As a result of mastering this chapter, the student must: know
- indicators of variation and their relationship;
- basic laws of distribution of characteristics;
- the essence of the consent criteria; be able to
- calculate indices of variation and goodness-of-fit criteria;
- determine distribution characteristics;
- evaluate the main numerical characteristics of statistical distribution series;
own
- methods of statistical analysis of distribution series;
- basics of analysis of variance;
- techniques for checking statistical distribution series for compliance with the basic laws of distribution.
Variation indicators
In the statistical study of characteristics of various statistical populations, it is of great interest to study the variation of the characteristic of individual statistical units of the population, as well as the nature of the distribution of units according to this characteristic. Variation - these are differences in individual values of a characteristic among units of the population being studied. The study of variation is of great practical importance. By the degree of variation, one can judge the limits of variation of a characteristic, the homogeneity of the population for a given characteristic, the typicality of the average, and 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, presented in the form of statistical distribution series, represent an ordered distribution of units of the population being studied into groups according to grouping (variing) criteria. If a qualitative characteristic is taken as the basis for the grouping, then such a distribution series is called attributive(distribution by profession, gender, color, etc.). If a distribution series is built on a quantitative basis, then such a series is called variational(distribution by height, weight, salary, etc.). To construct a variation series means to organize the quantitative distribution of population units by characteristic values, count the number of population units with these values (frequency), and 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 frequency (relative frequency).
There are two types of variation series: discrete and interval. Discrete series- This is a variation series, the construction of which is based on characteristics with discontinuous changes (discrete characteristics). The latter include the number of employees at the enterprise, tariff category, number of children in the family, etc. A discrete variation series represents a table that consists of two columns. The first column indicates the specific value of the attribute, and the second column indicates the number of units in the population with a specific value of the attribute. If a characteristic has a continuous change (amount of income, length of service, cost of fixed assets of the enterprise, etc., which within certain limits can take on any values), then for this characteristic it is possible to construct interval variation series. When constructing an interval variation series, the table also has two columns. The first indicates the value of the attribute in the interval “from - to” (options), the second indicates the number of units included in the interval (frequency). Frequency (repetition frequency) - the number of repetitions of a particular variant of attribute values. Intervals can be closed or open. Closed intervals are limited on both sides, i.e. have both a lower (“from”) and an upper (“to”) boundary. Open intervals have one boundary: either upper or lower. If the options are arranged in ascending or descending order, then the rows are called ranked.
For variation series, there are two types of frequency response options: accumulated frequency and accumulated frequency. The accumulated frequency shows how many observations the value of the characteristic took values less than a given one. The accumulated frequency is determined by summing the frequency values of a characteristic for a given group with all frequencies of previous groups. The accumulated frequency characterizes the proportion of observation units whose attribute values do not exceed the upper limit of the given group. Thus, the accumulated frequency shows the proportion of options in the totality that have a value no greater than the given one. Frequency, frequency, absolute and relative densities, accumulated frequency and frequency are characteristics of the magnitude of the variant.
Variations in the characteristics 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, coefficients of oscillation, 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 characteristic possessed by members of the population being studied is quantified. However, cases of coincidence of arithmetic means with different distribution patterns are possible, therefore, as statistical characteristics of variation series, the so-called structural means are calculated - mode, median, as well as quantiles, which divide the distribution series into equal parts (quartiles, deciles, percentiles, etc. ).
Fashion - This is the value of a characteristic that occurs in the distribution series more often than its other values. For discrete series, this is the option with the highest frequency. In interval variation series, in order to determine the mode, it is necessary to first determine the interval in which it is located, the so-called modal interval. In a variation series with equal intervals, the modal interval is determined by the highest frequency, in series with unequal intervals - but by the highest distribution density. The formula is then used to determine the mode in rows at equal intervals
where Mo is the fashion value; xMo - lower limit of the modal interval; h- modal interval width; / Mo - frequency of the modal interval; / Mo j is the frequency of the premodal 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 obey the normal distribution law. Homogeneous populations, as a rule, are characterized by single-vertex distributions. Multivertex also indicates the heterogeneity of the population being studied. The appearance of two or more vertices makes it necessary to regroup the data in order to identify more homogeneous groups.
In an interval variation series, the mode can be determined graphically using a histogram. To do this, draw two intersecting lines 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 onto the abscissa axis. The value of the feature on the x-axis corresponding to the perpendicular is the mode. In many cases, when characterizing a population, preference is given to the mode rather than the arithmetic mean as a generalized indicator.
Median - This is the central value of the attribute; it is possessed by the central member of the ranked series of the distribution. In discrete series, to find the value of the median, its serial number is first determined. To do this, if the number of units is odd, one is added to the sum of all frequencies, and the number is divided by two. If there are an even number of units in a row, there will be two median units, so in this case the median is defined as the average of the values of the two median units. 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 interval series, after determining the serial number of the median, the medial interval is found using 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 median value; x Me - lower limit of the median interval; h- width of the median interval; - the sum of the frequencies of the distribution series; /D - accumulated frequency of the pre-median interval; / Me - frequency of the median interval.
The median can be found graphically using a 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. Next, from the point of intersection of the indicated line with the cumulate, a perpendicular is lowered to the abscissa axis. The value of the attribute 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 either side of it.
- 2. It has the property of minimality, which means that the sum of absolute deviations of the attribute values from the median represents a 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 in advance the value of the median of the new distribution.
These properties of the median are widely used when designing the location of public service points - schools, clinics, gas stations, water pumps, etc. For example, if it is planned to build a clinic in a certain block of the city, then it would be more expedient to locate it at a point in the block that halves not the length of the block, but the number of residents.
The ratio of the mode, median and arithmetic mean indicates the nature of the distribution of the characteristic in the aggregate and allows us to assess the symmetry of the distribution. If x Me then there is a right-sided asymmetry of the series. With 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 median value; Mo - meaning of fashion; 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 calculate characteristic values similar to the median. Such characteristic values divide all distribution units into equal numbers; they are called quantiles or gradients. Quantiles are divided 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^- lower limit of the first quartile range; h- width of the first quarter interval; /, - frequencies of the interval series;
Cumulative 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. Q 2 = Me.
By analogy, the third quartile is calculated, having first found the third quarterly interval:
where is the lower limit of the third quartile range; h- width of the third quartile interval; /, - frequencies of the interval series; /X" - accumulated frequency in the interval preceding
G
third quartile interval; Jq is the 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 range:
where Aq is the value of the interquartile range; Q 3 - third quartile value; Q, is the value of the first quartile.
Deciles divide the population into 10 equal parts. A decile is a value of a characteristic in a distribution series that corresponds to tenths of the population size. By analogy with quartiles, the first decile shows that 10% of the population units are less than its value, and 90% are greater, and the ninth decile reveals that 90% of the population units are less than its value, and 10% are greater. The ratio of the ninth and first deciles, i.e. The decile coefficient is widely used in the study of income differentiation to measure the ratio of the income levels of the 10% most affluent and 10% of the least affluent population. Percentiles divide the ranked population into 100 equal parts. The calculation, meaning, and application of percentiles are similar to deciles.
Quartiles, deciles and other structural characteristics can be determined graphically by analogy with the median using cumulates.
To measure the size of variation, the following indicators are used: range of variation, average linear deviation, standard deviation, dispersion. The magnitude of the variation range 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 the amplitude of fluctuations in the values of a characteristic is:
Where R- the value of the range of variation; x max - maximum value of the attribute; x tt - minimum value of the attribute.
When calculating the range of variation, the value of the vast majority of series members is not taken into account, while the variation is associated with each value of the series member. Indicators that are averages obtained from deviations of individual values of a characteristic from their average value do not have this drawback: the average linear deviation and the standard deviation. There is a direct relationship between individual deviations from the average and the variability of a particular trait. The stronger the fluctuation, the greater the absolute size of the deviations from the average.
The average linear deviation is the arithmetic mean of the absolute values of deviations of individual options from their average value.
Average Linear Deviation for Ungrouped Data
where /pr is the value of the average linear deviation; x, - is the value of the attribute; X - p - number of units in the population.
Average linear deviation of the grouped series
where / vz - the value of the average linear deviation; x, is the value of the attribute; X - the average value of the characteristic for the population being studied; / - the number of population units in a separate group.
In this case, the signs of deviations are ignored, 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 various formulas: for grouped and ungrouped data. Due to its convention, the average linear deviation, separately from other indicators of variation, is used in practice relatively rarely (in particular, to characterize the fulfillment of contractual obligations regarding the uniformity of delivery; in the analysis of foreign trade turnover, the composition of employees, the rhythm of production, product quality, taking into account the technological features of production and etc.).
The standard deviation characterizes how much on average the individual values of the characteristic being studied deviate from the average value of the population, and is expressed in units of measurement of the characteristic being studied. The standard deviation, being one of the main measures of variation, is widely used in assessing the limits of variation of a characteristic in a homogeneous population, in determining the ordinate values of a 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 of ungrouped data is calculated using the following algorithm: each deviation from the mean is squared, all squares are summed, after which the sum of squares is divided by the number of terms of the series and the square root is extracted from the quotient:
where a Iip is the value of the standard deviation; Xj- attribute value; X- the average value of the characteristic for the population being studied; p - number of units in the population.
For grouped analyzed data, the standard deviation of the data is calculated using the weighted formula 
Where - standard deviation value; Xj- attribute value; X - the average value of the characteristic for the population being studied; f x - the number of population units in a particular group.
The expression under the root in both cases is called variance. Thus, dispersion is calculated as the average square of deviations of attribute values from their average value. For unweighted (simple) attribute values, the variance is determined as follows:
For weighted characteristic values
There is also a special simplified method for calculating variance: in general 
for unweighted (simple) characteristic values 
for weighted characteristic values 
using the zero-based method
where a 2 is the dispersion value; x, - is the value of the attribute; X - average value of the characteristic, h- group interval value, t 1 - weight (A =
Dispersion has its own 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 characteristic being studied.
The dispersion has the following properties.
- 1. The variance of a constant value is zero.
- 2. Reducing all values of a characteristic by the same value A does not change the value of the dispersion. This means that the average square of deviations can be calculated not from given values of a characteristic, but from their deviations from some constant number.
- 3. Reducing any characteristic values in k times reduces the dispersion by k 2 times, and the standard deviation is in k times, i.e. all values of the attribute 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 differing to one degree or another from the arithmetic mean, then it will always be greater than the average square of the deviations calculated from the arithmetic mean. The average square of the deviations will be greater by a very certain amount - by the square of the difference between the average and this conventionally taken value.
Variation of an alternative characteristic consists in the presence or absence of the studied property in units of the population. Quantitatively, the variation of an alternative attribute is expressed by two values: the presence of a unit of the studied property is denoted by one (1), and its absence is denoted by zero (0). The proportion of units that have the property being studied 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 possessing this property (P) by the proportion of units not possessing this property (G). The greatest variation of the population is achieved in cases where part of the population, constituting 50% of the total volume of the population, has a characteristic, and another part of the population, also equal to 50%, does not have this characteristic, and the dispersion reaches a maximum value of 0.25, t .e. P = 0.5, G= 1 - P = 1 - 0.5 = 0.5 and o 2 = 0.5 0.5 = 0.25. The lower limit of this indicator is zero, which corresponds to a situation in which there is no variation in the aggregate. The practical application of the variance of an alternative characteristic is to construct confidence intervals when conducting sample observations.
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, there is often a need to compare variations of various characteristics. 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 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 variability of the same characteristic in several populations with different arithmetic means, variation indicators are used - the coefficient of oscillation, the linear coefficient of variation and the coefficient of variation, which show the measure of fluctuations of extreme values around the average.
Oscillation coefficient:
Where V R - oscillation coefficient value; R- 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 characteristic for the population being studied.
Coefficient of variation:
Where V a - coefficient of variation value; a is the value of the standard deviation; X - the average value of the characteristic for the population being studied.
The coefficient of oscillation is the percentage ratio of the range of variation to the average value of the characteristic being studied, and the linear coefficient of variation is the ratio of the average linear deviation to the average value of the characteristic being studied, expressed as a percentage. The coefficient of variation is the percentage of the standard deviation to the average value of the characteristic being studied. As a relative value, expressed as a percentage, the coefficient of variation is used to compare the degree of variation of various characteristics. Using the coefficient of variation, the homogeneity of a statistical population is assessed. 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 more 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 general indicator of this population. In addition, coefficients of variation are used to compare the variability 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 coefficient value, the more significant the variation of the characteristic.
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 unequally interval variation series, the distribution density is also calculated. It is defined as the quotient of the corresponding frequency or frequency divided by the value of the interval. 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 is 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.
A special place in statistical analysis belongs to the determination of the average level of the characteristic or phenomenon being studied. The average level of a trait is measured by average values.
The average value characterizes the general quantitative level of the characteristic being studied and is a group property of the statistical population. It levels out, weakens random deviations of individual observations in one direction or another and highlights the main, typical property of the characteristic being studied.
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 (vital movement of the population, 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, plan and determine the population’s needs for various types of medical care (average number of requests or visits per resident per year, average length of stay of a patient in a hospital, average duration of examination patient, average availability of doctors, beds, etc.).
3. To characterize the sanitary and epidemiological state (average air dust content in the workshop, average area per person, average consumption of proteins, fats and carbohydrates, etc.).
4. To determine medical and physiological indicators in normal and pathological conditions, when processing laboratory data, to establish the reliability of the results of a sample study in social, hygienic, clinical, and experimental studies.
The calculation of average values is performed on the basis of variation series. Variation series is a qualitatively homogeneous statistical population, the individual units of which characterize the quantitative differences of the characteristic or phenomenon being studied.
Quantitative variation can be of two types: discontinuous (discrete) and continuous.
A discontinuous (discrete) attribute 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 it can be limited to kilograms, and for newborns - grams; height, blood pressure, time spent seeing a patient, and etc.).
The digital value of each individual characteristic or phenomenon included in the variation series is called a variant and is designated by the letter V . Other notations are also found in the mathematical literature, for example x or y.
A variation 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.
As the number of observations increases, repeating variant values tend to occur. In this case it is created 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 compiled with ranking.
Interval variation series compiled in order to simplify subsequent calculations performed without the use of a computer, with a very large number of observation units (more than 1000).
Continuous variation series includes option values, which can be any value.
If in a variation series the values of a characteristic (variants) are given in the form of individual specific numbers, then such a series is called discrete.
The general characteristics of the values of the characteristic reflected in the variation series are the average values. Among them, the most used are: arithmetic mean M, fashion Mo and median Me. Each of these characteristics is unique. They cannot replace each other and only together they represent the features of the variation series quite fully and in a condensed form.
Fashion (Mo) name the value of the most frequently occurring options.
Median (Me) – this is the value of the option dividing the ranked variation series in half (on each side of the median there is half of the option). 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 option values is arithmetic mean value( M ). In mathematical literature it is denoted .
Arithmetic mean (M, ) is a general quantitative characteristic of a certain characteristic of the phenomena being studied, constituting a qualitatively homogeneous statistical population. There are simple and weighted arithmetic averages. The simple arithmetic mean is calculated for a simple variation series by summing all the options and dividing this sum by the total number of options included in this variation 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, the weighted arithmetic mean is determined. The formula for calculating it:
,
Where: M - arithmetic weighted average;
Σ Vp - the sum of the products of the variant by 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:
· sum of deviations from the average ( 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 (divisor);
· 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. Averages that are close in value can be obtained from series with varying degrees of scattering. The closer the individual options are to each other in terms of their quantitative characteristics, the less dispersion (oscillation, variability) series, the more typical its average.
The main parameters that allow us to assess the variability of a trait are:
· Scope;
· Amplitude;
· Standard deviation;
· Coefficient of variation.
The variability of a trait can be approximately judged by the range and amplitude of the variation series. The range indicates the maximum (V max) and minimum (V min) options in the series. Amplitude (A m) is the difference between these options: A m = V max - V min.
The main, generally accepted measure of the variability of a variation series is dispersion (D ). But the most often used is a more convenient parameter calculated on the basis of dispersion - the standard deviation ( σ ). It takes into account the magnitude of the deviation ( d ) of each variation series from its arithmetic mean ( d=V - M ).
Since deviations from the average 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 dispersion of a variation series is the mean square of deviations of a 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 criteria.
Since dispersion is expressed as the square of deviations, its value cannot be used in comparison with the arithmetic mean. For these purposes it is used standard deviation, which is designated by the sign “Sigma” ( σ ). It characterizes the average deviation of all variants of a 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:
The specified formula is applied when the number of observations ( n ) more than 30. With a smaller number n the standard deviation value will have an error associated with the mathematical offset ( 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 a random variable X relative to its mathematical expectation based on an unbiased estimate of its variance.
With values n > 30 standard deviation ( σ ) and standard deviation ( s ) will be the same ( σ =s ). Therefore, in most practical manuals these criteria are considered to have different meanings. In Excel, standard deviation can be calculated using the function =STDEV(range). And in order to calculate the standard deviation, you need to create an appropriate formula.
The mean square or standard deviation allows you to determine how much the values of a characteristic may differ from the average 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 smaller than in cities located inland. Therefore, the standard deviation of daytime temperatures for the coastal city will be less than for the second city. In practice, this means that the average air temperature of each specific 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 allows you to evaluate possible temperature deviations from the average with the required level of probability.
According to probability theory, 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 varying characteristic are within M ± 1 σ , 95.5% - within M ± 2 σ and 99.7% - within M ± 3 σ .
The value of the standard deviation allows us to judge the nature of the homogeneity of the variation series and the study group. If the value of the standard deviation is small, then this indicates a fairly high homogeneity of the phenomenon being studied. The arithmetic mean in this case should be considered quite characteristic for a given variation series. However, too small a sigma value makes one think about an artificial selection of observations. With a very large sigma, the arithmetic mean characterizes the variation series to a lesser extent, which indicates significant variability of the characteristic or phenomenon being studied or the heterogeneity of the group under study. However, comparison of the value of the standard deviation is possible only for features of the same dimension. Indeed, if we compare the diversity of weights of newborn children and adults, we will always get higher sigma values in adults.
Comparison of the variability of features of different dimensions can be done using coefficient of variation. It expresses diversity as a percentage of the mean, allowing comparisons between different traits. The coefficient of variation in the medical literature is indicated by the sign “ WITH ", and in mathematical " v"and calculated by the formula:
.
Values of the coefficient of variation of less than 10% indicate small scattering, from 10 to 20% - about average, more than 20% - about strong scattering around the arithmetic mean.
The arithmetic mean is usually calculated based on data from a sample population. 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 part of the possible units of observation is studied, that is, the sample population. Information about all possible units representing the phenomenon being studied can be obtained by studying the entire population, which is not always possible. At the same time, for the purpose of generalizing 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 being studied, the results obtained on the basis of the sample population must be transferred to the general population using statistical methods.
To determine the degree of agreement between a sample study and the general population, it is necessary to estimate the magnitude of the error that inevitably arises during sample observation. This error is called " The error of representativeness"or "Average error of the arithmetic mean." It is actually 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 a general population. Since the sample mean is a random variable, such a forecast is performed with a level of probability acceptable to the researcher. In medical research it is at least 95%.
The representativeness error cannot be confused with registration errors or attention errors (slips, miscalculations, typos, etc.), which should be minimized by adequate methods and tools used during the experiment.
The magnitude of the representativeness error depends on both the sample size and the variability of the trait. The larger the number of observations, the closer the sample is to the population and the smaller the error. The more variable the sign, the greater the statistical error.
In practice, to determine the representativeness error in variation series, the following formula is used:
,
Where: m – error of representativeness;
σ – standard deviation;
n– number of observations in the sample.
The formula shows that the size of the average error is directly proportional to the standard deviation, i.e., the variability of the characteristic being studied, and inversely proportional to the square root of the number of observations.
When performing statistical analysis based on calculating relative values, constructing a variation series is not necessary. 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 on which the indicator is calculated;
n– number of observations in the sample population.
However, the specified 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%. 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 population is performed by indicating two values – the minimum and maximum. These extreme values of possible deviations, within which the desired average value of the population may fluctuate, are called “ Trust boundaries».
The postulates of probability theory have proven that with a normal distribution of a characteristic with a probability of 99.7%, the extreme values of deviations of the average will not be greater than the value of triple the representativeness error ( M ± 3 m ); in 95.5% – no more than twice the average error of the average value ( M ± 2 m ); in 68.3% – no more than one average error ( M ± 1 m ) (Fig. 9).
| P% |
Rice. 9. Probability density of normal distribution.
Note that the above statement is only true for a feature that obeys the normal Gaussian distribution law.
Most experimental studies, including in the field of medicine, are associated with measurements, the results of which can take on 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 such distribution, which has 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 the 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 ranges from to , which is never encountered in practice. However, the normal distribution very often works well as an approximation.
Whether weight, height and other physiological parameters of the human body are measured, the results are always influenced by a very large number of random factors (natural causes and measurement errors). Moreover, as a rule, the effect of each of these factors is insignificant. Experience shows that the results in such cases will be approximately normally distributed.
2. Many distributions associated with random sampling become normal as the volume of the latter increases.
3. The normal distribution is well suited as an approximation of other continuous distributions (for example, skewed).
4. The normal distribution has a number of favorable mathematical properties, which largely ensure 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 a normal distribution model. For this purpose, statistics have developed methods that are commonly called “Nonparametric”.
The choice of a statistical method that is suitable for processing data from a particular experiment should be made depending on whether the obtained data belongs to the normal distribution law. Testing the hypothesis for the subordination of a sign to the normal distribution law is carried out using a frequency distribution histogram (graph), as well as a number of statistical criteria. Among them:
Asymmetry criterion ( b );
Criterion for testing for kurtosis ( g );
Shapiro–Wilks test ( W ) .
An analysis of the nature of the data distribution (also called a test for normality of distribution) is carried out for each parameter. To confidently judge whether the distribution of a parameter corresponds to 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 check for normality using the Shapiro-Wilks criterion, it is necessary to find the value of this criterion using statistical tables at the required level of significance and depending on the number of observation units (degrees of freedom). Appendix 1. The normality hypothesis is rejected at small values of this criterion, as a rule, at w <0,8.
Variation series is a series of numerical values of a characteristic.
The main characteristics of the variation series: v – variant, p – frequency of its occurrence.
Types of variation series:
according to the frequency of occurrence of the options: simple - the option occurs once, weighted - the option occurs two or more times;
by location of options: ranked - options are arranged in descending and ascending order, unranked - options are written in no particular order;
by combining an option into groups: grouped - options are combined into groups, ungrouped - options are not combined into groups;
by size options: continuous - options are expressed as an integer and fractional number, discrete - options are expressed as an integer, complex - options are represented by a relative or average value.
A variation series is compiled and formalized for the purpose of calculating average values.
Variation series recording form:
8. Average values, types, calculation methods, application in healthcare
Average values– a cumulative generalizing characteristic of quantitative characteristics. Application of averages:
1. To characterize the organization of work of medical institutions and evaluate their activities:
a) in the clinic: indicators of doctors’ workload, average number of visits, average number of residents in the area;
b) in a hospital: the average number of days a bed is open per year; average length of hospital stay;
c) in the center of hygiene, epidemiology and public health: average area (or cubic capacity) per person, average nutritional standards (proteins, fats, carbohydrates, vitamins, mineral salts, calories), sanitary norms and standards, etc.;
2. To characterize physical development (main anthropometric characteristics, morphological and functional);
3. To determine the medical and physiological parameters of the body in normal and pathological conditions in clinical and experimental studies.
4. In special scientific research.
The difference between average values and indicators:
1. Coefficients characterize an alternative characteristic that occurs only in a certain part of the statistical population, which may or may not occur.
Average values cover characteristics that are common to all members of the team, but to varying degrees (weight, height, days of treatment in the hospital).
2. Coefficients are used to measure qualitative characteristics. Average values – for varying quantitative characteristics.
Types of averages:
arithmetic mean, its characteristics are standard deviation and mean error
mode and median. Fashion (Mo)– corresponds to the value of the characteristic that occurs more often than others in a given population. Median (Me)– the value of a characteristic that occupies the median value in a given population. It divides the series into 2 equal parts according to the number of observations. Arithmetic mean (M)– unlike the mode and median, it is based on all observations made, therefore it is an important characteristic for the entire distribution.
other types of averages that are used in special studies: root mean square, cubic, harmonic, geometric, progressive.
Arithmetic mean characterizes the average level of the statistical population.
For a simple series, where
∑v – amount option,
n – number of observations.
for a weighted series, where
∑vр – the sum of the products of each option and the frequency of its occurrence
n – number of observations.
Standard deviation arithmetic mean or sigma (σ) characterizes the diversity of a characteristic
- for a simple row
Σd 2 – the sum of the squares of the difference between the arithmetic mean and each option (d = │M-V│)
n – number of observations
- for a weighed series
∑d 2 p – the sum of the products of the squares of the difference between the arithmetic mean and each option and the frequency of its occurrence,
n – number of observations.
The degree of diversity can be judged by the magnitude of the coefficient of variation 
. More than 20% is strong diversity, 10-20% is medium diversity, less than 10% is weak diversity.
If we add and subtract one sigma (M ± 1σ) to the arithmetic mean value, then with a normal distribution, at least 68.3% of all variants (observations) will be within these limits, which is considered the norm for the phenomenon being studied. If k 2 ± 2σ, then 95.5% of all observations will be within these limits, and if k M ± 3σ, then 99.7% of all observations will be within these limits. Thus, the standard deviation is a standard deviation that allows us to predict the probability of the occurrence of such a value of the characteristic being studied that is within the specified boundaries.
Average error of the arithmetic mean or representativeness bias. For a simple, weighted series and the rule of moments:
.
To calculate average values, it is necessary: homogeneity of the material, a sufficient number of observations. If the number of observations is less than 30, n-1 is used in the formulas for calculating σ and m.
When assessing the result obtained by the size of the average error, a confidence coefficient is used, which makes it possible to determine the probability of a correct answer, that is, it indicates that the resulting value of the sampling error will not be greater than the actual error made as a result of continuous observation. Consequently, with an increase in the confidence probability, the width of the confidence interval increases, which, in turn, increases the confidence of the judgment and the supportability of the result obtained.
Rows built by quantitative basis, are called variational.
The distribution series consist of options(characteristic values) and frequencies(number of groups). Frequencies expressed as relative values (fractions, percentages) are called frequencies. The sum of all frequencies is called the volume of the distribution series.
By type, the distribution series are divided into discrete(constructed based on discontinuous values of the characteristic) and interval(based on continuous values of the characteristic).
Variation series represents two columns (or rows); one of which provides individual values of a varying characteristic, called variants and denoted by X; and in the other - absolute numbers showing how many times (how often) each option occurs. The indicators in the second column are called frequencies and are conventionally denoted by f. Let us note once again that the second column can also use relative indicators that characterize the share of the frequency of individual options in the total sum of frequencies. These relative indicators are called frequencies and are conventionally denoted by ω The sum of all frequencies in this case is equal to one. However, frequencies can also be expressed as percentages, and then the sum of all frequencies gives 100%.
If variants of a variation series are expressed in the form of discrete quantities, then such a variation series is called discrete.
For continuous characteristics, variation series are constructed as interval, that is, the values of the attribute in them are expressed “from... to...”. In this case, the minimum values of the characteristic in such an interval are called the lower limit of the interval, and the maximum - the upper limit.
Interval variation series are also constructed for discrete characteristics that vary over a large range. Interval series can be with equal And unequal at intervals.
Let's consider how the value of equal intervals is determined. Let us introduce the following notation:
i– interval size;
- the maximum value of the characteristic for population units;
– the minimum value of the characteristic for population units;
n – number of allocated groups.
, if n is known.
If the number of groups to be identified is difficult to determine in advance, then to calculate the optimal interval size with a sufficient population size, the formula proposed by Sturgess in 1926 can be recommended:
n = 1+ 3.322 log N, where N is the number of units in the aggregate.
The size of unequal intervals is determined in each individual case, taking into account the characteristics of the object of study.
Statistical sample distribution call a list of options and their corresponding frequencies (or relative frequencies).
The statistical distribution of the sample can be specified in the form of a table, in the first column of which the options are located, and in the second - the frequencies corresponding to these options ni, or relative frequencies Pi .
Statistical distribution of the sample
Interval series are variation series in which the values of the characteristics underlying their formation are expressed within certain limits (intervals). Frequencies in this case refer not to individual values of the attribute, but to the entire interval.
Interval distribution series are constructed based on continuous quantitative characteristics, as well as discrete characteristics that vary within significant limits.
An interval series can be represented by the statistical distribution of a sample indicating the intervals and their corresponding frequencies. In this case, the sum of the frequencies of the variants falling within this interval is taken as the frequency of the interval.
When grouping by quantitative continuous characteristics, determining the size of the interval is important.
In addition to the sample mean and sample variance, other characteristics of the variation series are also used.
Fashion The variant that has the highest frequency is called.
Statistical distribution series– this is an ordered distribution of population units into groups according to a certain varying characteristic.Depending on the characteristic underlying the formation of the distribution series, there are attributive and variational distribution series.
The presence of a common characteristic is the basis for the formation of a statistical population, which represents the results of the description or measurement of the general characteristics of the objects of study.
The subject of study in statistics is changing (varying) characteristics or statistical characteristics.
Types of statistical characteristics.
Distribution series are called attributive built according to quality criteria. Attributive– this is a sign that has a name (for example, profession: seamstress, teacher, etc.).
The distribution series is usually presented in the form of tables. In table 2.8 shows the attribute distribution series.
Table 2.8 - Distribution of types of legal assistance provided by lawyers to citizens of one of the regions of the Russian Federation.
Variation series are distribution series, built on a quantitative basis. Any variation series consists of two elements: options and frequencies.
Variants are considered to be the individual values of a characteristic that it takes in a variation series.
Frequencies are the numbers of individual options or each group of a variation series, i.e. These are numbers showing how often certain options occur in a distribution series. The sum of all frequencies determines the size of the entire population, its volume.
Frequencies are frequencies expressed as fractions of a unit or as a percentage of the total. Accordingly, the sum of the frequencies is equal to 1 or 100%. The variation series allows one to estimate the form of the distribution law based on actual data.
Depending on the nature of the variation of the trait, there are discrete and interval variation series.
An example of a discrete variation series is given in table. 2.9.
Table 2.9 - Distribution of families by the number of occupied rooms in individual apartments in 1989 in the Russian Federation.
Variation series
A certain quantitative characteristic is studied in the general population. A sample of volume is randomly extracted from it n, that is, the number of sample elements is equal to n. At the first stage of statistical processing, ranging samples, i.e. number ordering x 1 , x 2 , …, x n Ascending. Each observed value x i called option. Frequency m i is the number of observations of the value x i in the sample. Relative frequency (frequency) w i is the frequency ratio m i to sample size n: .When studying variation series, the concepts of accumulated frequency and accumulated frequency are also used. Let x some number. Then the number of options , whose values are less x, is called the accumulated frequency: for x i
A characteristic is called discretely variable if its individual values (variants) differ from each other by a certain finite value (usually an integer). The variation series of such a characteristic is called a discrete variation series.
Table 1. General view of a discrete variation frequency series
| Characteristic values | x i | x 1 | x 2 | … | x n |
| Frequencies | m i | m 1 | m 2 | … | m n |
A characteristic is called continuously varying if its values differ from each other by an arbitrarily small amount, i.e. a sign can take any value in a certain interval. A continuous variation series for such a characteristic is called interval.
Table 2. General view of the interval variation series of frequencies
Table 3. Graphic images of the variation series
| Row | Polygon or histogram | Empirical distribution function | |
| Discrete |  |  |  |
| Interval |  |  |  |
For graphical representation of variation series, the polygon, histogram, cumulative curve and empirical distribution function are most often used.
In table 2.3 (Grouping of the Russian population by average per capita income in April 1994) is presented interval variation series.
It is convenient to analyze distribution series using a graphical image, which allows one to judge the shape of the distribution. A visual representation of the nature of changes in the frequencies of the variation series is given by polygon and histogram.
The polygon is used when depicting discrete variation series.
Let us, for example, graphically depict the distribution of housing stock by type of apartment (Table 2.10).
Table 2.10 - Distribution of the housing stock of the urban area by type of apartment (conditional figures).
Rice. Housing distribution area
Not only the frequency values, but also the frequencies of the variation series can be plotted on the ordinate axes.
The histogram is used to depict an interval variation series. When constructing a histogram, the values of the intervals are plotted on the abscissa axis, and the frequencies are depicted by rectangles built on the corresponding intervals. The height of the columns in the case of equal intervals should be proportional to the frequencies. A histogram is a graph in which a series is depicted as bars adjacent to each other.
Let us graphically depict the interval distribution series given in table. 2.11.
Table 2.11 - Distribution of families by size of living space per person (conditional figures).
| N p/p | Groups of families by size of living space per person | Number of families with a given size of living space | Cumulative number of families |
| 1 | 3 – 5 | 10 | 10 |
| 2 | 5 – 7 | 20 | 30 |
| 3 | 7 – 9 | 40 | 70 |
| 4 | 9 – 11 | 30 | 100 |
| 5 | 11 – 13 | 15 | 115 |
| TOTAL | 115 | ---- | |
Rice. 2.2. Histogram of the distribution of families by the size of living space per person
Using the data of the accumulated series (Table 2.11), we construct cumulate distribution.
Rice. 2.3. Cumulative distribution of families by size of living space per person
The representation of a variation series in the form of a cumulate is especially effective for variation series whose frequencies are expressed as fractions or percentages of the sum of the frequencies of the series.
If we change the axes when graphically depicting a variation series in the form of cumulates, then we get ogiva. In Fig. 2.4 shows the ogive constructed on the basis of the data in Table. 2.11.
A histogram can be converted into a distribution polygon by finding the midpoints of the sides of the rectangles and then connecting these points with straight lines. The resulting distribution polygon is shown in Fig. 2.2 with a dotted line.
When constructing a histogram of the distribution of a variation series with unequal intervals, it is not the frequencies that are plotted along the ordinate, but the distribution density of the characteristic in the corresponding intervals.
The distribution density is the frequency calculated per unit interval width, i.e. how many units in each group are per unit of interval value. An example of calculating the distribution density is presented in table. 2.12.
Table 2.12 - Distribution of enterprises by number of employees (conditional figures)
| N p/p | Groups of enterprises by number of employees, people. | Number of enterprises | Interval size, people. | Distribution density |
| A | 1 | 2 | 3=1/2 | |
| 1 | Up to 20 | 15 | 20 | 0,75 |
| 2 | 20 – 80 | 27 | 60 | 0,25 |
| 3 | 80 – 150 | 35 | 70 | 0,5 |
| 4 | 150 – 300 | 60 | 150 | 0,4 |
| 5 | 300 – 500 | 10 | 200 | 0,05 |
| TOTAL | 147 | ---- | ---- |
Can also be used to graphically represent variation series cumulative curve. Using a cumulate (sum curve), a series of accumulated frequencies is depicted. Cumulative frequencies are determined by sequentially summing frequencies across groups and show how many units in the population have attribute values no greater than the value under consideration.
Rice. 2.4. Ogive of distribution of families by the size of living space per person
When constructing the cumulates of an interval variation series, variants of the series are plotted along the abscissa axis, and accumulated frequencies are plotted along the ordinate axis.
