = 15.62%

M(B) = -18%×0.02+2%×0.25+16%×0.40+27%×0.30+36%×0.03 = 22.14%

Krok 2 Imagine taking the data from the first formula.

As far as we can bachiti, the shares of Company A are characterized by a lower risk, stinks of lower stench in the form of a mean-square increase in profitability. The next step is to indicate how the profitability of the stink is lower, lower in the shares of Company B.

butt 2

The analyst has his own data on the supply of two valuable papers for the remaining 5 years, as presented in the tables.

Oskіlki exact rozpodіl pribykovostі nіdomy, and in razoradzhenny analytics є less than a selection from the general totality of data, we can razrahuvat standard vіdhilennya vybіrki on p_dstavі nemіschennoї variance.

Krok 1. Razrahuyemo ochіkuvanu pributkovіst for leather valuable paper as an arithmetic average vibrka.

X A \u003d (7 + 15 + 2 - 5 + 6) ÷ 5 \u003d 5%

X B \u003d (3 - 2 + 12 + 4 + 8) ÷ 5 \u003d 5%

Krok 2 Razrahuyemo standard vydhilennya pribytkovnosti kozhnoї s tsinnih papers for the formula for selection from the general sukupnostі danih.

It should be taken into account that the offending paper prices may equal the value of the profit of 5%. With this standard, the investment in the value paper B is lower, but for other equals, we will add the object of investing in the shortest profile of risk-pributkovist.

Standard entry in Excel

Two functions have been transferred to Excel for the calculation of the standard selection and general population.

For selection, use the "STDEV.V" function to speed up:

1. At the range of middle B1:F1
2. Choose a weekend B2.
3. fx at the splintering vіknі " Function insertion"Return Category" New alphabetical list» and select a function « STDEV.».
4. In the field" Number1» choose the range of the middle B1:F1, field " Number2OK».

For general victoriousness, the "STDEV.G" function is used:

1. At the range of middle B1:F1 the value of the vertical value X is entered.
2. Choose a weekend B2.
3. Press the button at the command line fx at the splintering vіknі " Function insertion"Return Category" New alphabetical list» and select a function « STDEV.G».
4. In the field" Number1» choose the range of the middle B1:F1, field " Number2» fill empty and press the button « OK».

Interpretation

In investing, the standard income is surplus as the world's volatility. What is the meaning of this, then it is more risk, connected with the investment in the entire asset, and navpaki. With other equal parameters, the priority should be given to the asset, for which the indicator will be minimal.

Dispersion- the arithmetic mean of the squares of the skin value is the sign of the global mean. Fallow in the form of weekend data, the dispersion can be unimportant (simple) or well-known.

The dispersion is covered by the following formulas:

For non-grouped data

For grouped data

The order of distribution of dispersion is important:

1. calculate the arithmetic mean

2. selected option according to the average

3. build up at the square of the skin variant in the middle

4. multiply the squares of the breath by the waves (frequencies)

5. subsumu otrimani create

6. I’ll take the sum divided by the sum of the vag

The formula for determining the variance can be converted to the following formula:

simple

The order of spreading dispersion is simple:

1. calculate the arithmetic mean

2. square the arithmetic mean

3. create a square skin option in a row

4. we know the sum of the squares of the option

5. divide the sum of the squares of the variant from the їхної number, tobto. designate the middle square

6. mark the difference between the average square of the signs and the square of the average

Also, the formula for the chosen variance can be converted into the following formula:

tobto. variance of the average difference of the squared value of the sign of the square of the arithmetic mean. In the case of a reworked formula, the addendum procedure for rozrahunka is turned off, individual values ​​are signs of x and pardon is included in the rozrahunka, it is tied to rounded vodhilen

Dispersion may have a number of powers, the deacons from them allow you to ask for a calculation:

1) the dispersion of a constant value is equal to zero;

2) if all options change the value of the sign to the same number, then the variance will not change;

3) if all options change the value of the signs of change in the same number of times (times), then the variance will change in times

Mean square deviation S- is the square root of the dispersion:

For non-grouped data:

· for variation series:

The range of variations, the mean linear and the mean quadratic variation with named quantities. The stench may be alone alone in the world, as if it were an individual sign.

The variance and mean quadratic variation are the most widely used indicators. It is explained that they are included in the majority of theorems of the theory of imovirnosti, as the foundation of mathematical statistics. In addition, the dispersion can be arranged in warehouses, which allow estimating the influx of various factors, which influence the variation of signs.

Razrahunok pokazniki v variatsii for banks, zgrupovanih for razmirom surplus, is shown in the table.

Rozmir pributka, mln. rub.	Number of banks	rozrahunkov showcases
3,7 - 4,6 (-)		4,15	8,30	-1,935	3,870	7,489
4,6 - 5,5		5,05	20,20	- 1,035	4,140	4,285
5,5 - 6,4		5,95	35,70	- 0,135	0,810	0,109
6,4 - 7,3		6,85	34,25	+0,765	3,825	2,926
7,3 - 8,2		7,75	23,25	+1,665	4,995	8,317
Together:			121,70		17,640	23,126

The average linear and average quadratic deviations show the signs in the average fluctuations in the value of signs in one and the last marriage. Yes, at to this particular type the average value of the colivannya rozmіru pributka to become: for the average liniynym vіdhilennyam 0.882 million rubles; for the average quadratic allowance - 1.075 million rubles. The average quadratic deviation is larger than the average linear deviation. Yakshcho rozpodіl signs, close to normal, between S and d іsnuє vzaєmozv'yazok: S=1,25d, or d=0,8S. Mean quadratic variation shows how the main mass of single marriages is arithmetic mean. Independently, in the form of a subdivision of 75, the value of the signs is taken in the interval x 2S, and if 89 all values ​​are taken in the interval x 3S (P.L. Chebishev's theorem).

One of the main tools of statistical analysis is the analysis of the mean square deviation. The Danish pokaznik allows for the assessment of standard care for vibrancy or general marriage. Let's find out how to figure out the RMS deviation formula in Excel.

It is very significant that it is the root-mean-square inspiration and it looks like this formula. Tsya value is the square root of the arithmetic mean number of squares of the difference of all values ​​in the row of that arithmetic mean. It is also the same name of this showcase - standard recognition. Call insults as equals.

Ale, of course, that in Excel koristuvachev it is not possible to ce virakhovuvati, shards for the new work of the program. Let's find out how to measure standard inspiration in Excel.

Rozrahunok at Excel

You can expand the value shown in Excel with the help of two special functions STDEV.(for vibirkovu sukupnistyu) that STDEV.G(for general shopping). The principle of їхної ії is absolutely the same, but there are three ways to revise them, which will be discussed below.

Option 1: function master

Method 2: Formulas tab

Method 3: manually entering formulas

Use the same way, if you don’t need it, it will be necessary to call out all the arguments. For which next run the formula manually.

Like bachimo, the mechanism of root-mean-square expansion in Excel is simpler. Koristuvachev it is better to enter numbers from the collection, or send them to the commissaries, as if to avenge them. Usі rozrahunki vykonuє the program itself. It’s more folded to understand that it’s a showcase, that it’s rozrakhovuetsya, and as a result, rozakhunka can be zastosuvat in practice. Ale rozuminnya tsgogo already belong more to the sphere of statistics, nizh nauchannya work with software security.

X i - vipadkovі (current) values;

X̅– the average value of the depression values ​​for the vibrancy is calculated according to the following formula:

Otzhe, variance is the average square of the difference . So the average value is taken care of, then we take it difference between skin output and average values, based on the square , It is folded and then divided by the number of values ​​in the given totality.

Retailing between the meanings and the average inspires the world of inspiration. Have a square to be built in order to make sure that all ventilations become turned off positive numbers and to avoid the interrelationship of positive and negative impressions in case of their summation. Let's sweat, gazing at the squares of life, we'll just spread the arithmetic mean.

The solution to the magic word "dispersion" is found in three words: middle - square - idhilen.

Mean square deviation (RMS)

Vityaguyuchi s variance square root, otrimuemo so called " mid-square deviation". Names are shaking "standard deviation" or "sigma" (according to the name of the Greek letter σ .). The formula for the root-mean-square ventilation can be seen:

Otzhe, variance - ce sigma of the square, or - mean square deviation of the square.

The mid-square difference, obviously, also characterizes the world of the rise of data, but now (on the basis of dispersion) it can be compared with the differences of the day, since they have the same ones (they sing out the formula of the rise). The range of variations is the difference between the extreme values. The mid-square view, like the world of insignificance, also takes the fate of rich statistical roses. Z її to help establish the accuracy level of various estimates and forecasts. If the variation is already large, then the standard choice will be large, and the forecast will be inaccurate, which will hang, for example, at even wider intervals of dovirchih.

Therefore, in the methods of statistical processing of data in the assessment of objects of non-violence, it is necessary to use the rule of two or three sigmas due to the necessary accuracy of the set task.

To match the two-sigma rule and the three-sigma rule, we use the Laplace formula:

F - F,

de Ф(x) – Laplace function;

Minimum value

β = maximum value

s = sigma value

a = mean value

In this case, there is a private form of the Laplace formula, if between α and β the value of the case value X is the same in the center of the subdivision a = M(X) per decimal value d: a = a-d, b = a+d. Abo (1) Formula (1) shows the possibility of a given adjustment d of the falloff value X with the normal law of distribution under the mathematical adjustment M(X) = a. If formula (1) is taken sequentially d = 2s and d = 3s, then we take: (2), (3).

two sigma rule

Mayzhe vіrogіdno (with a confidence factor of 0.954) it is possible to prove that the values ​​of the vertical value of X with the normal law of the distribution of the value of X with the normal law of the distribution of the value of X are subdivided in the form of a mathematical refinement M(X) = a by a value not greater than 2s (two mean square values). Confidence of imovirnistyu (Pd) is called the imovirnіst of podias, which are mentally accepted for authenticity (їх imovirnіst is close to 1).

The rule of two sigma is illustrated geometrically. On fig. 6 shows the Gauss curve with the center rozpodіlu a. The area, surrounded by the entire curve and the line Ox, is 1 (100%), and the area of ​​the curvilinear trapezium is between the abscissas a–2s and a+2s, zgidno with the two-sigma rule, is 0.954 (95.4% in flat conditions). The area of ​​shaded plots is 1-0.954 = 0.046 (5% of all areas). Qi dilyanka is called the critical area of ​​the value of the fall value. The values ​​of a vipadkovoї value, like a critical area, are small and practically mentally accepted for impossibility.

Imovirnist mentally impossible meaning is called equal to the importance of the vipadkovy value. Rіven znachennostі pov'yazaniy іz dovіrchoyu ymovіrnistyu formula:

de q – significance level, expressions in vіdsotkah.

three sigma rule

In case of virishennі power, which means more suppleness, if you trust the imovirnіst (Pd) equal to 0.997 (more precisely - 0.9973), replace the two-sigma rule, zgіdno with formula (3), beat the rule trio sigma.

Zgidno three sigma rule with a confidence level of 0.9973, the critical area will be the area of ​​the value of the sign posture interval (a-3s, a + 3s). Significance equals 0.27%.

Otherwise, seeming, the ability of those that the absolute value of the ventilation to outweigh the average square of the ventilation, is even smaller, the most expensive 0.0027 = 1-0.9973. Tse means that less than 0.27% of the fall may be the case. So, according to the principle of the impossibility of small-sized pods, they are considered practically impossibly. Tobto. The vibration is high current.

Why do you think the essence of the rules of three sigma:

If the vipad value is normal, then the absolute value of її vodhilennya in mathematical ochіkuvannya is selected from the trio mean square variance (RMS).

In practice, the rule of three sigmas should be set as follows: if the value of the difference was reached, the value of the difference is normal; in another case, it won't fit normally.

The level of significance is taken fallow, depending on the permitted level of risk and the set task. To assess non-violence sound, a less accurate selection is accepted, supplementing the two-sigma rule.

The most thorough characteristic of variation is the mean quadratic variation, which is called the standard (or standard deviations). mean square deviation() is closer to the square root of the middle square if the value of the sign is equal to the arithmetic mean:

Mean square deviation is simple:

The middle value of the quadratic deviation is set for grouping data:

Between the mean quadratic and mean linear respirations in the minds normal rozpodіlu May be the same spіvvіdnoshnya: ~ 1.25.

The core of the quadratic widhilenna, being the main absolutely mira of the war, vicoristovy, the knowledge of the crooked roses of normal, in the rosrahuks, who are successful with the organized vib.

Dispersion, її see, root mean square deviation.

Dispersion of the fall value- the world's rozkidu tsієї vipadkovoї value, so її vіdhilennya vіd mathematical ochіkuvannya. In statistics, the meaning of chi is often victorious. The square root of the dispersion is called the root-mean-square adjustment, the standard adjustment or the standard adjustment.

Global dispersion (σ2) Vymіryuє variatsiyu znaki vsіy sukupnostі pіd vplіv vsіh faktorіv, scho zumovіl tsyu variatsіyu. At the same time, using the grouping method, one can see and simulate a variation, zoomed in by a group sign, that variation, which is due to the influx of unmatched factors.

Intergroup dispersion (σ 2 m.gr) characterize the systematic variation, so that the importance of the value of the reported signs, which is blamed on the signs - the factor underlying the grouping.

Root-mean-square ventilation(Synonyms: mean quadratic deviation, mean square deviation, quadratic deviation; similar terms: standard deviation, standard expansion) - in the theory of dynamics and statistics of the greatest extensions, the indicator of the difference in the value of the mathematical value of the mathematical value When subdivided arrays of selections, the value of the arithmetic mean of the number of selections is the arithmetic mean of the number of selections.

Mean-square differences are measured in units of the same magnitude and vicoristed when the standard pardon of the arithmetic mean is calculated, when there are mutual intervals, when the statistical reversal of hypotheses, when the relationship is vypadkovymi values. It is shown as the square root of the variance of the magnitude.

RMS deviation:

standard care(Estimation of the root-mean-square deviation of the drop value xїї mathematical scaling based on the unbiased estimate of її variance):

de - Dispersion; - i-th element of the selection; - Obsyag vibirki; - Arithmetic mean of choices:

Slid signify that insulting assessments are misplaced. It’s impossible to induce an unbiased assessment with a bark. The proteo estimate for the correction of the estimate of the unbiased variance is possible.

The essence, the sphere of stosuvannya and the order of the designation of the mode and the median.

Krіm statisticheskih averages in statistics for the visual characteristics of the magnitude of the variation signs and internal life and rows are subdivided by structural averages, which are represented, mainly, fashion and median.

Fashion- tse variant of the row, which is most likely to be sharpened. Fashion zastosovuetsya, for example, when I change the clothes, it seems that the buyers will be the most drunk. The fashion for a discrete series is the variant that can make the highest frequency. When calculating the modal for the interval variation series, it is necessary to firstly assign the modal interval (behind the maximum frequency), and then the value of the modal value is the sign behind the formula:

- - Modi value

- lower boundary of the modal interval

- - Interval value

- - Modal interval frequency

- - Frequency of the interval that precedes the modal

- - Frequency of the interval following the modal

Median the meaning of the signs that lie at the basis of the ranged series and divide the whole series into two equal parts for the number of parts.

To determine the median in a discrete series for the presence of frequencies, the number of frequencies is calculated on the sum of the frequencies, and then the value of the variant falls on it. (However, if a number of sorts is not paired with an unpaired number of signs, then the number of the median is calculated according to the formula:

M e \u003d (n (number of signs in marriage) + 1) / 2,

at times of a paired number, the sign of the median is closer to the middle of the two, it means to be in the middle of the row).

When calculating median for the interval variation series, the median interval is assigned, in the boundaries of which the median is found, and then the value of the median for the formula:

- - Shukana median

- the lower boundary of the interval, which is to avenge the median

- - Interval value

- - Sum of frequencies or number of members in a series

The sum of the accumulated frequencies of the intervals to override the median

- - Median interval frequency

butt. Know the mode and the median.

Solution:
In this case, the modal interval changes at the boundaries of the century group of 25-30 years, but the highest frequency falls on this interval (1054).

Let's calculate the modi value:

Tse means that the modal age of students reaches 27 years.

Let's calculate the median. The median interval is found in the age group of 25-30 years, so that in the middle of this interval the variant was sorted out, so that the marriage is divided into two equal parts (? f i / 2 = 3462 / 2 = 1731). We give the formula the necessary numerical data and the minimum value of the median:

Tse means that one half of the students in May is up to 27.4 years old, and the other half is over 27.4 years old.

Crimium modi and median can be used as indicators, like quartiles, so that the rankings can be divided into 4 equal parts, decile- 10 parts and percentiles - per 100 parts.

The understanding of the vibrational caution is that sphere of yoga zastosuvannya.

Vibirkove watchfulness stagnation physically impossible through the great array of danih abo economically low-cost. Physical impossibility may be, for example, the number of passengers, market prices, family budgets. Ekonomichna nedotsіlnіst maє mіsce in otsіntsі yakostі commodities, scho z їkh znischennyam, for example, tasting, tasting tsegli on mіtsnіst thinly.

Statistical singles, selected for guardianship, become the vibratory sequence of the vibrator, and their array - the general sequence (GS). With whom the number of ones in the selection means n, and for all GS - N. Setting n/n is called an external rosemary or part of a vibratory.

The quality of the results of the vibratory caution should be determined by the representativeness of the sample, that is, because of the way it is represented in the HS. To ensure the representativeness of the selection, it is necessary to trim principle of vipadkovnosti selection, some kind of message, that the inclusion of a single HS in a vibrator is impossible to get in any other official of the Crimea in a fit.

Use 4 ways of vertical selection at the vibrator:

1. Vlasne vipadkovy ie “lotto method”, if the statistical values ​​are assigned ordinal numbers, which are entered on the sing objects (for example, barila), then they will be mixed in the sing volume (for example, at the bear) and chosen navmannya. The real way is to use an additional generator of vipadkovy numbers or mathematical tables of vipadkovy numbers.
2. Mechanical vіdbіr, for yakim vіdbієtsya skin ( N/n)-th value of the general marriage. For example, if you don’t have to pay 100,000 values, and you need to choose 1,000, then you can spend the skin 100,000 / 1000 = 100-a value. Moreover, if the stench is not ranked, then the first one is chosen in the first hundred, and the numbers of the others will be a hundred more. For example, as if the first one appeared alone No. 19, then the offensive may be No. 119, then No. 219, then No. 319 then. As the only one of the general marriage ranking, then number 50 is selected first, then number 150, then number 250 and so on.
3. A selection of values ​​from a heterogeneous array of data is conducted stratification(Rozsharovanim) way, if the general sukupnіst is broken up in advance into homogeneous groups, until such zastosovuetsya vipadkovy or mechanical vіdbіr.
4. A special way to fold the vibrator serial Vіdbіr, in which one vipadkovo chi mechanically select the other values, like the series (the sequence of the same number for the last time), in the middle of which one should lead a succesfull guard.

Yakіst vibrkovih guards to fallow and vіd vibrator type: repeated or non-repetitive.

At re-selection statistic values ​​that have been squandered from the selection, but after the series of victories turn into a general collection, looming the chance to spend in a new selection. At the same time, all the values ​​of the general marriage have the same possibility of being included in the selection.

Non-repeated vote it means that the statistical values ​​that were taken from the selection, otherwise the first series after the selection does not turn into the general collection, and for other values, the remaining values ​​are moved to the next selection.

Non-repetitive sampling gives accurate results, more often than not. But there are situations, if it is not possible to stop the traffic (passenger traffic, the rest of the drink is still) and then a repeated selection is carried out.

Boundary pardon of vibirka guarded, middle pardon of vibirka, the order of the severance.

Let's take a look at the report listing more ways of molding vibratory marriage and pardon, which are blamed for this. representativeness .
Vlasne-vipadkova The selection is based on the selection of individuals from the general sukupnosti navmannya without any elements of systemicity. Technically, the wet-violent selection is carried out by the method of foaling (for example, drawing lotteries) or after the table of vipadkovy numbers.

Wet-vipadkovy vіdvіr "at a clean look" in the practice of vibrancy guarding rarely stagnates, but in the middle of other types of vіdbora, the main ambush vibrating vigilance is realized in the new one. Let's take a look at the nutritional theory of the vibirka method and the pardon formulas for a simple vibirka vibirka.

Pardon vibirkovogo posterezhennya- the difference between the value of the parameter in the general marriage, and the same value calculated for the results of the vibratory guard. For the middle kіlkіsnoї signs pardon vibіrka is marked

The indicator is called the boundary pardon of the vibirka.
Vibіrkov average є vipadkovoy value, as you can accept different values in staleness, in addition, they ate alone at the vibrator. Otzhe, pardon vibirki є vipadkovymi values ​​and can take on different meanings. That is why the average is due to possible pardons - middle pardon vibirka, yak to deposit vіd:

The volume of the selection: the greater the number, the smaller the value of the average pardon;

Steps to change the additional signs: if there is less variation in signs, then, and dispersion, then less is the average pardon of the vibirka.

At vipadkovy repeated selection the middle pardon is granted:
.
In practice, the general dispersion is not exactly visible, but in theory of imovirnosti it was brought that
.
Oskіlki the value for dosing great n is close to 1, vvozhatimutsya, scho. Today, the average pardon of the vibrator may be repaid:
.
Ale in small vibes (when n<30) коэффициент необходимо учитывать, и среднюю ошибку малой выборки рассчитывать по формуле
.

At vipadkovy non-repetitive vibrating hover the formulas are corrected by the value. This is the middle pardon of the no-repeated vote:
that .
Because zavzhd less, then the multiplier () zavzhd less than 1. Tse means that the average pardon with a non-repetitive selection is zavzhd less, lower with a repeated one.
Mechanical vibration zastosovuetsya, if the general order is in any way ordered (for example, lists of selections in alphabetical order, telephone numbers, numbers of houses, apartments). Vіdbіr zdіysnyuєtsya through the song interval, which is equal to the healthy value of the number of vibrators. So for 2% selection, skin 50 units = 1 / 0.02 are selected, with 5% skin 1 / 0.05 = 20 units of the general marriage.

The ear of the ear is chosen in different ways: in the middle of the interval, in the middle of the interval, in the middle of the interval, in the reverse order. A smut with tsioma - get rid of a systematic pardon. For example, with 5% selection, the 13th is the first one, then the next 33, 53, 73, etc.

For the accuracy of the mechanical vibration, it is close to the wet-falling vibration. This is why the average pardon of a mechanical vicarious selection is the formula for a wet-ripple selection.

At typical selection sutured sukupnіst is broken up in front of the same, the same type of group. For example, in case of obstezhennі pіdpriєmstv tse can but galuzі, pіdgaluzі, schodo population - districts, social chi age groups. We will then develop an independent selection from the skin group by a mechanical or wet-weather method.

A typical selection gives more accurate results in other ways. Typization of the general marriage for the safety of representation in the selection of the skin typological group, which allows you to include the inclusion of intergroup dispersion in the middle pardon of the selection. Also, in case of a significant pardon of a typical selection, it is necessary to respect only the average of group dispersions with the rule of folding dispersions (). Today is the middle pardon of the vibirka:
upon re-selection
,
with non-repetitive selection
,
de - the average of the internal group dispersions in the sample.

Serial (or nested) species zastosovuєtsya at times, if the general sukupnіst is divided into a series of groups on the cob of vibratory trimming. These series can be packaging of finished products, student groups, teams. The series for cladding are selected mechanically in a damp-falling way, and in the middle of the series, a single succulent cladding is carried out. Therefore, the middle pardon of a vibrka is less likely to fall due to the inter-group (inter-series) dispersion, as it is calculated according to such a formula:

de r – number of selected series;
- middle i-th series.

The middle pardon of the serial vote is insured:

upon re-selection:
,
with non-repetitive selection:
,
de R – total number of series.

Combinations wіdbіr Let's take a look at ways to select.

The middle pardon of the vibirka for any way of choosing to lie down is the head rank of the absolute number of the vibirka in the smaller world - the vіd vіdsotka vibіrki. It is acceptable that 225 guards be carried out for the first couple with a general marriage of 4,500 individuals and for another - 225,000 individuals. The variances in both types are equal to 25. The same in the first type with 5% selection

In another case, with a 0.1% choice, it won’t be more expensive:

Such a rank, with a change in the number of votes in 50 times, the pardon of the vote increased insignificantly, the numbers of the vote did not change.
Let's assume that the number of choices was increased to 625 guards. At this point of view, the pardon of vibirka is good:

The increase in the selection by 2.8 times for one and the same number of general marriages reduces the pardon of the selection more by 1.6 times.

Methods and methods for molding vibratory marrow.

At the statistics, there are various ways of forming vibratory aggregates, which are charged with the assignments of the deposit and deposit, depending on the specifics of the object of extraction.

The main mental conduction of vibratory obstezhennia is the neglect of the blame for systematic pardons, which is blamed on the violation of the principle of equal opportunities to consume the skin unity of the general marital union. The advancement of systematic pardons is achieved through the establishment of scientifically grounded methods in the formation of vibratory marriage.

Establish the following ways to select individuals from the general marriage:

1) individual selection - the selection is made up of okremі singles;

2) group selection - at the same time, the group is consumed in a similar way, or in a series of single ones, which are twisted;

3) combinations of selection - all combinations of individual and group selection.
Ways of selection are determined by the rules of forming a vibratory marriage.

Vibrka may be:

• vlasne-vipadkova I believe in the fact that vibratory marriage is settled as a result of a vipadical (non-navmis) selection of others from the general marriage. Whenever the number of people chosen by the vibrator is chosen, the marriage of loneliness will sound, coming out of the accepted part of the vibe. A part of the selection is an indication of the number of individuals of the vibratory marriage n of the number of individuals of the general marriage N, that is.

• mechanical I believe in the fact that the vibrancy is the only one in the vibrator's sukupnіst viroblyayetsya from the general sukupnіst, divided into equal intervals (groups). With the change of the interval in the general population, the vital size of the vibrating part is good. Thus, at 2% vibrancy, skin 50-a unit (1:0.02) is selected, at 5% vibrancy - skin 20-unit (1:0.05), etc. In this manner, apparently to the accepted part of the selection, the general marriage is mechanically divided into equally large groups. From the skin group of the vibrator, only one individual is selected.
• typical - in case of general succulence, the cluster is divided into homogeneous types of groups. Then, from the skin type group, wet-violent or mechanical vibrating, an individual selection is carried out alone at vibrating mating. An important feature of the typical selection is those that give exact results against other methods of selecting singles from the selection;
• serial- for such a general order, divide it into the same group for the general order - series. Series are chosen for vibratory marriage. In the middle of the series, it is necessary to carry out a strong guard of individuals who have been consumed to the series;
• combined- Vibirka can be two-step. At this general's order, groups break up. Then we will form the selection of a group, and in the middle of the rest, we will select a group of singles.

In statistics, there are such ways of selecting individuals from a selection of women:

• single stage vibirka - skin vibrated alone at the same time, it is necessary to marry for a given sign (weather-vipadkova and serial vibirki);
• richness infrequently selection - selection is selected from the general marriage of okremy groups, that is how groups are selected from okremi singles (typical selection with a mechanical method of selection of singles from the selection of groups).

The cream of that is divided:

• repeated vote- Behind the scheme turned kuli. With all the skin loneliness that squandered at the vibrator, if the series turns into a general order, then there may be a chance to sip at the vibrator again;
• non-repetitive vote- Behind the scheme of the unturned kuli. You can get more accurate results with the same obsyazі vibirki.

The appointment of the necessary obligatory selection (student's table selection).

One of the scientific principles of the theory of the vibrating method is the safety of a sufficient number of selected individuals. Theoretically, the need to adhere to this principle is presented in the proofs of boundary theorems of the theory of imovirnosti, which allow us to establish, which obyag one should choose from the general totality, so that it is sufficient to ensure the representativeness of the selection.

The zrota pardon of the vibyrki, and the edges, the zbilhennya of the accurate borosovyki, the zy zy zy zbilshennyams of the vibyrki, also to the herd of the vibric storage, the virishovati of the vibrica vibyris is a dimensions of the Substinosta, the puppy of the pyresty, the boula of the pyremn, the pins of the pier Razrahunok nebhіdnogo vybіrka obyagu dlyaetsya for additional formulas, vyvedenih z formulas boundary pardons vіbіrki (A), similar to that kind of that method of selection. So, for a vipadic re-obligation of vibrating (n) it is possible:

The essence of the formula is in tsomu, which is directly proportional to the square of the confidence coefficient (t2) and variance variance signs (?2) and wrapped in proportion to the square of the boundary pardon of the vibrka (?2). Zocrema, for the sake of increasing the border pardon for two, the number of votes may need to be changed fourfold. Three three parameters, two (t і ?) are given by the last one.

With whom the doslidnik walks out zvdan vibirkovogo obstezhennia maє virishiti nutrition: for which kіlkisny podednanny it is better to turn on digital parameters for ensuring the optimal option? In one way, it is possible to have more power over the superiority of taking away the results (t), lower the world of accuracy (?), In the other - navpaki. There is no more variation in the power of the value of the boundary pardon of the vibrator, the signs of the indicator at the stage of designing the vibrator caution are not, in practice it is customary to set the value of the marginal pardon of the vibrator, sound, no more than 10% of the predicted mean value. Before the establishment of the transferred average level, you can approach it in different ways: win the data of similar earlier conducted rounds, or speed up the data of the basis of the selection and grow a small sample of the selection.

It is most convenient to set the time of the design of the vibratory guard the third parameter in the formula (5.2) - the dispersion of the vibratory aggregate. In this case, it is necessary to win all the information, which is in the order of the last, taken away in the previous similar and trial examinations.

Food service the necessary number of the selections is folded, as if the number of the selections are constricted, the sign of the single selection. In this case, the average skin level is a sign of their variations, as a rule, they are different, and therefore, there is a difference in nutrition about those, dispersion, which is a sign of superiority, it is possible to mark that task of obstezhennya.

When designing a vibratory warning, the value of the admissible pardon of vibrating is given before the appointment of a specific follow-up and that imovirnіst vibrating for the results of the warning.

Zagal the formula for the boundary pardon of the vibirkovo average value allows you to designate:

The value of the possible births of indications in the general marriage in the indications of the marriage of marriage;

The number of selections is necessary, which ensures the necessary accuracy, if between the possible pardons not to exceed the given value;

Imovirnist of the fact that at the holding of a pardon for the mother, the pardon is set between.

Rosepodil Student theoretically, it is a one-parameter family of absolutely uninterrupted roses.

Rows of dynamics (intervals, moments), zmikannya rows of dynamics.

Rows of dynamics- the value of statistical indications, as presented in the chronological sequence.

Leather dynamic range to replace two warehouses:

1) indications of periods of time (rocks, quarters, months, days of date);

2) pokazniki, which characterize the object, which is kept, for timchasov period, or on a special date, as they are called equals in a row.

Rivnі a number of turns both absolute and average values. Depending on the nature of the indications, there will be dynamic series of absolute, visual and average values. The series of dynamics from visual and average values ​​will be based on similar series of absolute values. Razrіznyayut intervals and momentary series of dynamics.

Dynamic interval series avenge the value of showings for the singing period of the hour. In the interval series, equals can be summed up, obsessed with appearances for the last three years, or the results are accumulated.

Dynamic moment series enter the value of the indications for the song hour (date to the hour). At the moment ranks of the last one, there may be more than a difference of appearances, which shows the change is low between the songs of the dates, there is no real money here. The accumulated results will not be squandered here.

The most important mental correctness encourages dynamic rows - the summation of equal rows, which lie down to different periods. Rivnі povinnі buti are presented in the same values, due to the mother of the place, however, is the guilt of the hoarying of the various parts of the apparition.

In order to to avoid the creation of real dynamics, the statistical analysis should be carried out in front of the expansion (zooming the rows of dynamics), in order to pass on the statistical analysis of the dynamic rows. Under the zmikannym rows of dynamics, it is possible to unite up to one row of two and more rows, even if they are insured for a different methodology, or not to allow territorial cordons too. Zmikanny ryadіv dynamiki can also allow the reduction of the absolute rіvnіv rіvnіv іnіnіv іnіnіv і to zagalї basis, scho leveling inconsistency іvnіv іnіvіnіv.

Understanding the totality of the series of dynamics, coefficients, growth rates and growth.

Rows of dynamics- A number of statistical indicators that characterize the development of phenomena of nature and success in the hour. Statistical collections that are published by the State Statistics Committee of Russia, avenge a large number of rows of dynamics in tabular form. A number of dynamics make it possible to reveal the regularity of the development of the manifestations that are being developed.

There are two types of pokazniks in a row of dynamics. indicators of the hour(Rocks, quarters, months and in) or moments and hours (on the cob of fate, on the cob of skin month, etc.). Rivniv indicators in a row. Indicators of the rіvnіv rows of dynamics can be expressed by absolute values ​​(production rate in tons or rubles), illustrative values ​​(number of the city's population in %) and average values ​​(average wages of labor workers for the rocks). In a tabular form, a row of dynamics should have two columns or two rows.

Correctly pobudova ryadіv dynamikі v perbachaє vykonannya low vymog:

1. usі pokazniki low dynamics may be but scientifically grounded, reliable;
2. oskaniki low dynamics may but zastavnі for an hour, tobto. may be charged for the same period to the hour of the same date;
3. Pozniki low dynamics may buti zastavnі territory;
4. pokazniki low dynamics may buti zastavnі for zmіst, tobto. calculation for a single methodology, in the same way;
5. ostentatious low dynamics may buti zastavnі on the stake of the state, scho vrakhovuyutsya. Usі pokazniki low dynamics due to but induced in one and the very loneliness of vimіryuvannya.

Statistical indicators can characterize either the results of the end-of-life process in the period, or the end of the end-of-life event, the song hour, tobto. indications can be interval (periodic) and momentary. Vidpovidno backwards, the series of dynamics can be either interval or momentary. Momentary rows of dynamics at their own pace can be with equal and uneven intervals of the hour.

Often the series of dynamics can be converted into a series of average values ​​and a series of visual values ​​(Lance and basic). Such rows of dynamics are called reverse rows of dynamics.

The technique of rozrahunka of the average level of the average dynamics of the price, is guided by the low dynamics. On the butts, you can see the rows of dynamics and the formulas for raising the middle level.

Absolute growth (Δy) to show how the number of single changes in the offensive row is equal to the front row (column 3 - lanciug absolute increments) or in the rows to the cob row (column 4 - basic absolute increments). The formula for the rozrahunka can be written as follows:

When the absolute values ​​are changed, the series will obviously be "changed", "decreased".

Indicators of the absolute increase to say that, for example, 1998. Product "A" increased in popularity against 1997. for 4 thousand t, and against 1994 p. - by 34 thousand. t.; behind the curtain of rock divas. tab. 11.5 gr. 3 and 4.

Growth coefficient it shows that the scaling times of changing the riven are low against the front ones (column 5 - lanceg coefficients of growth and decrease) which are equal to the cob cob riven (column 6 - basic coefficients of growth and reduction). The formula for the rozrahunka can be written as follows:

Tempi sprout show how the scallops of the vіdsotkіv to become the offensive rіven are low against the front ones (column 7 - lanceg tempi growth) and the counter cob rіvn (column 8 - basic tempi growth). The formula for the rozrahunka can be written as follows:

Primir, 1997 obsyagi vyrobnitstva product "A" against 1996 r. becoming 105.5% (

Growth rate show how much of the growth rate of the daylight period is against the forward (column 9 - lance growth rate) versus the cob cob rate (column 10 - basic growth rate). The formula for the rozrahunka can be written as follows:

T pr \u003d T p - 100% or T pr \u003d absolute increase / equal to the front period * 100%

Primir, 1996 vs. 1995 product "A" was more viroble by 3.8% (103.8% - 100%) chi (8:210) x100%, and against 1994 p. - by 9% (109% - 100%).

If the absolute equals change in the series, then the rate will be less than 100% and the rate of decrease will be obviously the rate of decrease (the growth rate will be minus sign).

Absolutely significant 1% increase(Gr. 11) shows how many individuals need to be worked in this period, so that the cost of the previous period is 1%. At the butt, 1995 treba bulo virobiti 2.0 yew. t., and 1998 p. - 2.3 yew. t., tobto. significantly more.

You can calculate the absolute value of 1% growth in two ways:

Divide the rive of the previous period by 100;

Lantsyugovі absolute increments of the difference on the vіdpovіdnі lanceugovі growth rates.

Absolute value 1% increase =

In dynamics, especially during a three-year period, an important cumulative analysis of the rate of growth is observed in the growth rate of skin viscosities and the increase in the decrease.

It is worthy of note that the method of analysis of the series of dynamics has been reviewed, it is established both for the series of dynamics, equal to those of absolute values ​​(t, thousand rubles, the number of workers, etc.), ,% ash content of coal and in) or average values ​​(average yield per c/ha, average wage is low).

The order of the analyzed analytical indicators that are calculated for the skin of the row in the rows with the front or the cob row, when analyzing the rows of the dynamics, it is necessary to calculate the average for the period of the analytical indicators: the average row of the row, the average increase in the absolute growth rate .

The method of rozrahunka of the middle level is low, the dynamics were more pronounced. In the interval series of dynamics analyzed by us, the average value of the series is calculated according to the formula of the arithmetic mean simple:

Seredniorіchny obsyag vyrobnitstva goods for 1994-1998. becoming 218.4 thousand. t.

The average absolute increase is also calculated according to the formula of the arithmetic average:

The average absolute growth changed over the years from 4 to 12 thousand tons (div. gr. 3), and the average increase in production in the period 1995-1998. becoming 8.5 thousand. t.

Methods of analysis of the average growth rate and the average growth rate should be taken into account. Let's look at them from the butt of the indications in the tables of river indicators are low.

Average rіven low dynamics.

Series of dynamics (or hour series)- tse numerical value of a single statistical indicator in the last moments or in a period of an hour (to sort it out in chronological order).

The numerical value of the third statistical indicator, which makes the dynamics low, is called equal to a number and ring out with a letter y. First member in a row y 1 call the cob abo baseline, and the rest y n - kіntsevim. Moments or a period of an hour, until which lie equal, signify through t.

A series of dynamics, as a rule, represent a table or a graph, moreover, the hour scale will be on the abscissa axis t, and along the y-axis - the scale equal to the series y.

Average indicators low dynamics

Skin row of dynamics can be seen as sukupnіst n small at the hour of indications, yakі can be taken as average values. Such narrow (middle) signs are especially necessary for the second change of that number of signs in different periods, in different countries only.

The peculiar characteristic of low dynamics can be persh for everything average row. The method of rozrahunka of the middle equal to lie in the fact that the momentary series is either interval (periodic).

In times interval low yogo average equal to the formula of a simple arithmetic mean value equal to low, tobto.

=
What a mess momentary a row to avenge n rivniv ( y1, y2, …, yn) with equal intervals between dates (moments of the hour), such a series can easily be converted into a series of average values. With this indicator (rіven) on the cob of the skin period, at the same time, it is an indicator for the end of the previous period. Then the average value of the indicator for the skin period (interval between dates) can be covered as a total value at on the cob and put the end of the period, tobto. yak. There will be a number of such middle ones. As it was planned earlier, for the series of average values, the average rіven is calculated for the arithmetic average.

So, you can write:
.
After the transformation of the number book, we will take:
,

de Y1і Yn- First and last row; Yi- Promіzhnі rivnі.

Tsya average at home at the statistics yak middle chronological for momentary lavas. This name was taken away by the word "cronos" (hour, lat.), the shards are redeemed from indications that change in the hour.

For some nervous between dates, the chronological average for the moment series can be calculated as the arithmetic average from the average values ​​for the skin pair of moments, which are related to the magnitude of the averages (during the hour) between the dates, tobto.
.
In this vantage point it is reported that in the intervals between the dates there were equal values, and mi z two vіdomih ( yiі yi+1) we will determine the average, for which we will later raise the total average for the entire period that is being analyzed.
How is it conveyed, what is the skin meaning yi become imminent until the onset (i+ 1)- right moment, tobto. if the date of the change is exact, then the account can be calculated according to the formula of the arithmetic mean:
,

de - hour, with the stretch of some kind of rіven imminent.

Krіm average rіvnya in the ranks of the dynamics razrakhovuyutsya and іnshі іnshі іnshі avіdnі pokazynі - srednya zmіna іvnіv іnіv ryad (basic talantzugovymi svoj), avrednіy tempo zmeni.

Base average absolute change is a private view of the rest of the basic absolute change on the number of changes. Tobto

Lanzug average absolute change equal to a number of є frequent views of the sum of all lanzugian absolute changes for the number of changes, tobto

According to the sign of the average absolute changes, to judge the nature of the change in the appearance of the average: growth, decline and stability.

From the rules for the control of basic and Lantzugian absolute changes, it is clear that the basic and Lantzugian mean changes can be equal.

The order from the average absolute change is repaid and the average is the same as the basic and lanciug methods.

Basic average output change stands for the formula:

Lantsyug's middle class stands for the formula:

Naturally, the basis and the Lantzugian average change may be the same and the same and the same from the criteria values ​​1 fight about the nature of the change in the middle: growth, decline in stability.
Vіdnіmannyam 1 s base and lanzugovoj average pervnnoї zmіni utpovіdny average rate of change, by the sign of which one can judge the nature of the change of the pre-existing phenomenon, beaten by this series of dynamics.

Seasonality and seasonality index.

Seasonal kolivannya - stiykі vnutrishnyorіchnі kolyvannya.

The main principle of the state for otrimanya maximum effect - tse maximization of income and minimization of costs. Vychayuchi seasonal colivannya, vyrishuetsya zavdannya maximum eligibility in skin equanimity.

When cultivating seasonal koliva, two mutually dependent plants are violating:

1. Revealing the specifics of the development of the appearance in the internal dynamics;

2. Vymіryuvannya seasonal kolivani z budovoyu model і seasonal whilі;

To win over seasonal koliva, count the numbers of seasonality. At the outrageous looking stench, the values ​​of the weekend equals low dynamics to the theoretical equals, which act as a basis for the equal.

The fluctuations in seasonality are superimposed on the seasonal fluctuations, and the seasonality index is averaged.

In this period for the skin period of the river cycle, the most significant indications are indicated for the average seasonal indices:

The average index of seasonal crops in the country in the fall of the fall seasons of the main trend in the development.

Depending on the nature of the trend, the formula for the average seasonality index can be taken as follows:

1.For a series of internal dynamics with a clearly expressed main trend in development:

2. For a series of internal dynamics in a trend, which moves or decreases, daily, or insignificant:

De - hot middle;

Methods of analysis of the main trend.

On the development of the manifestations, hour by hour, the factors of difference are poured in, according to the character, by the force of the injection. Deyakі їх to wear vypadkovy character, іnші mаyut practіyne practіyne vplіv і form at the lavas of the dynamics of the same trend of development.

An important task of statistics is the manifestation of the average dynamics of the trend, which is influenced by different trends. With this method, a number of dynamics are processed using the methods of enlargement of intervals, the average average, and the analytical variance and int.

Interval enlargement method foundations for enlarged periods of time, to which equal dynamics can be seen, tobto. is a substitute for the data, which are worth a hundred of hourly periods, tributes for larger periods. Particularly effective if the cobs lie in a row until short intervals. For example, rows of pokaznikіv, scho suyuyuschih podіy podіy, zamіnuyuyutsya ranks, scho lie down to tizhnevyh, monthly and so on. Allow me to show more clearly "all development of the apparition". The average, calculated at enlarged intervals, allows you to directly reveal the nature (accelerated growth) of the main trend in development.

Cosmic mean method similar to the previous one, but in this case, the actual equals are replaced by the middle equals, loosened for successively loose (kovzayushchy) enlarged intervals, which are hoarse m rivniv are low.

For example, how to accept m=3, then, on the back of the head, the middle of the first three equals in a row, then - from the same number of equals, and then starting from the other after the rachunk, then - starting from the third, etc. Otzhe, mid-range scho “kovza” with low dynamics, passing one term. Reimbursement from m the members of the midline are visible to the middle (center) of the skin interval.

This method is usuvaє less vipadkovі kolivannya. If there is a row of seasonal whistle, then it will be saved and after smoothing by the method of kovznoy average.

Analytical review. With the method of using vipadkovyh coliving and showing the trend, there is a stagnation of equivalence in a row behind analytical formulas (or analytical virivnyuvannya). This is the essence of replacing empirical (actual) equals with theoretical equals, like insurance for equal equals, taken as a mathematical model of the trend, de theoretical equals are considered as a function of the hour: . For this skin, the actual rіven is taken as the sum of two warehouses:

The task of analytical virіvnyuvannya is brought to the offensive:

1. Designated on the basis of actual data to the type of hypothetical function, the building most adequately depicts the trend in the development of the studied indicator.

2. The value of the empirical data of the parameters in the assigned function (equality)

3. Rozrahunok for knowledge of theoretical (virtivnyakh) equals.

Vybir tієї chi іnshої funktії zdіysnyuїtsya, zvіch, z urakhuvannya graphic image of empirical data.

How to serve as a model equal to regression, the parameters of which are developed according to the method of least squares

It is lower to induce the most victorious for viriving the dynamic rows of equal regression from those designated for the selection of such tendencies, the development of the stench is the most suitable.

For the knowledge of the parameters of the guidance, they use special algorithms and computer programs. Zocrema, to find the parameters of the straight line, you can use the following algorithm:

If you number the period or time and hour so that St = 0 happened, then put more algorithms to simply forgive and change to

The lines on the graph are posted on one straight line, which will pass on the closest line to the actual lines of this dynamic row. The sum of the squares of the respirations is the expiration of the influx of the vipadkovy factors.

Z її dopomogoyu rozrahuyemo middle (standard) pardon equal:

Here n is the number of guards, and m is the number of parameters of equals (we have two of them - b 1 and b 0).

The main trend (trend) shows how to add systematic factors to low dynamics, and the increase in the rate of the trend () to serve as a world of excess factors.

To assess the quality of the victorious model of the dynamic series, it is also necessary to Fisher's F test. Vіn є vіdshennya dvoh variances, variance setting itself, viklikanо regressієyu, tobto. doslіdzhuvanim chinnik, dispersion, viklikanoї vipadkovymi reasons, tobto. excess dispersion:

For a flaming look, the formula for which criterion can be given as follows:

where n is the number of guards, tobto. number of equals in a row,

m is the number of parameters in the row, y is the actual row row,

Vibration of the row - the middle row of the row.

More away, lower, more, the model is not zavzhd may appear to be complete. Її can be recognized as such only in that moment, if the criterion F in her crosses the critical boundary. Qia boundary is installed for additional table F-rozpodіlu.

The essence and classification of indexes.

Under the index of the statistics, there is a visual indicator, which characterizes the change in the size of a being in an hour, space, or in a parity with a standard.

The main element of the index index is the indexed value. Under the indexed value, the meanings of the signs of statistical marriage, the change of which is the object of marriage, are understood.

For additional indexes, there are three main tasks:

1) assessment of the change of a folding box;

2) the appointment of a number of officials to replace a folding box;

3) the equalization of the size of this phenomenon with the value of the past period, the value of the other territory, as well as with standards, plans, forecasts.

Indexes are classified according to 3 signs:

2) for equal ohoplennya elements in marriage;

3) for methods of rozrahunka zagalnyh іndekіv.

For zmist indexing values ​​indexes are subdivided into indexes of small (volume) indications and indexes of low indications. Indices of key performance indicators - index of physical performance of industrial products, physical performance of sales, number and index of performance indicators - indexes of prices, co-working capacity, productivity of work, average wages and wages.

For equal hoarying of single women, the indices are subdivided into two classes: individual and wild. For their characteristics, we introduce the following clever designations, adopted in practice for the index method:

q- kіlkіst (obsyag) whether a product in natural vrazі ; R- Variety of single virobleny products; z- compatibility of a single viroblen product; t- Spend time on the production of a single product (labor) ; w- Vibration of products in a variety of ways in one hour; v- Viroblennya products in natural virazhenni in one hour; T- Zagalni vitrati hour chi number of practitioners.

In order to differentiate, until a certain period, or the object is to be indexed, it is customary for the right-handed person to put subscript signs below the leading symbol. So, for example, in the indexes of dynamics, as a rule, for the regular (current, winter) periods, the successive sign 1 i is used for the periods, with which the alignment is carried out,

Individual indices serve to characterize the change of other elements of a folding case (for example, change of obligation to release products of the same type). The stench is the best values ​​of the dynamics, the victories of the goiter, the equivalence of indexing values.

The individual index of the physical obligation of the products is determined

З аналітичної точки зору наведені індивідуальні індекси динаміки аналогічні коефіцієнтам (темпам) зростання і характеризують зміну індексованої величини в поточному періоді в порівнянні з базисним, тобто показують, у скільки разів вона зросла (зменшилася) або скільки відсотків складає її зростання (зниження). The values ​​of the indexes are expressed by the coefficients in percentages.

Global (degrees) index changing all the elements of a folding thing

Aggregate indexє the main form of the index. Aggregate wine is called the fact that this number and banner are a set of "aggregate"

Average indexes, secondary.

Crimium of aggregate indexes in statistics zastosovuetsya іnsha їх form - the mean value of the index. Up to їх calculation are given only, if the information, which is in order, does not allow the total aggregate index to be decommissioned. So, even if there are daily data about prices, but also information about product variability in the in-line period and in the case of individual price indexes for leather goods, then it is not possible to calculate the overall price index as an aggregate one, but it is possible to calculate the average average. So, even though we do not know the number of different kinds of products, but we can see individual indexes and variability of products of the base period, we can assign a global index of physical production as an average value.

Average index - tse index, charges as the average value of individual indices. The aggregate index is the main form of the global index, so the middle index can be the same as the aggregate index. When calculating the average indexes, there are two forms of averages: arithmetic and harmonic.

The arithmetic mean index is the same as the aggregate index, as a rule, individual indices will be warehouses of the aggregate index. Only in this way, the value of the index, which is secured by the formula of the arithmetic mean, is more equal to the aggregate index.