Point variation series. An example of solving typical problems. Conditions and limitations for applying the odds ratio

A special place in statistical analysis belongs to the determination of the average level of the studied trait or phenomenon. The average level of a feature is measured by average values.

The average value characterizes the general quantitative level of the studied trait and is a group property of the statistical population. It levels, weakens the random deviations of individual observations in one direction or another and highlights the main, typical property of the trait under study.

Averages are widely used:

1. To assess the health status of the population: characteristics of physical development (height, weight, chest circumference, etc.), identifying the prevalence and duration of various diseases, analyzing demographic indicators (natural population movement, average life expectancy, population reproduction, average population, etc.).

2. To study the activities of medical institutions, medical personnel and assess the quality of their work, planning and determining the needs of the population in various types of medical care (average number of applications or visits per inhabitant per year, average duration of a patient's stay in a hospital, average duration of a patient's examination, average provision with doctors, beds, etc.).

3. To characterize the sanitary and epidemiological state (average dustiness of the air in the workshop, average area per person, average consumption of proteins, fats and carbohydrates, etc.).

4. To determine the medical and physiological parameters in the norm and pathology, in the processing of laboratory data, to establish the reliability of the results of a selective study in socio-hygienic, clinical, experimental studies.

Calculation of average values ​​is performed on the basis of variation series. Variation series- this is a qualitatively homogeneous statistical set, the individual units of which characterize the quantitative differences of the studied feature or phenomenon.

Quantitative variation can be of two types: discontinuous (discrete) and continuous.

A discontinuous (discrete) sign is expressed only as an integer and cannot have any intermediate values ​​(for example, the number of visits, the population of the site, the number of children in the family, the severity of the disease in points, etc.).

A continuous sign can take on any values ​​within certain limits, including fractional ones, and is expressed only approximately (for example, weight - for adults you can limit yourself to kilograms, and for newborns - grams; height, blood pressure, time spent on seeing a patient, etc.).

The digital value of each individual feature or phenomenon included in the variation series is called a variant and is indicated by the letter V . There are also other notations in the mathematical literature, for example x or y.

A variational series, where each option is indicated once, is called simple. Such series are used in most statistical problems in the case of computer data processing.

With an increase in the number of observations, as a rule, there are repeated values ​​of the variant. In this case, it creates grouped variation series, where the number of repetitions is indicated (frequency, denoted by the letter " R »).

Ranked variation series consists of options arranged in ascending or descending order. Both simple and grouped series can be composed with ranking.

Interval variation series are made up in order to simplify subsequent calculations performed without using a computer, with a very large number of observation units (more than 1000).

Continuous variation series includes variant values, which can be any value.

If in the variation series the values ​​of the attribute (options) are given in the form of separate specific numbers, then such a series is called discrete.

The general characteristics of the values ​​of the attribute reflected in the variation series are the average values. Among them, the most used are: the arithmetic mean M, fashion Mo and median me. Each of these characteristics is unique. They cannot replace each other, and only in the aggregate, quite fully and in a concise form, are the features of the variational series.

Fashion (Mo) name the value of the most frequently occurring options.

Median (me) is the value of the variant dividing the ranged variational series in half (on each side of the median there is a half of the variant). In rare cases, when there is a symmetrical variation series, the mode and median are equal to each other and coincide with the value of the arithmetic mean.

The most typical characteristic of variant values ​​is arithmetic mean value( M ). In mathematical literature, it is denoted .

Arithmetic mean (M, ) is a general quantitative characteristic of a certain feature of the studied phenomena, which make up a qualitatively homogeneous statistical set. Distinguish between simple arithmetic mean and weighted mean. The simple arithmetic mean is calculated for a simple variational series by summing all the options and dividing this sum by the total number of options included in this variational series. Calculations are carried out according to the formula:

Where: M - simple arithmetic mean;

Σ V - amount option;

n- number of observations.

In the grouped variation series, a weighted arithmetic mean is determined. The formula for its calculation:

Where: M - arithmetic weighted average;

Σ vp - the sum of products of a variant on their frequencies;

n- number of observations.

With a large number of observations in the case of manual calculations, the method of moments can be used.

The arithmetic mean has the following properties:

the sum of the deviations of the variant from the mean ( Σ d ) is equal to zero (see Table 15);

When multiplying (dividing) all options by the same factor (divisor), the arithmetic mean is multiplied (divided) by the same factor (divider);

If you add (subtract) the same number to all options, the arithmetic mean increases (decreases) by the same number.

Arithmetic averages, taken by themselves, without taking into account the variability of the series from which they are calculated, may not fully reflect the properties of the variation series, especially when comparison with other averages is necessary. Average values ​​close in value can be obtained from series with different degrees of dispersion. The closer the individual options are to each other in terms of their quantitative characteristics, the less scattering (fluctuation, variability) series, the more typical its average.

The main parameters that allow assessing the variability of a trait are:

· scope;

Amplitude;

· Standard deviation;

· The coefficient of variation.

Approximately, the fluctuation of a trait can be judged by the scope and amplitude of the variation series. The range indicates the maximum (V max) and minimum (V min) options in the series. The amplitude (A m) is the difference between these options: A m = V max - V min .

The main, generally accepted measure of the fluctuation of the variational series are dispersion (D ). But the more convenient parameter is most often used, calculated on the basis of the variance - the standard deviation ( σ ). It takes into account the deviation value ( d ) of each variant of the variational series from its arithmetic mean ( d=V - M ).

Since the deviations of the variant from the mean can be positive and negative, when summed they give the value "0" (S d=0). To avoid this, the deviation values ​​( d) are raised to the second power and averaged. Thus, the variance of the variational series is the average square of the deviations of the variant from the arithmetic mean and is calculated by the formula:

It is the most important characteristic of variability and is used to calculate many statistical tests.

Because the variance is expressed as the square of the deviations, its value cannot be used in comparison with the arithmetic mean. For these purposes, it is used standard deviation, which is denoted by the sign "Sigma" ( σ ). It characterizes the average deviation of all variants of the variation series from the arithmetic mean in the same units as the mean itself, so they can be used together.

The standard deviation is determined by the formula:

This formula is applied for the number of observations ( n ) is greater than 30. With a smaller number n the value of the standard deviation will have an error associated with the mathematical bias ( n - 1). In this regard, a more accurate result can be obtained by taking into account such a bias in the formula for calculating the standard deviation:

standard deviation (s ) is an estimate of the standard deviation of the random variable X relative to its mathematical expectation based on an unbiased estimate of its variance.

For values n > 30 standard deviation ( σ ) and standard deviation ( s ) will be the same ( σ=s ). Therefore, in most practical manuals, these criteria are treated as having different meanings. In Excel, the calculation of the standard deviation can be done with the function =STDEV(range). And in order to calculate the standard deviation, you need to create an appropriate formula.

The root mean square or standard deviation allows you to determine how much the values ​​of a feature can differ from the mean value. Suppose there are two cities with the same average daily temperature in summer. One of these cities is located on the coast, and the other on the continent. It is known that in cities located on the coast, the differences in daytime temperatures are less than in cities located inland. Therefore, the standard deviation of daytime temperatures near the coastal city will be less than that of the second city. In practice, this means that the average air temperature of each particular day in a city located on the continent will differ more from the average than in a city on the coast. In addition, the standard deviation makes it possible to estimate possible temperature deviations from the average with the required level of probability.

According to the theory of probability, in phenomena that obey the normal distribution law, there is a strict relationship between the values ​​of the arithmetic mean, standard deviation and options ( three sigma rule). For example, 68.3% of the values ​​of a variable attribute are within M ± 1 σ , 95.5% - within M ± 2 σ and 99.7% - within M ± 3 σ .

The value of the standard deviation makes it possible to judge the nature of the homogeneity of the variation series and the group under study. If the value of the standard deviation is small, then this indicates a sufficiently high homogeneity of the phenomenon under study. The arithmetic mean in this case should be recognized as quite characteristic of this variational series. However, a too small sigma makes one think of an artificial selection of observations. With a very large sigma, the arithmetic mean characterizes the variation series to a lesser extent, which indicates a significant variability of the studied trait or phenomenon or the heterogeneity of the study group. However, comparison of the value of the standard deviation is possible only for signs of the same dimension. Indeed, if we compare the weight diversity of newborns and adults, we will always get higher sigma values ​​in adults.

Comparison of the variability of features of different dimensions can be performed using coefficient of variation. It expresses diversity as a percentage of the mean, which allows comparison of different traits. The coefficient of variation in the medical literature is indicated by the sign " WITH ", and in the mathematical " v» and calculated by the formula:

The values ​​of the coefficient of variation less than 10% indicate a small scattering, from 10 to 20% - about the average, more than 20% - about a strong scattering around the arithmetic mean.

The arithmetic mean is usually calculated on the basis of sample data. With repeated studies under the influence of random phenomena, the arithmetic mean may change. This is due to the fact that, as a rule, only a part of the possible units of observation, that is, a sample population, is investigated. Information about all possible units representing the phenomenon under study can be obtained by studying the entire general population, which is not always possible. At the same time, in order to generalize the experimental data, the value of the average in the general population is of interest. Therefore, in order to formulate a general conclusion about the phenomenon under study, the results obtained on the basis of the sample population must be transferred to the general population by statistical methods.

In order to determine the degree of agreement between the sample study and the general population, it is necessary to estimate the amount of error that inevitably arises during sample observation. Such an error is called representativeness error” or “Mean error of the arithmetic mean”. It is, in fact, the difference between the averages obtained from selective statistical observation and similar values ​​that would be obtained from a continuous study of the same object, i.e. when studying the general population. Since the sample mean is a random variable, such a forecast is made with an acceptable level of probability for the researcher. In medical research, it is at least 95%.

The representativeness error should not be confused with registration errors or attention errors (misprints, miscalculations, misprints, etc.), which should be minimized by an adequate methodology and tools used in the experiment.

The magnitude of the error of representativeness depends on both the sample size and the variability of the trait. The larger the number of observations, the closer the sample to the general population and the smaller the error. The more variable the feature, the greater the statistical error.

In practice, the following formula is used to determine the representativeness error in variational series:

Where: m – representativeness error;

σ – standard deviation;

n is the number of observations in the sample.

It can be seen from the formula that the size of the average error is directly proportional to the standard deviation, i.e., the variability of the trait under study, and inversely proportional to the square root of the number of observations.

When performing statistical analysis based on the calculation of relative values, the construction of a variation series is not mandatory. In this case, the determination of the average error for relative indicators can be performed using a simplified formula:

Where: R- the value of the relative indicator, expressed as a percentage, ppm, etc.;

q- the reciprocal of P and expressed as (1-P), (100-P), (1000-P), etc., depending on the basis for which the indicator is calculated;

n is the number of observations in the sample.

However, the indicated formula for calculating the representativeness error for relative values ​​can only be applied when the value of the indicator is less than its base. In a number of cases of calculating intensive indicators, this condition is not met, and the indicator can be expressed as a number of more than 100% or 1000%o. In such a situation, a variation series is constructed and the representativeness error is calculated using the formula for average values ​​based on the standard deviation.

Forecasting the value of the arithmetic mean in the general population is performed with the indication of two values ​​- the minimum and maximum. These extreme values ​​​​of possible deviations, within which the desired average value of the general population can fluctuate, are called " Confidence boundaries».

The postulates of probability theory proved that with a normal distribution of a feature with a probability of 99.7%, the extreme values ​​of the deviations of the mean will not exceed the value of the triple error of representativeness ( M ± 3 m ); in 95.5% - no more than the value of the doubled average error of the average value ( M ±2 m ); in 68.3% - no more than the value of one average error ( M ± 1 m ) (Fig. 9).

P%

Rice. 9. Probability density of normal distribution.

Note that the above statement is true only for a feature that obeys the normal Gaussian distribution law.

Most experimental studies, including those in the field of medicine, are associated with measurements, the results of which can take almost any value in a given interval, therefore, as a rule, they are described by a model of continuous random variables. In this regard, most statistical methods consider continuous distributions. One of these distributions, which plays a fundamental role in mathematical statistics, is normal, or Gaussian, distribution.

This is due to a number of reasons.

1. First of all, many experimental observations can be successfully described using a normal distribution. It should be immediately noted that there are no distributions of empirical data that would be exactly normal, since a normally distributed random variable is in the range from to , which never occurs in practice. However, the normal distribution is very often a good approximation.

Whether measurements of weight, height and other physiological parameters of the human body are carried out - everywhere a very large number of random factors (natural causes and measurement errors) influence the results. And, as a rule, the effect of each of these factors is insignificant. Experience shows that the results in such cases will be distributed approximately normally.

2. Many distributions associated with a random sample, with an increase in the volume of the latter, become normal.

3. The normal distribution is well suited as an approximate description of other continuous distributions (for example, asymmetric ones).

4. The normal distribution has a number of favorable mathematical properties, which largely ensured its widespread use in statistics.

At the same time, it should be noted that in medical data there are many experimental distributions that cannot be described by the normal distribution model. To do this, statistics have developed methods that are commonly called "Nonparametric".

The choice of a statistical method that is suitable for processing the data of a particular experiment should be made depending on whether the data obtained belong to the normal distribution law. Hypothesis testing for the subordination of a sign to the normal distribution law is performed using a histogram of the frequency distribution (graph), as well as a number of statistical criteria. Among them:

Asymmetry criterion ( b );

Criteria for checking for kurtosis ( g );

Shapiro–Wilks criterion ( W ) .

An analysis of the nature of the distribution of data (it is also called a test for the normality of the distribution) is carried out for each parameter. In order to confidently judge the compliance of the parameter distribution with the normal law, a sufficiently large number of observation units (at least 30 values) is required.

For a normal distribution, the skewness and kurtosis criteria take the value 0. If the distribution is shifted to the right b > 0 (positive asymmetry), with b < 0 - график распределения смещен влево (отрицательная асимметрия). Критерий асимметрии проверяет форму кривой распределения. В случае нормального закона g =0. At g > 0 the distribution curve is sharper if g < 0 пик более сглаженный, чем функция нормального распределения.

To test for normality using the Shapiro-Wilks test, it is required to find the value of this criterion using statistical tables at the required level of significance and depending on the number of units of observation (degrees of freedom). Appendix 1. The hypothesis of normality is rejected for small values ​​of this criterion, as a rule, for w <0,8.

grouping- this is the division of the population into groups that are homogeneous in some way.

Service assignment. With the online calculator you can:

• build a variation series, build a histogram and a polygon;
• find indicators of variation (mean, mode (including graphically), median, range of variation, quartiles, deciles, quartile coefficient of differentiation, coefficient of variation and other indicators);

Instruction. To group a series, you must select the type of the resulting variation series (discrete or interval) and specify the amount of data (number of rows). The resulting solution is saved in a Word file (see the example of grouping statistical data).

If the grouping has already been done and the discrete variation series or interval series, then you need to use the online calculator Variation indicators. Testing the hypothesis about the type of distribution produced using the service Study of the form of distribution.

Types of statistical groupings

Variation series. In the case of observations of a discrete random variable, the same value can be encountered several times. Such values ​​\u200b\u200bof a random variable x i are recorded indicating n i the number of times it appears in n observations, this is the frequency of this value.
In the case of a continuous random variable, grouping is used in practice.

1. Typological grouping- this is the division of the studied qualitatively heterogeneous population into classes, socio-economic types, homogeneous groups of units. To build this grouping, use the Discrete variational series parameter.
2. Structural grouping is called, in which a homogeneous population is divided into groups that characterize its structure according to some varying feature. To build this grouping, use the Interval series parameter.
3. A grouping that reveals the relationship between the studied phenomena and their features is called analytical group(see analytical grouping of series).

Example #1. According to table 2, build the distribution series for 40 commercial banks of the Russian Federation. According to the obtained distribution series, determine: average profit per one commercial bank, credit investments on average per one commercial bank, modal and median value of profit; quartiles, deciles, range of variation, mean linear deviation, standard deviation, coefficient of variation.

Solution:
In chapter "Type of statistical series" choose Discrete Series. Click Paste from Excel. Number of groups: according to the Sturgess formula

Principles of building statistical groupings

A series of observations ordered in ascending order is called a variation series. grouping sign is the sign by which the population is divided into separate groups. It is called the base of the group. Grouping can be based on both quantitative and qualitative characteristics.
After determining the basis of the grouping, the question of the number of groups into which the study population should be divided should be decided.

When using personal computers for processing statistical data, the grouping of units of an object is carried out using standard procedures.
One such procedure is based on using the Sturgess formula to determine the optimal number of groups:

k = 1+3.322*lg(N)

Where k is the number of groups, N is the number of population units.

The length of the partial intervals is calculated as h=(x max -x min)/k

Then count the number of hits of observations in these intervals, which are taken as frequencies n i . Few frequencies, the values ​​of which are less than 5 (n i< 5), следует объединить. в этом случае надо объединить и соответствующие интервалы.
The midpoints of the intervals x i =(c i-1 +c i)/2 are taken as new values.

Example #3. As a result of a 5% self-random sample, the following distribution of products by moisture content was obtained. Calculate: 1) the average percentage of humidity; 2) indicators characterizing the variation in humidity.
The solution was obtained using a calculator: Example No. 1

Build a variation series. Based on the found series, construct a distribution polygon, a histogram, and a cumulate. Determine the mode and median.
Download Solution

Example. According to the results of selective observation (sample A appendix):
a) make a series of variations;
b) calculate the relative frequencies and accumulated relative frequencies;
c) build a polygon;
d) compose an empirical distribution function;
e) plot the empirical distribution function;
f) calculate numerical characteristics: arithmetic mean, variance, standard deviation. Solution

Based on the data given in Table 4 (Appendix 1) and corresponding to your option, perform:

1. Based on the structural grouping, construct a variational frequency and cumulative distribution series using equal closed intervals, assuming the number of groups is 6. Present the results in a table and graphically.
2. Analyze the variational distribution series by calculating:
   • arithmetic mean value of the feature;
   • mode, median, 1st quartile, 1st and 9th decile;
   • standard deviation;
   • the coefficient of variation.
3. Draw conclusions.

Required: to rank the series, build an interval distribution series, calculate the mean, mean variance, mode and median for the ranged and interval series.

Based on the initial data, construct a discrete variational series; present it in the form of a statistical table and statistical graphs. 2). Based on the initial data, construct an interval variation series with equal intervals. Choose the number of intervals yourself and explain this choice. Present the resulting variation series in the form of a statistical table and statistical graphs. Indicate the types of tables and graphs used.

In order to determine the average duration of customer service in a pension fund, the number of customers of which is very large, a survey of 100 customers was conducted according to the scheme of self-random non-repetitive sampling. The survey results are presented in the table. Find:
a) the boundaries within which, with a probability of 0.9946, the average service time for all clients of the pension fund is concluded;
b) the probability that the share of all fund clients with a service duration of less than 6 minutes differs from the share of such clients in the sample by no more than 10% (in absolute value);
c) resampling volume, at which with a probability of 0.9907 it can be argued that the share of all fund clients with a service duration of less than 6 minutes differs from the share of such clients in the sample by no more than 10% (in absolute value).
2. According to task 1, using Pearson's X 2 test, at the significance level α = 0.05, test the hypothesis that the random variable X - customer service time - is distributed according to the normal law. Construct on one drawing a histogram of the empirical distribution and the corresponding normal curve.
Download Solution

Given a sample of 100 items. Necessary:

1. Build a ranked variational series;
2. Find the maximum and minimum terms of the series;
3. Find the range of variation and the number of optimal intervals for constructing an interval series. Find the length of the interval of the interval series;
4. Build an interval series. Find the frequencies of the elements of the sample falling into the composed gaps. Find the midpoints of each interval;
5. Construct a histogram and a polygon of frequencies. Compare with normal distribution (analytically and graphically);
6. Plot the empirical distribution function;
7. Calculate sample numerical characteristics: sample mean and central sample moment;
8. Calculate approximate values ​​of standard deviation, skewness and kurtosis (using MS Excel analysis package). Compare approximate calculated values ​​with exact ones (calculated using MS Excel formulas);
9. Compare selected graphic characteristics with the corresponding theoretical ones.

Download Solution

We have the following sample data (10% sample, mechanical) on the output and the amount of profit, million rubles. According to the original data:
Task 13.1.
13.1.1. Build a statistical series of distribution of enterprises by the amount of profit, forming five groups at equal intervals. Plot distribution series plots.
13.1.2. Calculate the numerical characteristics of a series of distribution of enterprises by the amount of profit: arithmetic mean, standard deviation, variance, coefficient of variation V. Draw conclusions.
Task 13.2.
13.2.1. Determine the boundaries within which, with a probability of 0.997, the amount of profit of one enterprise in the general population is concluded.
13.2.2. Using Pearson's x2-criterion, at a significance level α, test the hypothesis that the random variable X - the amount of profit - is distributed according to the normal law.
Task 13.3.
13.3.1. Determine the coefficients of the sample regression equation.
13.3.2. Establish the presence and nature of the correlation between the cost of manufactured products (X) and the amount of profit per enterprise (Y). Plot a scatterplot and a regression line.
13.3.3. Calculate the linear correlation coefficient. Using Student's t-test, check the significance of the correlation coefficient. Draw a conclusion about the closeness of the relationship between factors X and Y using the Chaddock scale.
Guidelines. Task 13.3 is performed using this service.
Download Solution

Task. The following data represents the amount of time spent by clients in concluding contracts. Build an interval variation series of the presented data, a histogram, find an unbiased estimate of the mathematical expectation, a biased and unbiased estimate of the variance.

Example. According to table 2:
1) Build distribution series for 40 commercial banks of the Russian Federation:
A) by the amount of profit;
B) by the amount of credit investments.
2) According to the obtained distribution series, determine:
A) average profit per commercial bank;
B) credit investments on average per commercial bank;
C) modal and median value of profit; quartiles, deciles;
D) modal and median value of credit investments.
3) According to the distribution series obtained in paragraph 1, calculate:
a) range of variation;
b) average linear deviation;
c) standard deviation;
d) coefficient of variation.
Record the necessary calculations in tabular form. Analyze the results. Draw your own conclusions.
Plot the resulting distribution series. Determine the mode and median graphically.

Solution:
To build a grouping with equal intervals, we will use the service Grouping of statistical data.

Figure 1 - Entering parameters

Description of parameters
Number of lines: amount of raw data. If the dimension of the series is small, indicate its number. If the selection is large enough, then click the Paste from Excel button.
Number of groups: 0 - the number of groups will be determined by the Sturgess formula.
If a specific number of groups is specified, specify it (for example, 5).
Row type: Discrete series.
Significance level: for example, 0.954 . This parameter is set to define the confidence interval for the mean.
Sample: For example, 10% mechanical sampling is done. Specify the number 10. For our data, we specify 100 .

As a result of mastering this chapter, the student must: know

• indicators of variation and their relationship;
• basic laws of distribution of features;
• the essence of the consent criteria; be able to
• calculate rates of variation and goodness of fit;
• determine the characteristics of distributions;
• evaluate the main numerical characteristics of statistical distribution series;

own

• methods of statistical analysis of distribution series;
• basics of dispersion analysis;
• methods for checking statistical distribution series for compliance with the basic laws of distribution.

Variation indicators

In the statistical study of the features of various statistical populations, it is of great interest to study the variation of the feature of individual statistical units of the population, as well as the nature of the distribution of units according to this feature. Variation - these are the differences in the individual values ​​of the trait among the units of the studied population. The study of variation is of great practical importance. By the degree of variation, one can judge the boundaries of the variation of the trait, the homogeneity of the population for this trait, the typicality of the average, the relationship of factors that determine the variation. Variation indicators are used to characterize and organize statistical populations.

The results of the summary and grouping of statistical observation materials, drawn up in the form of statistical distribution series, represent an ordered distribution of units of the studied population into groups according to a grouping (variable) attribute. If a qualitative trait is taken as the basis for grouping, then such a distribution series is called attributive(distribution by profession, gender, color, etc.). If the distribution series is built on a quantitative basis, then such a series is called variational(distribution by height, weight, wages, etc.). To construct a variational series means to order the quantitative distribution of population units according to the values ​​of the attribute, to count the number of population units with these values ​​(frequency), to arrange the results in a table.

Instead of the frequency of a variant, it is possible to use its ratio to the total volume of observations, which is called the frequency (relative frequency).

There are two types of variation series: discrete and interval. Discrete series- this is such a variational series, the construction of which is based on signs with a discontinuous change (discrete signs). The latter include the number of employees in the enterprise, the wage category, the number of children in the family, etc. A discrete variational series is a table that consists of two columns. The first column indicates the specific value of the attribute, and the second - the number of population units with a specific value of the attribute. If a sign has a continuous change (the amount of income, length of service, the cost of fixed assets of an enterprise, etc., which within certain limits can take on any values), then for this sign it is possible to construct interval variation series. The table when constructing an interval variation series also has two columns. The first indicates the value of the feature in the interval "from - to" (options), the second - the number of units included in the interval (frequency). Frequency (repetition frequency) - the number of repetitions of a particular variant of the attribute values. Intervals can be closed and open. Closed intervals are limited on both sides, i.e. have a border both lower (“from”) and upper (“to”). Open intervals have any one border: either upper or lower. If the options are arranged in ascending or descending order, then the rows are called ranked.

For variational series, there are two types of frequency response options: cumulative frequency and cumulative frequency. The cumulative frequency shows how many observations the value of the feature took on values ​​less than the specified value. The cumulative frequency is determined by summing the values ​​of the characteristic frequency for a given group with all the frequencies of the previous groups. The accumulated frequency characterizes the proportion of units of observation in which the values ​​of the feature do not exceed the upper limit of the day group. Thus, the accumulated frequency shows the specific weight of the variant in the aggregate, which have a value not greater than the given one. Frequency, frequency, absolute and relative densities, cumulative frequency and frequency are characteristics of the magnitude of the variant.

Variations in the sign of statistical units of the population, as well as the nature of the distribution, are studied using indicators and characteristics of the variation series, which include the average level of the series, the average linear deviation, the standard deviation, dispersion, oscillation coefficients, variation, asymmetry, kurtosis, etc.

Average values ​​are used to characterize the distribution center. The average is a generalizing statistical characteristic, in which the typical level of a trait possessed by members of the studied population is quantified. However, there may be cases where the arithmetic means coincide with a different nature of the distribution, therefore, as statistical characteristics of the variation series, the so-called structural averages are calculated - mode, median, as well as quantiles that divide the distribution series into equal parts (quartiles, deciles, percentiles, etc.).

Fashion - this is the value of the feature that occurs more frequently in the distribution series than its other values. For discrete series, this is the variant with the highest frequency. In interval variational series, in order to determine the mode, it is necessary first of all to determine the interval in which it is located, the so-called modal interval. In a variational series with equal intervals, the modal interval is determined by the highest frequency, in series with unequal intervals - but by the highest distribution density. Then, to determine the mode in rows with equal intervals, apply the formula

where Mo is the value of fashion; x Mo - the lower limit of the modal interval; h- modal interval width; / Mo - modal interval frequency; / Mo j - frequency of the pre-modal interval; / Mo+1 is the frequency of the post-modal interval, and for a series with unequal intervals in this calculation formula, instead of the frequencies / Mo, / Mo, / Mo, distribution densities should be used Mind 0 _| , Mind 0> UMO+"

If there is a single mode, then the probability distribution of the random variable is called unimodal; if there is more than one mode, it is called multimodal (polymodal, multimodal), in the case of two modes - bimodal. As a rule, multimodality indicates that the distribution under study does not follow the normal distribution law. Homogeneous populations, as a rule, are characterized by unimodal distributions. Multivertex also indicates the heterogeneity of the studied population. The appearance of two or more vertices makes it necessary to regroup the data in order to isolate more homogeneous groups.

In an interval variation series, the mode can be determined graphically using a histogram. To do this, two intersecting lines are drawn from the top points of the highest column of the histogram to the top points of two adjacent columns. Then, from the point of their intersection, a perpendicular is lowered to the abscissa axis. The feature value on the abscissa corresponding to the perpendicular is the mode. In many cases, when characterizing the population as a generalized indicator, preference is given to the mode, rather than the arithmetic mean.

Median - this is the central value of the feature; it is possessed by the central member of the ranked distribution series. In discrete series, to find the value of the median, its serial number is first determined. To do this, with an odd number of units, one is added to the sum of all frequencies, the number is divided by two. If there are an even number of 1s, there will be 2 median 1s in the series, so in this case the median is defined as the average of the values ​​of the 2 median 1s. Thus, the median in a discrete variation series is the value that divides the series into two parts containing the same number of options.

In the interval series, after determining the ordinal number of the median, the median interval is found by the accumulated frequencies (frequencies), and then, using the formula for calculating the median, the value of the median itself is determined:

where Me is the value of the median; x Me - the lower limit of the median interval; h- median interval width; - the sum of the frequencies of the distribution series; /D - the accumulated frequency of the pre-median interval; / Me - the frequency of the median interval.

The median can be found graphically using the cumulate. To do this, on the scale of accumulated frequencies (frequencies) of the cumulate, from the point corresponding to the ordinal number of the median, a straight line is drawn parallel to the abscissa axis until it intersects with the cumulate. Further, from the point of intersection of the indicated straight line with the cumulate, a perpendicular is lowered to the abscissa axis. The value of the feature on the x-axis corresponding to the drawn ordinate (perpendicular) is the median.

The median is characterized by the following properties.

• 1. It does not depend on those attribute values ​​that are located on both sides of it.
• 2. It has the property of minimality, which means that the sum of the absolute deviations of the attribute values ​​from the median is the minimum value compared to the deviation of the attribute values ​​from any other value.
• 3. When combining two distributions with known medians, it is impossible to predict the median value of the new distribution in advance.

These properties of the median are widely used in designing the location of public service points - schools, clinics, gas stations, water pumps, etc. For example, if it is planned to build a polyclinic in a certain quarter of the city, then it is more expedient to locate it at a point in the quarter that bisects not the length of the quarter, but the number of inhabitants.

The ratio of the mode, median and arithmetic mean indicates the nature of the distribution of the trait in the aggregate, allows you to evaluate the symmetry of the distribution. If x Me then there is a right-hand asymmetry of the series. With a normal distribution X - Me - Mo.

K. Pearson, based on the alignment of various types of curves, determined that for moderately asymmetric distributions, the following approximate relationships between the arithmetic mean, median and mode are valid:

where Me is the value of the median; Mo - fashion value; x arithm - the value of the arithmetic mean.

If there is a need to study the structure of the variation series in more detail, then the characteristic values ​​are calculated, similar to the median. Such feature values ​​divide all distribution units into equal numbers, they are called quantiles or gradients. Quantiles are subdivided into quartiles, deciles, percentiles, etc.

Quartiles divide the population into four equal parts. The first quartile is calculated similarly to the median using the formula for calculating the first quartile, having previously determined the first quarterly interval:

where Qi is the value of the first quartile; xQ^- the lower limit of the first quartile interval; h- width of the first quarterly interval; /, - frequencies of the interval series;

Accumulated frequency in the interval preceding the first quartile interval; Jq ( - frequency of the first quartile interval.

The first quartile shows that 25% of the population units are less than its value, and 75% are more. The second quartile is equal to the median, i.e. Q2 = me.

By analogy, the third quartile is calculated, having previously found the third quarterly interval:

where is the lower limit of the third quartile interval; h- width of the third quartile interval; /, - frequencies of the interval series; /X"- accumulated frequency in the interval preceding

G

third quartile interval; Jq - frequency of the third quartile interval.

The third quartile shows that 75% of the population units are less than its value, and 25% are more.

The difference between the third and first quartiles is the interquartile interval:

where Aq is the value of the interquartile interval; Q 3 - the value of the third quartile; Q, - the value of the first quartile.

Deciles divide the population into 10 equal parts. A decile is a value of a feature in a distribution series that corresponds to tenths of the population. By analogy with quartiles, the first decile shows that 10% of the population units are less than its value, and 90% are more, and the ninth decile reveals that 90% of the population units are less than its value, and 10% are more. The ratio of the ninth and first deciles, i.e. decile coefficient, widely used in the study of income differentiation to measure the ratio of income levels of 10% of the most wealthy and 10% of the least wealthy population. Percentiles divide the ranked population into 100 equal parts. The calculation, meaning and use of percentiles are similar to deciles.

Quartiles, deciles and other structural characteristics can be determined graphically by analogy with the median using the cumulate.

To measure the size of the variation, the following indicators are used: the range of variation, the average linear deviation, the standard deviation, and the variance. The magnitude of the range of variation depends entirely on the randomness of the distribution of the extreme members of the series. This indicator is of interest in cases where it is important to know what is the amplitude of fluctuations in the values ​​of the attribute:

Where R- the value of the range of variation; x max - the maximum value of the feature; x tt - the minimum value of the attribute.

When calculating the range of variation, the value of the vast majority of the series members is not taken into account, while the variation is associated with each value of the series member. This shortcoming is devoid of indicators that are averages obtained from the deviations of individual trait values ​​from their average value: the average linear deviation and the standard deviation. There is a direct relationship between individual deviations from the average and the fluctuation of a particular trait. The stronger the volatility, the greater the absolute size of the deviations from the average.

The average linear deviation is the arithmetic average of the absolute values ​​of the deviations of individual options from their average value.

Mean Linear Deviation for Ungrouped Data

where / pr - the value of the average linear deviation; x, - - the value of the feature; X - P - number of population units.

Grouped Series Average Linear Deviation

where / vz - the value of the average linear deviation; x, - the value of the feature; X - the average value of the trait for the studied population; / - the number of population units in a separate group.

Deviation signs are ignored in this case, otherwise the sum of all deviations will be equal to zero. The average linear deviation depending on the grouping of the analyzed data is calculated using different formulas: for grouped and non-grouped data. The average linear deviation, due to its conditionality, separately from other indicators of variation, is used relatively rarely in practice (in particular, to characterize the fulfillment of contractual obligations in terms of the uniformity of supply; in the analysis of foreign trade turnover, the composition of employees, the rhythm of production, product quality, taking into account technological features of production, etc.).

The standard deviation characterizes how much the individual values ​​of the studied trait deviate on average from the average value for the population, and is expressed in units of the studied trait. The standard deviation, being one of the main measures of variation, is widely used in assessing the boundaries of the variation of a trait in a homogeneous population, in determining the values ​​of the ordinates of the normal distribution curve, as well as in calculations related to the organization of sample observation and establishing the accuracy of sample characteristics. The standard deviation for ungrouped data is calculated according to the following algorithm: each deviation from the average is squared, all squares are summed, after which the sum of squares is divided by the number of terms in the series and the square root is taken from the quotient:

where a Iip - the value of the standard deviation; Xj- feature value; X- the average value of the attribute for the studied population; P - number of population units.

For grouped analyzed data, the standard deviation of the data is calculated using the weighted formula

Where - the value of the standard deviation; Xj- feature value; X - the average value of the trait for the studied population; fx- the number of population units in a particular group.

The expression under the root in both cases is called the variance. Thus, the variance is calculated as the average square of the deviations of the trait values ​​from their average value. For unweighted (simple) feature values, the variance is defined as follows:

For weighted characteristic values

There is also a special simplified way to calculate the variance: in general terms

for unweighted (simple) feature values for weighted characteristic values
using the method of counting from conditional zero

where a 2 - the value of the dispersion; x, - - the value of the feature; X - the average value of the feature, h- group interval value, t 1 - weight (A =

Dispersion has an independent expression in statistics and is one of the most important indicators of variation. It is measured in units corresponding to the square of the units of measurement of the trait under study.

The dispersion has the following properties.

• 1. The dispersion of a constant value is zero.
• 2. Reducing all values ​​of the feature by the same value of A does not change the value of the variance. This means that the mean square of deviations can be calculated not from the given values ​​of the attribute, but from their deviations from some constant number.
• 3. Decreasing all values ​​of the feature in k times reduces the dispersion in k 2 times, and the standard deviation - in k times, i.e. all attribute values ​​can be divided by some constant number (say, by the value of the series interval), the standard deviation can be calculated, and then multiplied by a constant number.
• 4. If we calculate the average square of deviations from any value And at differs to some extent from the arithmetic mean, then it will always be greater than the mean square of the deviations calculated from the arithmetic mean. In this case, the mean square of deviations will be larger by a well-defined value - by the square of the difference between the average and this conditionally taken value.

The variation of an alternative feature is the presence or absence of the studied property in the units of the population. Quantitatively, the variation of an alternative attribute is expressed by two values: the presence of the studied property in a unit is denoted by one (1), and its absence is denoted by zero (0). The proportion of units that have the property under study is denoted by P, and the proportion of units that do not have this property is denoted by G. Thus, the variance of an alternative attribute is equal to the product of the proportion of units that have a given property (P) by the proportion of units that do not have this property (G). The greatest variation of the population is achieved in cases where a part of the population, which is 50% of the total volume of the population, has a feature, and the other part of the population, also equal to 50%, does not have this feature, while the variance reaches a maximum value of 0.25, i.e. P = 0.5, G= 1 - P \u003d 1 - 0.5 \u003d 0.5 and o 2 \u003d 0.5 0.5 \u003d 0.25. The lower limit of this indicator is equal to zero, which corresponds to a situation in which there is no variation in the aggregate. The practical application of the variance of an alternative feature is to build confidence intervals when conducting a sample observation.

The smaller the variance and standard deviation, the more homogeneous the population and the more typical the average will be. In the practice of statistics, it often becomes necessary to compare variations of various features. For example, it is interesting to compare variations in the age of workers and their qualifications, length of service and wages, cost and profit, length of service and labor productivity, etc. For such comparisons, indicators of the absolute variability of characteristics are unsuitable: it is impossible to compare the variability of work experience, expressed in years, with the variation of wages, expressed in rubles. To carry out such comparisons, as well as comparisons of the fluctuation of the same attribute in several populations with different arithmetic means, variation indicators are used - the oscillation coefficient, the linear coefficient of variation and the coefficient of variation, which show the measure of fluctuations of the extreme values ​​around the average.

Oscillation factor:

Where V R - the value of the oscillation coefficient; R- the value of the range of variation; X -

Linear coefficient of variation".

Where vj- the value of the linear coefficient of variation; I- the value of the average linear deviation; X - the average value of the trait for the population under study.

The coefficient of variation:

Where Va- the value of the coefficient of variation; a - the value of the standard deviation; X - the average value of the trait for the population under study.

The oscillation coefficient is the percentage of the range of variation to the mean value of the trait under study, and the linear coefficient of variation is the ratio of the mean linear deviation to the mean value of the trait under study, expressed as a percentage. The coefficient of variation is the percentage of the standard deviation to the average value of the trait under study. As a relative value, expressed as a percentage, the coefficient of variation is used to compare the degree of variation of various traits. Using the coefficient of variation, the homogeneity of the statistical population is estimated. If the coefficient of variation is less than 33%, then the population under study is homogeneous, and the variation is weak. If the coefficient of variation is greater than 33%, then the population under study is heterogeneous, the variation is strong, and the average value is atypical and cannot be used as a generalizing indicator of this population. In addition, the coefficients of variation are used to compare the fluctuation of one trait in different populations. For example, to assess the variation in the length of service of workers at two enterprises. The higher the value of the coefficient, the more significant the variation of the feature.

Based on the calculated quartiles, it is also possible to calculate the relative indicator of quarterly variation using the formula

where Q 2 And

The interquartile range is determined by the formula

The quartile deviation is used instead of the range of variation to avoid the disadvantages associated with using extreme values:

For unequal interval variational series, the distribution density is also calculated. It is defined as the quotient of the corresponding frequency or frequency divided by the interval value. In unequal interval series, absolute and relative distribution densities are used. The absolute distribution density is the frequency per unit length of the interval. Relative distribution density - the frequency per unit length of the interval.

All of the above is true for distribution series whose distribution law is well described by the normal distribution law or is close to it.

variational called distribution series built on a quantitative basis. The values ​​of quantitative characteristics in individual units of the population are not constant, more or less differ from each other.

Variation- fluctuation, variability of the value of the attribute in units of the population. Separate numerical values ​​of the trait occurring in the studied population are called options values. The insufficiency of the average value for a complete characterization of the population makes it necessary to supplement the average values ​​with indicators that make it possible to assess the typicality of these averages by measuring the fluctuation (variation) of the trait under study.

The presence of variation is due to the influence of a large number of factors on the formation of the trait level. These factors act with unequal force and in different directions. Variation indicators are used to describe the measure of trait variability.

Tasks of the statistical study of variation:

• 1) the study of the nature and degree of variation of signs in individual units of the population;
• 2) determination of the role of individual factors or their groups in the variation of certain features of the population.

In statistics, special methods for studying variation are used, based on the use of a system of indicators, With by which variation is measured.

The study of variation is essential. The measurement of variations is necessary when conducting sample observation, correlation and variance analysis, etc. Ermolaev O.Yu. Mathematical statistics for psychologists: Textbook [Text] / O.Yu. Ermolaev. - M.: Flint Publishing House of the Moscow Psychological and Social Institute, 2012. - 335p.

According to the degree of variation, one can judge the homogeneity of the population, the stability of individual values ​​of features and the typicality of the average. On their basis, indicators of the closeness of the relationship between the signs, indicators for assessing the accuracy of selective observation are developed.

There is variation in space and variation in time.

Variation in space is understood as the fluctuation of the values ​​of a feature in units of the population representing separate territories. Under the variation in time is meant the change in the values ​​of the attribute in different periods of time.

To study the variation in the distribution series, all variants of the attribute values ​​are arranged in ascending or descending order. This process is called series ranking.

The simplest signs of variation are minimum and maximum- the smallest and largest value of the attribute in the aggregate. The number of repetitions of individual variants of feature values ​​is called the frequency of repetition (fi). It is convenient to replace frequencies with frequencies - wi. Frequency - a relative indicator of frequency, which can be expressed in fractions of a unit or a percentage and allows you to compare variation series with a different number of observations. Expressed by the formula:

where Xmax, Xmin - the maximum and minimum values ​​of the attribute in the aggregate; n is the number of groups.

To measure the variation of a trait, various absolute and relative indicators are used. The absolute indicators of variation include the range of variation, the average linear deviation, variance, standard deviation. The relative indicators of fluctuation include the coefficient of oscillation, the relative linear deviation, the coefficient of variation.

An example of finding a variation series

Exercise. For this sample:

• a) Find a variation series;
• b) Construct the distribution function;

No.=42. Sample items:

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

Solution.

• a) construction of a ranked variational series:
  • 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 5 5 5 5 5 6 6 6 7 7 7 7 7 7 7 8 8 8 8 8 8 9 9
• b) construction of a discrete variational series.

Let's calculate the number of groups in the variation series using the Sturgess formula:

Let's take the number of groups equal to 7.

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

For the convenience of constructing the table, we will take the number of groups equal to 8, the interval will be 1.

Rice. 1 The volume of sales of goods by the store for a certain period of time