Definition of variation series. Analysis of variation series
All values of the property under study that occur in the population under study are called the value of the attribute (option, option), and a change in this value by varying. Options are denoted in small letters of the Latin alphabet with indices corresponding to the serial number of the group - x i .
A number that shows how many times each characteristic value occurs in the population being studied frequency and denote f i . The sum of all frequencies of the series is equal to the volume of the population being studied.
Very often you need to count accumulated frequency (S). The accumulated frequency for each characteristic value shows how many units of the population have a characteristic value no greater than given value. The accumulated frequency is calculated by sequentially adding the following attribute values to the frequency of the first value of the frequency sign:
The accumulated frequency begins to be calculated from the very first value of the attribute
The sum of frequencies is always equal to one or 100%. Replacing frequencies with frequencies allows one to compare variation series with different numbers of observations.
The frequencies of the series (f i) in some cases can be replaced by the frequencies (ω i).
If the variation series is given at unequal intervals, then for a correct idea of the nature of the distribution it is necessary to calculate the absolute or relative density of the distribution.
Absolute distribution density (p f ) represents the frequency value per unit size of the interval of a separate group of the series:
R f = f/ i.
Relative distribution density (p ω ) represents the frequency value per unit size of the interval of a separate group of the series:
R ω = ω / i.
For series with unequal intervals, only these characteristics give a more correct idea of the nature of the distribution than frequency and frequency.
Statistical sample distribution name a list of options (sign values) and their corresponding frequencies or distribution densities, relative frequencies or relative densities distributions.
Different distribution series are characterized by different sets of frequency characteristics:
minimal – attribute series (frequency, frequency),
for discrete ones, four characteristics are used (frequency, frequency, accumulated frequency, accumulated frequency),
for interval ones – all five (frequency, frequency, accumulated frequency, accumulated frequency, absolute and relative distribution densities).
Rules for constructing an interval variation series
Graphic representation of variation series
The first stage of studying a variation series is to construct its graphical representation. A graphical representation of variation series facilitates their analysis and allows one to judge the shape of the distribution. To graphically represent a variation series in statistics, a histogram, polygon, and cumulate distribution are constructed.
A discrete variation series is depicted as a so-called frequency polygon.
To display an interval series, a frequency distribution polygon and a frequency histogram are used.
Graphs are constructed in a rectangular coordinate system.
| Parameter name | Meaning |
| Article topic: | Variation series |
| Rubric (thematic category) | Production |
Observed values of a random variable X 1 , X 2 , …, x k are called options.
Frequency options X i is usually called the number n i (i=1,…,k), showing how many times this option occurs in the sample.
Frequency(relative frequency, fraction) options x i (i=1,…,k) is usually called the ratio of its frequency n i to sample size n.
Frequencies and frequencies are called scales.
Cumulative frequency It is customary to call the number of options whose values are less than a given one X:
Cumulative frequency It is customary to call the ratio of the accumulated frequency to the sample volume:
Variation series(statistical series) – it is customary to call a sequence of options written in ascending order and their corresponding weights.
The variation series should be discrete(sampling of values of a discrete random variable) and continuous (interval)(sample of values of a continuous random variable).
The discrete variation series has the form:
| … | ||||
| … |
When the number of options is large or the feature is continuous ( random value can take any values in a certain interval), are interval variation series.
To construct an interval variation series, carry out grouping option - they are divided into separate intervals:
The number of intervals is sometimes determined using Sturges formulas:
Then the number of options falling into each interval is counted - frequencies n i(or frequency n i/n). If the option is on the boundary of the interval, then it is attached to the right interval.
The interval variation series has the form:
| Options | … | |||
| Frequencies | … |
Empirical (statistical) distribution function it is customary to call a function whose value at a point X is equal to the relative frequency of the variant taking on a value less than X(cumulative frequency for X):
Frequency polygon called a broken line whose segments connect points with coordinates ( X 1 ; n 1), (X 2 ; n 2), …, (x k; n k). It is constructed in a similar way frequency polygon, which is a statistical analogue of a distribution polygon.
It is worth saying that for a continuous variation series a polygon can be constructed if the values X 1 , X 2 , …, x k take the midpoints of the intervals.
An interval variation series is usually depicted graphically using histograms.
bar chart– a stepped figure consisting of rectangles whose bases are partial intervals of length h= x i +1 – x i, i= 0,…,k-1, and the heights are equal to the frequencies (or frequencies) of the intervals n i (w i).
Cumulates(cumulative curve) – a curve of accumulated frequencies (frequencies). For discrete series The cumulate represents a broken line connecting the points or , . For interval series the cumulate begins from a point whose abscissa is equal to the beginning of the first interval, and the ordinate is equal to the accumulated frequency (frequency) equal to zero. Other points of this broken line correspond to the ends of the intervals.
Variation series - concept and types. Classification and features of the category "Variation Series" 2017, 2018.
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The simplest transformation of statistical data is to order them by size. A sample of volume from the general population, ordered in non-decreasing order of elements, i.e. , called variation series: . In the case where the volume of observations... .
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Variation series, their elements.
A researcher interested in the tariff category of mechanical workers
workshop, conducted a survey of 100 workers. Let us place the observed values
prize in ascending order. This operation is called ranking
statistical data. As a result, we get the following series, which calls
Xia ranked:
1,1,..1, 2,2..2, 3,3,..3, 4,4,..4, 5,5,..5, 6,6,..6.
From the ranked series it follows that the studied characteristic (tariff
rank) took on six different values: 1, 2, 3, 4, 5 and 6.
Further different meanings we'll call it the prize option-
mi, and under by varying - understand changes in attribute values.
Depending on the values accepted by the sign, the signs are divided
on discretely varying and continuously varying.
Tariff category is a discretely varying feature. Number, impressions-
The number of times option x occurs in a number of observations is called hour-
totoy option m x .
Instead of the frequency of option x, we can consider its relationship to the general
number of observations n, which is called frequency option and its relationship denotes w x .
w x =m x /n=m x /åm x
A table that allows you to judge the distribution of frequencies (or frequencies) between options is called discrete variation series.
Along with the concept of frequency, the concept is used accumulated frequency,
which denote t x nak. The accumulated frequency shows how many times
In the observations, the sign took values less than the given value x. Attitude
the addition of the accumulated frequency to the total number of observations n is called accumulated
frequency and denote w x nak. It's obvious that
w x nak =m x nak /n=m x nak /åm x .
Accumulated frequencies (frequencies for a discrete variation series, calculated in the following table:
| X | m x | m x nak | w x nak |
| 0+4=4 | 0,04 | ||
| 4+6=10 | 0,10 | ||
| 10+12=22 | 0,22 | ||
| 22+16=38 | 0,38 | ||
| 38+44=82 | 0,82 | ||
| 82+18=100 | 1,00 | ||
| Above 6 |
Let it be necessary to study the output per worker - machine operator of a mechanical shop in the reporting year as a percentage of the previous year. Here, the characteristic x being studied is production in the reporting year as a percentage of the previous one. This is a continuously varying feature. To identify characteristic features Variations in the values of the attribute will be combined into groups of workers whose output fluctuates within 10%. We present the grouped data in the table:
| Research Sign x | Number of workers m | Share of workers w | Accumulated frequency m x nak | w x nak |
| 80-90 | 8/117 | 8/117 | ||
| 90-100 | 15/117 | 8+15=23 | 23/117 | |
| 100-110 | 46/117 | 23+46=69 | 69/117 | |
| 110-120 | 29/117 | 69+29=98 | 98/117 | |
| 120-130 | 13/117 | 98+13=111 | 111/117 | |
| 130-140 | 3/117 | 111+3=114 | 114/117 | |
| 140-150 | 3/117 | 114+3=117 | 117/117 | |
| å |
In the frequency table, m shows how many observations the characteristic took on values belonging to a particular interval. This frequency is called interval, and its ratio to the total number of observations is interval frequency w. A table that allows one to judge the frequency distribution between intervals of variation in the values of a characteristic is called interval variation series.
An interval variation series is constructed based on observational data for non-
a continuously varying trait, as well as a discretely varying one, if
the number of variants observed is large. A discrete variation series is built
only for a discretely varying trait
Sometimes an interval variation series is conditionally replaced by a discrete one.
Then the middle value of the interval is taken as option x, and the corresponding
varying interval frequency - for t x.
To determine the optimal constant interval h, they often use Sturgess formula:
h=(x max – x min)/(1+3.322*lg n).
Construction of int.var.rows
Frequencies m show how many observations the characteristic took on values belonging to a particular interval. This frequency is called interval frequency, and its ratio to the total number of observations is called interval frequency w. A table that allows one to judge the distribution of frequencies (or frequencies) between intervals of variation in the values of a characteristic is called an interval variation series.
An interval variation series is constructed based on observational data for a continuously varying trait, as well as for a discretely varying one, if the number of observed variants is large. A discrete variation series is constructed only for a discretely varying characteristic.
Sometimes an interval variation series is conditionally replaced by a discrete one. Then the middle value of the interval is taken as option x, and the corresponding interval frequency is taken as mx
To construct an interval variation series, it is necessary to determine the size of the interval, establish a full scale of intervals, and group the observation results in accordance with it.
To determine the optimal constant interval h, the Sturgess formula is often used:
h = (xmax - xmin) /(1+ 3.322 log n) .
where xmax xmin are the maximum and minimum options, respectively. If, as a result of calculations, h turns out to be a fractional number, then either the nearest whole number or the nearest simple fraction should be taken as the value of the interval.
It is recommended to take the value a1=xmin-h/2 as the beginning of the first interval; the beginning of the second interval coincides with the end of the first and is equal to a2=a1 +h; the beginning of the third interval coincides with the end of the second and is equal to a3=a2 + h. The construction of intervals continues until the beginning of the next interval in order is greater than xmax. After establishing the interval scale, the observation results should be grouped.
5) Concept, forms of expression and types of statistical indicators.
Statistical indicator represents a quantitative characteristic of socio-economic phenomena and processes in conditions of qualitative certainty. The qualitative certainty of the indicator lies in the fact that it is directly related to internal content the phenomenon or process being studied, its essence.
System of statistical indicators is a set of interrelated indicators that has a single-level or multi-level structure and is aimed at solving a specific statistical problem.
Unlike a characteristic, a statistical indicator is obtained by calculation. This can be a simple counting of population units, summing their characteristic values, comparing 2 or several values, or more complex calculations.
There is a distinction between a specific statistical indicator and a category indicator.
Specific statistical indicator characterizes the size, magnitude of the phenomenon or process being studied in a given place and in given time. However, in theoretical works and at the design stage of statistical observation, they also operate with absolute indicators or category indicators.
Indicators-categories reflect the essence, general distinctive properties of specific statistical indicators of the same type without indicating the place, time and numerical value. All statistical indicators are divided according to the coverage of population units into individual and free, and according to form - into absolute, relative and average.
Individual indicators characterize a separate object or a separate unit of a population - an enterprise, a firm, a bank, etc. An example is the number of industrial and production personnel of an enterprise. Based on the comparison of two individual absolute indicators characterizing the same object or unit, an individual relative indicator is obtained.
Summary indicators in contrast to individual ones, they characterize a group of units, representing part of a statistical population or the entire population as a whole. These indicators are divided into volumetric and calculated.
Volume indicators are obtained by adding the characteristic values of individual units of the population. The resulting value, called the volume of a feature, can act as a volumetric absolute indicator, or can be compared with another volumetric absolute value or the volume of a population. In the last 2 cases, volumetric relative and volumetric average indicators are obtained.
Estimated indicators, calculated using various formulas, serve to solve individual statistical problems of analysis - measuring variation, characteristics of structural changes, assessing relationships, etc. They are also divided into absolute, relative or average.
This group includes indices, correlation coefficients, sampling errors and other indicators.
The coverage of population units and the form of expression are the main, but not the only classification characteristics of statistical indicators. An important classification feature is also the time factor. Socio-economic processes and phenomena are reflected in statistical indicators either at a certain point in time, usually at a certain date, the beginning or end of a month, year, or for a certain period - day, week, month, quarter, year. In the first case, the indicators are momentary, in the second – interval.
Depending on belonging to one or two objects of study, they distinguish single-object And interobject indicators. If the former characterize only one object, then the latter are obtained as a result of a comparison of two quantities relating to different objects.
From the point of view of spatial certainty, statistical indicators are divided into general territorial, characterizing the object or phenomenon being studied in the country as a whole, regional and local, relating to any part of the territory or a separate object.
6) Types and relationships of relative indicators.
Relative indicator is the result of dividing one absolute indicator by another and expresses the relationship between the quantitative characteristics of socio-economic processes and phenomena. Therefore, relative to absolute indicators, relative indicators or indicators in the form of relative values are derivatives.
When calculating a relative indicator, the absolute indicator found in the numerator of the resulting ratio is called current or comparable. The indicator with which the comparison is made and which is in the denominator is called the basis or basis of comparison. Relative measures can be expressed as percentages, ppm, ratios, or can be named numbers.
All relative indicators used in practice are divided into:
·speakers; plan; ·implementation of the plan; structures; · coordination; · intensity and level of eco-development; · comparisons.
Relative danamics indicator represents the ratio of the level of the process or phenomenon under study for this period time to the level of the same process or phenomenon in the past.
OPD=current indicator/previous. Or a baseline.
The value calculated in this way shows how many times the current level exceeds the previous one or what share of the latter it is. If this indicator is expressed as a multiple ratio, it is called growth rate, when multiplying this coefficient by 100% we get growth rate.
Relative structure index represents the relationship between the structural parts of the object being studied and their whole. The relative structure indicator is expressed in fractions of a unit or as a percentage. Calculated values (d i), respectively called fractions or specific gravity, show which share or which specific gravity has the i-th part in the total.
Relative indicators of coordination characterize the relationship between the individual parts of the whole. In this case, the part that has the greatest share or is a priority from an economic, social or any other point of view is selected as a basis for comparison. As a result, we get how many units of each structural part there are per 1 unit of the basic structural part.
Relative intensity index characterizes the degree of distribution of the process or phenomenon being studied in its inherent environment. This indicator is calculated when the absolute value is insufficient to formulate substantiated conclusions about the scale of the phenomenon, its size, saturation, and distribution density. It can be expressed as a percentage, ppm, or a named quantity. A variety of relative intensity indicators are relative indicators of the level of environmental development, characterizing production per capita and playing important role in assessing the development of the state's economy. In terms of the form of expression, these indicators are close to the average indicators, which often leads to their confusion or identification. The only difference between them is that when calculating the average indicator, we are dealing with a set of units, each of which is a carrier of the averaged characteristic.
Relative Comparison Index represents the ratio of absolute indicators of the same name that characterize different objects (enterprises, firms, regions, districts, etc.)
Variation indicators
The study of variation (change in the values of a characteristic within a population) has great importance in statistics and socio-economic research in general. Absolute and relative indicators of variation, characterizing the variability of the values of a varying characteristic, make it possible, in particular, to measure the degree of connection and interrelation, to assess the degree of homogeneity of the population, the typicality and stability of the average, and to determine the magnitude of the possible error of sampling observation.
Absolute indicators of variation include the range of variation, average linear deviation, dispersion, standard deviation and quarterly deviation.
The range of variation shows by what amount the value of a quantitatively varying characteristic changes
R=xmax-xmin, where xmax(xmin) is the maximum (minimum) value of the characteristic in the aggregate (in the distribution series).
The average linear deviation d is defined as the average value of the deviations of the attribute variants from the average to the first power, taken modulo:
The average linear deviation is relatively rarely used to assess the variation of a trait. Typically the variance and standard deviation are calculated.
If it is necessary to compare the variability of several characteristics in one population or the same characteristic in several populations with different indicators of the center of distribution, then relative indicators of variation are used.
These include the following indicators:
1. Oscillation coefficient:
2. Relative linear deviation:
3. Coefficient of variation:
4. Relative quartile variation indicator:
The most commonly used measure of relative variation is the coefficient of variation. This indicator is used not only for a comparative assessment of variation, but also as a characteristic of the homogeneity of the population. A population is considered homogeneous if<0,33.
Forms.
1. Stat. reporting is an organizational form in which observation units provide information about their activities in the form of forms, regulatory apparatus.
The peculiarity of reporting is that it must be justified, enforceable and legally confirmed by the signature of the manager or responsible person.
2. Specially organized observation is the most striking and simple example of this form of observation of phenomena. census. The census is usually carried out at regular intervals, simultaneously throughout the entire study area at the same time.
Russian statistical bodies conduct censuses of the population of certain types of subsistence and organizations, material resources, perennial plantings, public health construction objects, etc.
4. Register form of observation - based on maintaining a statistical register. In the register each unit of observation is characterized by a number of indicators. In domestic statistical practice, the most widespread are the US-I registers and the sub-registers.
Population registration is carried out by the Civil Registry Office
Registration - USRPO led.org. statistics.
Kinds.
can be divided into groups according to the following. signs:
a) according to the time of registration
b) by coverage of units of society
By time reg. they are:
Current (continuous)
Intermittent (periodic and one-time)
At current obs. changes in phenomena and processes are recorded as they occur (registration of birth, death, marriage, divorce, etc.)
Periodic obs. carried out through def. time intervals (N population census every 10 years)
One time obs. carried out either not regularly, or only once (referendum)
By coverage units. Sov-ti stat-e observ. there are:
Solid
Not continuous
Continuous observation is a survey of all units of society
Continuous observation assumes that only part of the research is subject to observation.
There are several types of non-continuous observation:
Basic method array
Selective (on your own)
Monographic
This method is characterized by the fact that, as a rule, the most creatures are selected, usually the largest units. sov-ti in the cat. center means. part of all the signs.
With monographic observation, careful an. are subject to dept. units study the owl or maybe or typical for a given Soviet unit. or presenting new varieties of phenomena.
Multi-observation carried out with the aim of identifying or emerging trends in the development of this phenomenon.
Methods
Direct observation
Documentary observation
Directly called such obs. with cat The registrars themselves, by immediately measuring, counting, restraining the fact that is subject to registration, and on this basis make an entry in the form.
Documentary method of observation. based on the use of various documents as sources of information, usually accounting records (i.e. statistical reporting)
A survey is a method of persuasion with a cat. the necessary information will be obtained from the words of the respondent (i.e., the person being interviewed) (oral, correspondent, questionnaire, personal, etc.)
Determination of sampling errors.
In the process of conducting sample observation, two types of errors are distinguished: registration and representativeness.
Registration errors – deviations between the value of the indicator obtained during statistical observation and its actual value. These errors can appear during both continuous and incomplete observation. Registration errors occur due to incorrect or inaccurate information. The sources of this type of error can be a lack of understanding of the essence of the question, inattention of the registrar, omission or re-counting of individual observation units. Registration errors are divided into systematic, caused by reasons acting in any one direction and smoothing the survey results (rounding of numbers), and random, which are the result of the action of various random factors (rearranging neighboring numbers). Random errors have different directions and, with a sufficiently large volume of the population being surveyed, cancel each other out.
Representativeness errors – deviations of the values of the indicator of the surveyed population from its value in the original population. These errors are also divided into systematic, resulting from a violation of the principles of selection of units to be observed from the original population, and random, which arise if the selected population does not fully reproduce the entire population as a whole. The magnitude of the random error can be estimated.
Sampling bias– the difference between the value of a characteristic in the general population and its value calculated based on the results of sample observation. In the practice of sample surveys, the average and maximum sampling errors are most often determined.
The average sampling error is calculated differently for different sampling methods. If random or mechanical selection, then
For average: m = s 2 / (n) 1/2
For a fraction: m = (w(1-w)/n) 1/ 2, where
m - average sampling error
s 2 – general variance
n – sample size
If the sample population is formed on the basis of a typical sample and the selection of units is carried out in proportion to the size of typical groups, then the average error is equal to:
For medium: m = (s i 2 / n) 1/2
For share: m = (w i (1-w i) / n) 1/2 , Where
s i 2 – average of intragroup variances
w i is the proportion of units in this group that have the trait under study.
s i 2 = ås 2 n i / ån i
The average serial sampling error is:
For medium: m = (d x 2 / r) 1/2
For share: m = (d 2 w/r) 1/2
d 2 w – intergroup proportion variance
d x 2 – intergroup dispersion of a quantitative trait.
r – number of selected series/
d 2 x = å(x i -x) 2 / r
d 2 w = å(w i – w) 2 / r
If the selection of units from the general population is carried out in a non-repetitive manner, then an amendment is made to the average error formulas: (1-n/N) 1/2
Marginal sampling error D is calculated as the product of the confidence coefficient t and the average sampling error: D = t*m. D is related to the probability confidence level that guarantees it. This level determines the confidence coefficient t, and vice versa. The t values are given in special mathematical tables.
Determining the sample size.
The sample size is calculated, as a rule, at the design stage of the sample survey. The formulas for determining the sample size follow from the formulas for the maximum sampling errors.
The volume of actual random and mechanical repeated sampling is determined by the formulas:
For average n = t 2 s 2 / D 2
For a share n = t 2 w(1-w) / D 2
In case of non-repetitive sampling:
For average n = t 2 s 2 N / ND 2 +t 2 s 2
For a share n = t 2 w(1-w)N / ND 2 +t 2 w(1-w).
Quantities s 2 and w unknown before random observation. They are roughly found like this:
1. taken from previous surveys;
2. if the maximum and minimum values of a characteristic are known, then the standard deviation is determined according to the “three sigma” rule:
s = x max – x min / 6
3. when studying an alternative trait, if there is no information about its share in the general population, the maximum possible value w = 0.5 is taken
With typical selection, proportional to the size of typical groups, the sample size for each group is determined by the formula : n i = n*N i / N, Where
n i – sample size from the i-th group
N i– volume of the i group in the gene society.
When sampling is proportional to the variation of a characteristic, the sample size from each group is found as follows: n i = nN i s i /åN i s i .
With a typical resampling proportional to the size of the groups, the total sample size is found as follows:
For average n = t 2 s 2 i / D 2
For a share n = t 2 w(1-w) / D 2
In the case of non-repetitive typical sampling:
For average n = t 2 s 2 i N / D 2 N+t 2 s 2 i
For a share n = t 2 w(1-w)N / D 2 N+t 2 w(1-w)
Basic concepts and prerequisites for the use of correlation and regression analysis.
Correlation is a statistical dependence between random variables that do not have a strictly functional character, in which a change in one of the random variables leads to a change in the mathematical expectation of the other.
Correlation analysis– has as its task the quantitative determination of the close connection between two characteristics and between the effective and many factor characteristics. The closeness of the connection is quantitatively expressed by the magnitude of the correlation coefficients.
Correlation-regression analysis as a general concept includes measuring the tightness, direction of the connection and establishing an analytical expression (form) of the connection (regression analysis).
Regression analysis consists in determining the analytical expression of a relationship in which a change in one value (called a dependent or resultant characteristic) is due to the influence of one or more independent values (factors), and the set of all other factors that also influence the dependent value is assumed - is calculated for constant and average values. Regression can be single-factor (paired) and multi-factor (multiple).
The purpose of regression analysis is an assessment of the functional dependence of the conditional average value of the resultant characteristic (Y) on the factor (x 1, x 2, ... x k) characteristics.
The main premise of regression analysis is that only the resultant characteristic (U) obeys the normal distribution law, and the factor characteristics x 1, x 2,..., x k can have an arbitrary distribution law. In the analysis of time series, time t acts as a factor attribute. At the same time, in regression analysis it is assumed in advance that there are cause-and-effect relationships between the effective (U) factor (x 1, x 2,..., x k) characteristics. The regression equation, or statistical model of the relationship between socio-economic phenomena, expressed by the function Y x = f (x 1, x 2,..., x k), is quite adequate to the real simulated phenomenon or process if the following conditions are met requirements for their construction.
1. The set of initial data under study should be homogeneous and mathematically described by continuous functions.
2. The ability to describe the modeled phenomenon with one or more equations of cause-and-effect relationships.
3. All factor characteristics must have a quantitative (numerical) expression.
4. The presence of a sufficiently large volume of the sample population being studied.
5. Cause-and-effect relationships between phenomena and processes should be described by linear or reducible to linear forms of dependence.
6. Lack of quantitative restrictions on the parameters of the communication model.
7. Constancy of the territorial and temporal structure of the studied population.
The theoretical validity of relationship models built on the basis of correlation and regression analysis is ensured by compliance with the following basic conditions.
1. All characteristics and their joint distributions must obey the normal distribution law;
2. The variance of the modeled characteristic (V) must remain constant all the time when the value (V) and the values of the factor characteristics change.
3. Individual observations should be independent, i.e., the results obtained in the i -th observation should not be related to previous ones and contain information about subsequent observations, as well as influence them.
OBJECTIVES OF THE SUMMARY AND ITS CONTENTS
observation provides information on each unit of the object under study. The data obtained are not general indicators. With their help, it is impossible to draw conclusions about the object as a whole without preliminary data processing.
Therefore, the goal of the next stage of statistical research is to systematize the primary data and obtain, on this basis, a summary characteristic of the entire object using generalizing statistical patterns.
Summary - a set of sequential operations to generalize specific individual facts that form a set, in order to identify typical features and patterns inherent in the phenomenon being studied as a whole.
if during statistical observation data is collected about each unit of an object, then the result of the summary is detailed data reflecting the entire population as a whole
The statistical summary should be carried out on the basis of a preliminary theoretical analysis of phenomena and processes, so that during the summary one does not lose information about the phenomenon under study and all statistical results reflect the most important characteristic features of the object.
Based on the depth of material processing, the summary can be simple or complex.
A simple summary is the operation of calculating the total totals for a set of observation units.
A complex summary is a set of operations that includes grouping observation units, calculating totals for each group and for the entire object, and presenting the grouping results and summary in the form of statistical tables.
The summary is preceded by the development of its program, which consists of the following stages: selection of grouping characteristics; determining the order of group formation; development of a statistical system to characterize groups and the object as a whole; development of a system of statistical table layouts in which the summary results should be presented.
Summary of the form of material processing: decentralized and centralized.
With a decentralized summary (this is what is used, as a rule, when processing statistical reporting), the development of the material is carried out in successive stages. Thus, the reports of enterprises are compiled by the statistical authorities of the constituent entities of the Russian Federation, and the results for the region are sent to the State Statistics Committee of Russia, and there the results for the national economy of the country as a whole are determined.
With a centralized summary, all primary material enters one organization, where it is processed from start to finish. A centralized summary is usually used to process materials from one-time statistical surveys.
According to the technique of execution, the statistical summary is divided into mechanized and manual.
Mechanized summary - in which all operations are carried out using electronic computers. With manual summary, all main operations (calculation of group and general totals) are carried out manually.
To carry out the summary, a plan is drawn up that sets out organizational issues: by whom and when all operations will be carried out, the procedure for carrying them out, the composition of information to be published in periodicals.
Closing rows of din-ki
When analyzing din-ki rows, the need arises to close them - to combine two or more rows into one row. Closure is necessary in cases where the levels of the series are not comparable due to territorial changes, due to changes in prices and due to changes in the calculation method for the levels of the series. it is necessary to close (combine) the above two rows into one. This can be done using the comparability coefficient. Multiplying the data for the year by the resulting coefficient, we obtain a closed (comparable) series of dynamics of absolute values. 2. The method of closing the series of dynamics (method of reduction to one base) is that the levels of the year in which the changes occurred, as before the change , and after changes are taken as 100%, and the rest are recalculated as a percentage in relation to these levels, respectively.
30. Methods for aligning rows of din-ki
Any series of dynamics can theoretically be represented in the form of three components:
Trend (the main trend and development of the dynamic series);
Cyclic (periodic) fluctuations, including seasonal ones;
Random fluctuations.
One of the tasks that arises when analyzing dynamic series is to establish changes in the levels of the phenomenon being studied. In some cases, the pattern of changes in the levels of the din-ki series is quite clear, for example, either a systematic decrease in the levels of the series, or their increase. sometimes the levels of a series undergo very different changes (either increasing or decreasing). In this case, we can only talk about a general trend and development: either growth or decline.
Identification of the main trend and development (trend) is called time series alignment, and methods of identifying the main trend are called alignment.
Direct identification of a trend can be done by three methods.
* Md enlargement of intervals. This MD is based on the enlargement of time periods, which include the levels of the series. For example, a row of din-ki
daily output is replaced by a number of monthly projections, etc.
* Md moving average. In this method, the initial levels of the series are replaced by average values, which are obtained from a given level and several symmetrically surrounding it. The integer number of levels over which the average value is calculated is called the smoothing interval. The smoothing interval can be odd (3, 5, 7, etc. points) or even (2, 4, 6, etc. points). The averages are calculated using the sliding method, that is, by gradually excluding the first level from the accepted sliding period and including the next one. With odd smoothing, the resulting arithmetic mean value is assigned to the middle of the calculated interval.
“-” m-dics of smoothing by moving averages consists in the convention of determining smoothed levels for points at the beginning and end of the series.
* Analytical alignment is the most effective way to identify the main trend and development. In this case, the levels of the dynamics series are expressed as a function of time: Yt=f(t)
The purpose of analytic alignment of a series is to determine the analytic value f(t). In practice, using the existing time series, they set the form and find the parameters of the function f(t), and then analyze the behavior of deviations from the trend.
In economics, a function of the form is often used: Уi = а0 +∑ аi +ti
From a function of the form (3.12), most often when leveling, the linear function /(*) = ao + a1 *t or the parabolic f(t) = a0 +att + a2 t2 is used.
The coefficients ao,a,a2,...,ap in the formula are found by least squares.
According to this method, to find the parameters of a polynomial of pth degree, it is necessary to solve a system of so-called normal equations:
nao+a1∑t=∑Y
ao∑t+ a1∑t*t= ∑Y*t.
The trend shows how systematic factors influence the dynamics of the population. Fluctuation of levels around the trend serves as a measure of the impact of residual (random) factors. This measure of impact can be assessed
using the standard deviation formula.
Basic concepts of correlation and regression analysis.
Variation series: definition, types, main characteristics. Calculation method
mode, median, arithmetic mean in medical and statistical research
(show with a conditional example).
A variation series is a series of numerical values of the characteristic being studied, differing from each other in magnitude and arranged in a certain sequence (in ascending or descending order). Each numerical value of a series is called a variant (V), and the numbers showing how often a particular variant occurs in a given series are called frequency (p).
The total number of observation cases that make up the variation series is denoted by the letter n. The difference in the meaning of the characteristics being studied is called variation. If a varying characteristic does not have a quantitative measure, the variation is called qualitative, and the distribution series is called attributive (for example, distribution by disease outcome, health status, etc.).
If a varying characteristic has a quantitative expression, such variation is called quantitative, and the distribution series is called variational.
Variation series are divided into discontinuous and continuous - based on the nature of the quantitative characteristic; simple and weighted - based on the frequency of occurrence of the variant.
In a simple variation series, each option occurs only once (p=1), in a weighted series, the same option occurs several times (p>1). Examples of such series will be discussed further in the text. If the quantitative characteristic is continuous, i.e. Between integer quantities there are intermediate fractional quantities; the variation series is called continuous.
For example: 10.0 – 11.9
14.0 – 15.9, etc.
If the quantitative characteristic is discontinuous, i.e. its individual values (variants) differ from each other by an integer and do not have intermediate fractional values; the variation series is called discontinuous or discrete.
Using the heart rate data from the previous example
for 21 students, we will construct a variation series (Table 1).
Table 1
Distribution of medical students by heart rate (bpm)
Thus, to construct a variation series means to systematize and organize the available numerical values (variants), i.e. arrange in a certain sequence (in ascending or descending order) with their corresponding frequencies. In the example under consideration, the options are arranged in ascending order and expressed as integer discontinuous (discrete) numbers, each option occurs several times, i.e. we are dealing with a weighted, discontinuous or discrete variation series.
As a rule, if the number of observations in the statistical population we are studying does not exceed 30, then it is enough to arrange all the values of the characteristic being studied in an ascending variation series, as in Table. 1, or descending order.
At large quantities observations (n>30), the number of occurring variants can be very large, in this case an interval or grouped variation series is compiled, in which, to simplify subsequent processing and clarify the nature of the distribution, the variants are combined into groups.
Typically the number of group options ranges from 8 to 15.
There should be at least 5 of them, because... otherwise it will be too rough, excessive enlargement, which distorts the overall picture of variation and greatly affects the accuracy of average values. When the number of group variants is more than 20-25, the accuracy of calculating average values increases, but the characteristics of the variation of the characteristic are significantly distorted and mathematical processing becomes more complicated.
When compiling a grouped series, it is necessary to take into account
− option groups must be arranged in a certain order (ascending or descending);
− intervals in option groups must be the same;
− the values of the interval boundaries should not coincide, because it will be unclear which groups to classify individual variants into;
− must be taken into account quality features collected material when setting interval limits (for example, when studying the weight of adults, an interval of 3-4 kg is acceptable, and for children in the first months of life it should not exceed 100 g)
Let's construct a grouped (interval) series characterizing data on the pulse rate (beats per minute) of 55 medical students before the exam: 64, 66, 60, 62,
64, 68, 70, 66, 70, 68, 62, 68, 70, 72, 60, 70, 74, 62, 70, 72, 72,
64, 70, 72, 76, 76, 68, 70, 58, 76, 74, 76, 76, 82, 76, 72, 76, 74,
79, 78, 74, 78, 74, 78, 74, 74, 78, 76, 78, 76, 80, 80, 80, 78, 78.
To build a grouped series you need:
1. Determine the size of the interval;
2. Determine the middle, beginning and end of the groups of the variation series.
● The size of the interval (i) is determined by the number of supposed groups (r), the number of which is set depending on the number of observations (n) according to a special table
Number of groups depending on the number of observations:
In our case, for 55 students, you can create from 8 to 10 groups.
The value of the interval (i) is determined by the following formula –
i = V max-V min/r
In our example, the value of the interval is 82-58/8= 3.
If the interval value is a fractional number, the result should be rounded to a whole number.
There are several types of averages:
● arithmetic mean,
● geometric mean,
● harmonic mean,
● root mean square,
● average progressive,
● median
IN medical statistics Arithmetic averages are most often used.
The arithmetic mean (M) is a generalizing value that determines what is typical for the entire population. The main methods for calculating M are: the arithmetic mean method and the method of moments (conditional deviations).
The arithmetic mean method is used to calculate the simple arithmetic mean and the weighted arithmetic mean. The choice of method for calculating the arithmetic mean depends on the type of variation series. In the case of a simple variation series, in which each option occurs only once, the arithmetic mean simple is determined by the formula:
where: M – arithmetic mean value;
V – value of the varying characteristic (variants);
Σ – indicates the action – summation;
n – total number of observations.
An example of calculating the simple arithmetic average. Respiratory rate (number of breathing movements per minute) in 9 men aged 35 years: 20, 22, 19, 15, 16, 21, 17, 23, 18.
To determine the average level of respiratory rate in men aged 35 years, it is necessary:
1. Construct a variation series, arranging all options in ascending or descending order. We have obtained a simple variation series, because option values occur only once.
M = ∑V/n = 171/9 = 19 breaths per minute
Conclusion. The respiratory rate in men aged 35 years is on average 19 respiratory movements per minute.
If individual values of a variant are repeated, there is no need to write down each variant in a line; it is enough to list the occurring sizes of the variant (V) and next to it indicate the number of their repetitions (p). Such a variation series, in which the options are, as it were, weighed by the number of frequencies corresponding to them, is called a weighted variation series, and the calculated average value is the weighted arithmetic mean.
The weighted arithmetic mean is determined by the formula: M= ∑Vp/n
where n is the number of observations equal to the sum of frequencies – Σр.
An example of calculating the arithmetic weighted average.
The duration of disability (in days) in 35 patients with acute respiratory diseases (ARI) treated by a local doctor during the first quarter of the current year was: 6, 7, 5, 3, 9, 8, 7, 5, 6, 4, 9, 8, 7, 6, 6, 9, 6, 5, 10, 8, 7, 11, 13, 5, 6, 7, 12, 4, 3, 5, 2, 5, 6, 6, 7 days .
The method for determining the average duration of disability in patients with acute respiratory infections is as follows:
1. Let's construct a weighted variation series, because Individual values of the option are repeated several times. To do this, you can arrange all options in ascending or descending order with their corresponding frequencies.
In our case, the options are arranged in ascending order
2. Calculate the arithmetic weighted average using the formula: M = ∑Vp/n = 233/35 = 6.7 days
Distribution of patients with acute respiratory infections by duration of disability:
| Duration of disability (V) | Number of patients (p) | Vp |
| ∑p = n = 35 | ∑Vp = 233 |
Conclusion. The duration of disability in patients with acute respiratory diseases averaged 6.7 days.
Mode (Mo) is the most common option in the variation series. For the distribution presented in the table, the mode corresponds to an option equal to 10; it occurs more often than others - 6 times.
Distribution of patients by length of stay in a hospital bed (in days)
| V |
| p |
Sometimes it is difficult to determine the exact magnitude of a mode because there may be several “most common” observations in the data being studied.
Median (Me) is a nonparametric indicator that divides the variation series into two equal halves: on both sides of the median is located same number option.
For example, for the distribution shown in the table, the median is 10, because on both sides of this value there are 14 options, i.e. the number 10 occupies a central position in this series and is its median.
Given that the number of observations in this example is even (n=34), the median can be determined as follows:
Me = 2+3+4+5+6+5+4+3+2/2 = 34/2 = 17
This means that the middle of the series falls on the seventeenth option, which corresponds to a median equal to 10. For the distribution presented in the table, the arithmetic mean is equal to:
M = ∑Vp/n = 334/34 = 10.1
So, for 34 observations from table. 8, we got: Mo=10, Me=10, arithmetic mean (M) is 10.1. In our example, all three indicators turned out to be equal or close to each other, although they are completely different.
The arithmetic mean is the resultant sum of all influences; all options without exception, including extreme ones, often atypical for a given phenomenon or population, take part in its formation.
The mode and median, unlike the arithmetic mean, do not depend on the value of all individual values of the varying characteristic (the values of the extreme variants and the degree of dispersion of the series). The arithmetic mean characterizes the entire mass of observations, the mode and median characterize the bulk
- 1. Public health and healthcare as a science and area of practical activity. Main goals. Object, subject of study. Methods.
- 2. History of healthcare development. Modern healthcare systems, their characteristics.
- 3. State policy in the field of protecting public health (Law of the Republic of Belarus “On Health Care”). Organizational principles of the public health care system.
- 4. Nomenclature of healthcare organizations
- 6. Insurance and private forms of healthcare.
- 7. Medical ethics and deontology. Definition of the concept. Modern problems of medical ethics and deontology, characteristics. Hippocratic Oath, Doctor's Oath of the Republic of Belarus, Code of Medical Ethics.
- 10. Statistics. Definition of the concept. Types of statistics. Statistical data recording system.
- 11. Groups of indicators for assessing the health status of the population.
- 15.Unit of observation. Definition, characteristics of accounting characteristics
- 26. Time series, their types.
- 27. Time series indicators, calculation, application in medical practice.
- 28. Variation series, its elements, types, rules of construction.
- 29. Average values, types, calculation methods. Application in the work of a doctor.
- 30. Indicators characterizing the diversity of a trait in the population being studied.
- 31. Representativeness of the feature. Assessing the reliability of differences in relative and average values. The concept of Student's t test.
- 33. Graphic displays in statistics. Types of diagrams, rules for their construction and design.
- 34. Demography as a science, definition, content. The importance of demographic data for health care.
- 35. Population health, factors influencing public health. Health formula. Indicators characterizing public health. Analysis scheme.
- 36. Leading medical and social problems of population. Problems of population size and composition, mortality, fertility. Take from 37,40,43
- 37. Population statistics, study methods. Population censuses. Types of age structures of the population. Population size and composition, implications for healthcare
- 38. Population dynamics, its types.
- 39. Mechanical movement of the population. Study methodology. Characteristics of migration processes, their impact on population health indicators.
- 40. Fertility as a medical and social problem. Study methodology, indicators. Fertility levels according to WHO data. Current trends in the Republic of Belarus and in the world.
- 42. Population reproduction, types of reproduction. Indicators, calculation methods.
- 43. Mortality as a medical and social problem. Study methodology, indicators. Overall mortality levels according to WHO data. Modern tendencies. Main causes of population mortality.
- 44. Infant mortality as a medical and social problem. Factors determining its level. Methodology for calculating indicators, WHO assessment criteria.
- 45. Perinatal mortality. Methodology for calculating indicators. Causes of perinatal mortality.
- 46. Maternal mortality. Methodology for calculating the indicator. Level and causes of maternal mortality in the Republic of Belarus and the world.
- 52.Medical and social aspects of the neuropsychic health of the population. Organization of psychoneurological care.
- 60. Methodology for studying morbidity. 61. Methods for studying population morbidity, their comparative characteristics.
- Methodology for studying general and primary morbidity
- Indicators of general and primary morbidity.
- 63. Study of population morbidity according to special registration data (infectious and major non-epidemic diseases, hospitalized morbidity). Indicators, accounting and reporting documents.
- Main indicators of “hospitalized” morbidity:
- Main indicators for the analysis of morbidity with VUT.
- 65. Study of morbidity according to preventive examinations of the population, types of preventive examinations, procedure. Health groups. The concept of “pathological affection”.
- 66. Morbidity according to data on causes of death. Study methodology, indicators. Medical death certificate.
- Main morbidity indicators based on causes of death:
- 67. Forecasting morbidity rates.
- 68. Disability as a medical and social problem. Definition of the concept, indicators.
- Disability trends in the Republic of Belarus.
- 69. Mortality. Calculation method and analysis of lethality. Implications for the practical activities of doctors and healthcare organizations.
- 70. Standardization methods, their scientific and practical purpose. Calculation methods and analysis of standardized indicators.
- 72. Criteria for determining disability. The degree of expression of persistent disorders of body functions. Indicators characterizing disability.
- 73. Prevention, definition, principles, modern problems. Types, levels, directions of prevention.
- 76. Primary health care, definition of the concept, role and place in the system of medical care for the population. Main functions.
- 78.. Organization of medical care provided to the population on an outpatient basis. Main organizations: medical outpatient clinic, city clinic. Structure, tasks, areas of activity.
- 79. Nomenclature of hospital organizations. Organization of medical care in hospital settings of healthcare organizations. Indicators of provision of inpatient care.
- 80. Types, forms and conditions of medical care. Organization of specialized medical care, their tasks.
- 81. Main directions for improving inpatient and specialized care.
- 82. Protecting the health of women and children. Control. Medical organizations.
- 83. Modern problems of women's health. Organization of obstetric and gynecological care.
- 84. Organization of medical and preventive care for children. Leading problems in children's health.
- 85. Organization of health care for the rural population, basic principles of providing medical care to rural residents. Stages of organization.
- Stage II – territorial medical association (TMO).
- Stage III – regional hospital and regional medical institutions.
- 86. City clinic, structure, tasks, management. Key performance indicators of the clinic.
- Key performance indicators of the clinic.
- 87. Precinct-territorial principle of organizing outpatient care for the population. Types of plots.
- 88. Territorial therapeutic area. Standards. Contents of the work of a local therapist.
- 89. Office of infectious diseases of the clinic. Sections and methods of work of a doctor in the office of infectious diseases.
- 90. Preventive work of the clinic. Prevention department of the clinic. Organization of preventive examinations.
- 91. Dispensary method in the work of the clinic, its elements. Control card of dispensary observation, information reflected in it.
- 1st stage. Registration, examination of the population and selection of contingents for registration at the dispensary.
- 2nd stage. Dynamic monitoring of the health status of those being examined and carrying out preventive and therapeutic measures.
- 3rd stage. Annual analysis of the state of dispensary work in hospitals, assessment of its effectiveness and development of measures to improve it (see Question 51).
- 96. Department of medical rehabilitation of the clinic. Structure, tasks. The procedure for referral to the medical rehabilitation department.
- 97. Children's clinic, structure, tasks, sections of work.
- 98. Features of providing medical care to children on an outpatient basis
- 99. The main sections of the work of a local pediatrician. Contents of treatment and preventive work. Communication in work with other treatment and prevention organizations. Documentation.
- 100. Contents of preventive work of a local pediatrician. Organization of nursing care for newborns.
- 101. Comprehensive assessment of the health status of children. Medical examinations. Health groups. Medical examination of healthy and sick children
- Section 1. Information about the divisions and installations of the treatment and preventive organization.
- Section 2. Staff of the treatment and prevention organization at the end of the reporting year.
- Section 3. Work of doctors of the clinic (outpatient clinic), dispensary, consultations.
- Section 4. Preventive medical examinations and work of dental (dental) and surgical offices of a medical and preventive organization.
- Section 5. Work of medical and auxiliary departments (offices).
- Section 6. Operation of diagnostic departments.
- Section I. Activities of the antenatal clinic.
- Section II. Obstetrics in a hospital
- Section III. Maternal mortality
- Section IV. Information about births
- 145. Medical and social examination, definition, content, basic concepts.
- 146. Legislative documents regulating the procedure for conducting medical and social examinations.
- 147. Types of darkness. Composition of regional, district, inter-district, city and specialized MRECs. Organization of work, rights and responsibilities. The procedure for referral to MREK and examination of citizens.
- 148. Basic tasks and concepts of medical and social examination.
- 149. Rehabilitation, definition, types. Law of the Republic of Belarus “On the Prevention of Disability and Rehabilitation of Disabled Persons”.
- the series is formed from relative or average values.
27. Time series indicators, calculation, application in medical practice.
The absolute level of the series-value (levels) that make up the dynamic series (reflect
phenomena at a certain moment or time interval))
Absolute increase represents the difference between the next and previous levels.
Growth rate is the ratio of the next level to the previous one, multiplied by 100%.
Rate of increase is the ratio of the absolute increase (decrease) to the previous level, multiplied by 100%.
Value of 1% increase is determined by the ratio of absolute growth to the growth rate.
Visualization indicator (shows the ratio of each level of the series to one of them, usually the initial one, taken as 100%).
28. Variation series, its elements, types, rules of construction.
Variation series- a number of homogeneous statistical quantities characterizing the same quantitative accounting characteristic, differing from each other in their magnitude and arranged in a certain order (decreasing or increasing).
Elements of the variation series:
A) option -v- the numerical value of the changing quantitative characteristic being studied.
b) frequency -porf- repeatability of an option in a variation series, showing how often one or another option occurs in a given series.
V) total number of observations -n- sum of all frequencies: n=ΣΡ. If the total number of observations is more than 30, the statistical sample is considered big, if n is less than or equal to 30 - small.
Variation series are:
depending on the frequency of occurrence of the trait:
A) simple- series - each option occurs once, i.e. frequencies are equal to unity.
b) ordinary- a series in which options appear more than once.
V) grouped- a series in which options are combined into groups according to their size within a certain interval, indicating the frequency of repetition of all options included in the group.
A grouped variation series is used when there is a large number of observations and a large range of extreme values.
Processing the variation series consists of obtaining the parameters of the variation series (average value, standard deviation and average error of the average value).
3. depending on the number of observations:
a) even and odd
b) large (if the number of observations is more than 30) and small (if the number of observations is less than or equal to 30)
29. Average values, types, calculation methods. Application in the work of a doctor.
Average values give a generalizing characteristic of a statistical population according to a certain changing quantitative characteristic. average value characterizes the entire series of observations with one number, expressing the general measure of the characteristic being studied. It levels out random deviations of individual observations and gives a typical characteristic of a quantitative characteristic.
Requirements for average values:
1) qualitative homogeneity of the population for which the average value is calculated - only then will it objectively reflect the characteristic features of the phenomenon being studied.
2) the average value should be based on a mass generalization of the characteristic being studied, because only then does it express the typical dimensions of the trait
Average values are obtained from distribution series (variation series).
Types of averages:
A ) fashion(Mo) is the value of a characteristic that occurs more often than others in the aggregate. The mode is taken to be the variant that corresponds to the largest number of frequencies in the variation series.
b ) Median(Me) is the value of a characteristic that occupies the middle value in the variation series. It divides the variation series into two equal parts.
The magnitude of the mode and median are not affected by the numerical values of the extreme variants available in the variation series. They cannot always accurately characterize the variation series and are used relatively rarely in medical statistics. The arithmetic mean characterizes the variation series more accurately.
V ) Arithmetic mean(M, or) - calculated based on all numerical values of the characteristic being studied.
Other average values are used less frequently: geometric average (when processing the results of titration of antibodies, toxins, vaccines); root mean square (when determining the average diameter of a cell cut, the results of skin immunological tests); average cubic (to determine the average volume of tumors) and others.
In a simple variation series, where options occur only once, the simple arithmetic mean is calculated using the formula:
where V are the numerical values of the option, n is the number of observations,In a regular variation series, the weighted arithmetic mean is calculated using the formula:
Where V are the numerical values of the variant, p is the frequency of occurrence of the variant, n is the number of observations.
Averages of the same value can be obtained from series with different degrees of dispersion; therefore, to characterize a variation series, in addition to the average value, another characteristic is needed , allowing one to assess the degree of its variability.
Simple indicators characterizing the diversity of a trait in the population under study are
A) limit- minimum and maximum value of a quantitative characteristic
b) amplitude- the difference between the largest and smallest value of the option.
Application of average values:
a) to characterize physical development (height, weight, chest circumference, dynamometry)
b) to assess the state of human health by analyzing the physiological, biochemical parameters of the body (blood pressure, heart rate, body temperature)
c) to analyze the activities of medical organizations (average number of days a bed is open per year, etc.)
d) to evaluate the work of doctors (average number of visits per doctor, average number of surgical operations, average hourly workload of a doctor at a clinic appointment)
