When studying consumer demand in shoe departments shopping complex"Moscow" received the following data on the distribution of sales of men's summer shoes by size:
a) replace group frequencies with frequencies;
b) for each group, determine the cumulative frequencies;
c) construct the cumulative distribution. At the beginning of 2001, the operating credit institutions in the Russian Federation were distributed as follows by the amount of registered authorized capital:
Perform a frequency distribution analysis using frequency densities and cumulative frequencies. Draw your own conclusions. According to the State Statistics Committee of the Russian Federation at the beginning of the 2000/01 academic year, the number of students in various forms of education at state universities was distributed as follows (thousand people): daytime - 2442, evening - 259, correspondence - 1519, external study - 52.
Conduct a frequency distribution analysis and draw conclusions. For this:
a) state the initial data in the table;
b) replace group frequencies with frequencies;
c) for each group, determine the cumulative frequencies. According to the results of the winter examination session of one year of students, the following distribution of marks by points was obtained:
Define:
a) the average score for assessing students' knowledge;
b) modal performance score and median score;
c) draw conclusions about the nature of this distribution. According to task 7.1, determine the modal size of men's shoes, explain its content. The distribution of trading firms by the size of the monthly turnover is characterized by the following data:
Define:
a) the average monthly turnover per firm;
b) modal and median value of monthly turnover;
c) draw conclusions about the nature of this distribution. According to the State Statistics Committee of the Russian Federation, the number of people employed in the economy by age in 2000 was distributed as follows:
Determine the median, first and third quartiles, and first and tenth deciles. Explain their content. The distribution of the unemployed by the duration of the break in the work of the Nth region is characterized by the following data:
Determine the median and quarterly values ​​for the duration of the break in work, explain their content and carry out comparative analysis. The distribution of commercial banks by the amount of credit investments is characterized by the following data:
Determine the quartiles and deciles of the level of credit investments, explain their content. The distribution of the population by the size of the average per capita money income in Russia for 2000 is characterized by the following data:
To assess the degree of decile differentiation of the population, determine the deciles of per capita income. Explain their content. Distribution of juvenile delinquency in one of the regions of the Russian Federation for the 1st half of 2003:
Determine the indicators of variation:
a) scope;

d) coefficient of variation.
Assess the quantitative homogeneity of the population. The distribution of the number of words in a telegram in two post offices is characterized by the following data:

Order writing a unique work

Dear students,

I present to your attention sample tasks.

All tasks, one way or another, were solved in the classroom and can be solved by each of you. You can get ready. For questions, I created a group on Vkontakte http://vkontakte.ru/mesistat, you can contact here I will answer as much as possible.

Good luck to all. See you at the exam.

Sincerely,

"> Task 1.

"> The following data on the population of the Central Federal District of the Russian Federation as of 01.01.2002 by regions (million people) are known:

">1,5 1,2 2,2 1,6

">1,9 1,1 0,9 1,8

">1,6 0,8 1,3 2,1

">2,4 1,3 1,1 1,2

"> Using this data, build an interval variation series for the distribution of regions of the Central Federal District of the Russian Federation, selecting three groups of regions with equal open intervals.

"> Task 2.

"> The following data are available on the progress of 20 students of the group on the theory of statistics in the 2012 session:

">5,4, 3, 3, 5, 4, 4, 4, 3,4, 4, 5, 4, 4, 3, 2, 5, 3, 4, 4, 4, 3, 2,5, 2, 5, 5, 2, 3, 3.

">Build:

"> a) a series of distribution of students according to the grades received in the session, and depict it

"> graphically;

"> b) a series of distribution of students according to the level of performance, highlighting two groups in it

"> students: underachieving (2 points), doing well (3 points and above);

"> Task 3.

"> There is the following data on paper production in the Russian Federation:

">Year:1998 1999 2000 2001

"> Produced paper, thousand tons by years: 2453 , 2968 , 3326 , 3415

"> Calculate the relative indicators of the dynamics with a variable and constant base of comparison.

"> Task 4.

"> The volume of sales of joint-stock companies in 2003 in comparable prices increased by 5% compared to the previous year and amounted to 146 million rubles. Determine the volume of sales in 2002.

"> Task 5.

"> For three districts of the city, the following data are available (at the end of the year):

"> Determine the average size of a deposit in Sberbank in the whole city.

"> Task 6.

"> Based on the results of the winter examination session of one course of students, the following distribution of marks by points was obtained:

"> Define:

">a) the average score for assessing students' knowledge;

"> b) the modal score of academic performance and the median value of the score;

"> Task 7.

"> The distribution of trading firms by the size of the monthly turnover is characterized by the following data:

"> Define:

"> a) the average monthly turnover per firm;

">b) the modal and median value of the monthly turnover;

"> Task 8.

"> The distribution of construction firms by investment volume is characterized by the following data:

"> Define distribution characteristics:

"> a) average;

"> b) fashion;

"> c) standard deviation;

">Task " xml:lang="en-US" lang="en-US">9">.

"> "> The growth rate of the textile industry in the region for

"> 1999-2003 are characterized by the following data (as a percentage of the previous year):

">1999 2000 2001 2002 2003

">106,3 105,2 106,1 106,3 105,9

"> Determine the average annual growth rate and increase in output over the five years

"> (1999-2003).

">Task 1 " xml:lang="en-US" lang="en-US">0">.

"> "> The average annual growth rate of sown areas of agricultural

"> enterprises of the region amounted to 1256 in 1991-1995, and 8.2% in 1996-2000. Determine the average annual growth rate of sown areas of agricultural enterprises for 1991-2000.

">Task 1 " xml:lang="en-US" lang="en-US">1">.

"> "> There is the following data on retail turnover in all distribution channels in the region.

"> To study the general trend of the region's retail turnover by months for 2001 2003, perform: 1) transformation of the initial data by aggregating time periods: a) into quarterly levels; b) into annual levels; 2) smoothing quarterly levels of retail turnover using moving average.

We calculate the average work experience:

= = = = 6,5 years

We calculate the variance:

It should be borne in mind that dispersion is a measureless quantity and independent economic importance does not have. The variance is needed to calculate the standard deviation. In this case, the standard deviation is:

of the year.

Standard deviation shows that on average the options

deviate from the arithmetic mean (= 6.5) by 3.5 years with fluctuations in work experience individual workers from 0 to 15 years old.

To characterize the degree of fluctuation of a sign, it is necessary to express the standard deviation as a percentage of the arithmetic mean, i.e. calculate the coefficient of variation ( V):

.

The coefficient of variation indicates that the variability in the work experience of salespeople is very significant and heterogeneous.

5.7.4. Determine the first and third quartiles of the interval series according to the content of defective goods in the batch of goods received by the store:

Solution:

The first and third quartiles of the existing series are determined by the formulas:

= 14+2 = 14,3%;

= 18+2 =18,0%.

Consequently, in the distribution series according to data on defective goods in the incoming batch of goods to the store, the first quartile is 14.3%, and the third - 18.0%, i.e. 25% of goods contain defects that do not exceed 14.3%, and for 75% of goods, the percentage of defects does not exceed 18%.

5.7.5. Determine the 1st and 9th deciles of the interval series according to the moisture content of the batch of goods received by the store:

Solution:

The first and ninth deciles of the table data are determined by the formulas:

= 12+2 = 13%;

= 20+2 =20%.

Thus, the values ​​of deciles indicate that among 10% of a batch of goods with a minimum percentage of moisture, the maximum percentage is 13%, and among 10% of a batch of goods with the highest percentage of moisture, the minimum percentage is 20%, i.e. 1.54 times more.

5.7.6. There is data on the working time (years) of 24 workers in the factory shop:

Work experience in this workshop (years): 4; 3; 6; four; four; 2; 3; 5; four; four; 5; 2; 3; four; four; 5; 2; 3; 6; 5; four; 2; four; 3.

Required:

1. build a discrete distribution series,

2. give graphic image row,

3. calculate the indicators of the distribution center, the indicators of variation and the form of distribution.

Solution:

1. Discrete series of distribution of work experience in the factory shop:

2. Let us present a graphical representation of the constructed discrete variational series of the distribution of workers according to the time of work in the shop in the form of a frequency polygon:

The frequency polygon is closed, for this the extreme vertices are connected to points on the abscissa axis, spaced one division in the accepted scale (in this case X=1 and X=7 ).

3. The indicators of the distribution center include: arithmetic mean, mode and median.

The arithmetic mean () is determined by the following formula:

Fashion ( M 0) = 4 years (4 years occurs 9 times, i.e. this is the highest frequency f).

To determine the median, you need to determine the number of the interval in which it is located:

N Me = ;

Median ( M e) = 4 years (because numbers 12 and 13 correspond to 4 years).

Variation indicators include: range of variation ( R), mean linear deviation (), variance ( σ2), standard deviation ( σ ), the coefficient of variation ( V).

The range of variation is determined by the formula:

R = Xmax – Xmin= 6 – 2 = 4 years

To determine the average linear deviation and other indicators of variation, we will build an additional calculation table:

years

years

Consequently, individual values ​​differ on average from the arithmetic mean by 1.15 years, or 30.3%.

The root mean square deviation exceeds the mean linear deviation ( > ) in accordance with the majorance properties of averages.

The value of the coefficient of variation ( V= 30.3%) indicates that the population is fairly homogeneous.

As can be seen from the previously constructed polygon of the variation series, the distribution of shop workers by the time they work in the shop is asymmetrical, therefore, the asymmetry index is determined:

Therefore, the asymmetry is left-sided, insignificant.

5.7.7. Distribution of workers manufacturing enterprise the monthly salary is as follows:

Determine the coefficient of decile differentiation.

Formulate a conclusion.

Solution:

The coefficient of decile differentiation is determined by the formula:

To do this, we determine the place of deciles:

;

To calculate the numerical values ​​of deciles, we determine the intervals in which they are located, for which we calculate the accumulated frequencies and write the results in a table:

The table shows that the first decile is in the range of 15.0 - 16.0, the ninth decile is in the range of 18.0 - 19.0.

Let's calculate the numerical values ​​of the deciles:

thousand roubles. or 15292.1 rubles.

thousand roubles. or 18461.5 rubles.

Therefore, the lowest monthly wage of the top 10% of workers is 1.21 times higher than the highest monthly wage of the bottom 10% of workers.

5.7.8. The following data are available on the age composition of employees of enterprises consumer cooperation N - district (years): 18, 38, 28, 29, 26, 38, 34, 22, 28, 30, 22, 23, 35, 33, 27, 24, 30, 32, 28, 25, 29, 26 , 31, 24, 29, 27, 32, 25, 29, 29.

To analyze the distribution of employees of consumer cooperation enterprises by age, it is required:

1. build an interval distribution series;

2. calculate the indicators of the distribution center, the indicators of variation and the form of distribution;

3. formulate conclusions.

Solution:

1. The value of the grouping interval is determined by the formula:

n(number of intervals) - we take equal to 7.

The resulting interval distribution series is presented in the table:

2. Calculate the indicators of the distribution center ( , Mo, Me):

where: - the average value of the feature in the interval (the center of each interval).

To determine the numerical value of the mode ( Mo) according to our interval series, we determine that it is in the interval of 27-30 years, since the largest number of employees ( f= 10) is in this interval.

The mode value is determined by the formula:

Mo= x 0 +i =

To determine the numerical value of the median ( Me) we also first determine the interval in which it is located:

The median is also the interval of 27-30 years, since the numbers 15 and 16 of the series are in this interval.

= of the year.

To calculate the variation indicators, we will compile an auxiliary table:

of the year

of the year

.

Consequently, the age variation among employees of consumer cooperation enterprises is not significant, which confirms the sufficient homogeneity of the population.

The indicator of the asymmetry of the distribution of workers by age is determined by the formula:

.

Therefore, the asymmetry is right-sided, insignificant.

With right-sided asymmetry, there is a relationship between the indicators of the distribution center:

Mo< Ме <

For a given distribution, this relation is satisfied, i.e.

28,3 < 28,6 < 28,7.

For the existing distribution, taking into account the slight asymmetry, we determine the kurtosis (pointedness) indicator:

M 4 - central moment of the fourth order,

σ4- standard deviation in the fourth power.

= =

.

The negative value of the kurtosis indicates the flatness of this distribution.

5.8. Tasks for independent work

Task 1.

Based on the grouping of stores by turnover retail for the quarter determine:

· the average size of the turnover of the 1st store;

standard deviation;

· the coefficient of variation.

Record your solution in a table.

Task 2.

Distribution of juvenile delinquency in one of the regions Russian Federation for the 1st half of 2010:

Determine the indicators of variation:

a) scope;

c) standard deviation;

d) relative range of variation;

e) relative linear deviation.

Task 3.

The distribution of the number of words in a telegram in two post offices is characterized by the following data:

Define for each post office:

a) the average number of words in one telegram;

b) average linear deviation;

c) linear coefficient of variation;

d) compare the variation in the number of words in the telegram.

Task 4.

The distribution of the mileage of a trading company van is characterized by the following data:

Define:

a) the average length of run for 1 flight;

Task 5.

The distribution of the number of unemployed by age groups in the Nth region for 2008-2010 is characterized by the following data:

Define:

a) for each year, the average age of the unemployed;

b) standard deviation;

c) coefficient of variation.

Compare the variation in the age of the unemployed over two years.

Task 6.

The distribution of commercial banks by assets is characterized by the following data:

Determine the total variance in two ways:

a) ordinary;

b) according to the method of moments.

Task 7.

The turnover of a public catering enterprise per employee per quarter is characterized by the following data:

Determine for each enterprise: the coefficient of variation and compare the variation in the turnover of public catering in these enterprises. Draw your own conclusions.

Task 8.

The average value of the trait in the aggregate is 20, and the mean square of the individual values ​​of this trait is 400.

Task 9.

Specific gravity the main workers in the three shops of the enterprise amounted to: 80, 75 and 90% of the total number of workers.

Determine the variance and standard deviation of the share of the main workers for the enterprise as a whole, if the number of workers in the three shops was 100, 200 and 150, respectively.

Task 10.

The variance of the trait is 360000, the coefficient of variation is 50%.

What is the mean value of the feature?

Task 11.

When checking a batch of electric lamps out of 1000 pieces, 30 pieces turned out to be defective.

Determine the variance and standard deviation.

Task 12.

The distribution of employees of the enterprise by the amount of monthly income is as follows:

Determine the quartile differentiation coefficient.

Formulate a conclusion.

Task 13.

The following data are available on the distribution of grocery stores in the region by the amount of turnover per month:

It is required to calculate the average monthly turnover of stores in the region, the variance and the coefficient of variation.

Task 14.

The average value of the trait in the population is 13, and the mean square of the individual values ​​of this trait is 174.

Determine the coefficient of variation.

Task 15.

The output quality control of incoming components gave the following results:

Calculate the variance of the reject rate for each incoming batch.

Task 16.

The distribution of workers of two sections by length of service is as follows:

Determine in which area the composition of workers by length of service is more homogeneous.

Task 17.

According to the State Statistics Committee of the Russian Federation, the number of people employed in the economy by age in 2010 was distributed as follows:

Determine the median, first and third quartiles, and first and ninth deciles. Explain their content.

Task 18.

The distribution of the unemployed by the duration of a break in work in the N-th region is characterized by the following data:

Determine the median and quartile values ​​for the duration of a break at work, explain their content and make a comparative analysis.

Task 19.

The distribution of commercial banks by the amount of credit investments is characterized by the following data:

Determine the quartiles and deciles of the level of credit investments, explain their content.

Task 20.

The distribution of the population by the size of the average per capita cash income in Russia in 2010 is characterized by the following data:

To assess the degree of decile differentiation of the population, determine the deciles of per capita income. Explain their content.

Task 21.

The distribution of farms by sown area is characterized by the following data:

Determine the variance and standard deviation of crop areas by using the method of moments to calculate the arithmetic mean and variance.

Task 22.

The distribution of construction firms by investment volume is characterized by the following data:

Define distribution characteristics:

a) average value

c) standard deviation

d) coefficient of variation and asymmetry

e) quartile and decile deviation coefficients.

Draw conclusions about the homogeneity and nature of the distribution of construction firms.

Task 23.

In the study of the labor activity of employees of the organization (worked man-days per year), the average values ​​and values ​​of the central moments were obtained:

Using indicators of asymmetry and kurtosis, compare the nature of the distribution of men and women by labor activity. Draw your own conclusions.

____________________________________________________________________

??? QUESTIONS FOR SELF-CHECKING

1. The concept of general and systematic variation?

2. Types of variation indicators and for what purposes are they used?

3. Absolute indicators of variation and their calculation?

4. What is the standard deviation and the procedure for its calculation?

5. Mean quartile deviation and how to calculate it?

6. Types of relative indicators of variation?

7. What is the coefficient of variation, for what purposes is it used and how is it calculated?

8. Moments in distribution rows?

9. Initial moment of distribution and its order?

10. The central moment of the distribution and the definition of its order?

11. Rank indicators of variation: quartiles, deciles, percentiles?

12. Mean, mode and median in assessing the distribution skewness?

13. Definition of the coefficient of asymmetry?

14. The indicator of kurtosis of distribution and the definition of its errors?

15. The concept of normal, right-handed and left-handed distribution?

Relative indicator of asymmetry.

Asymmetry index based on the central moment of the third order
,

where is the central moment of the third order;
is the standard deviation cubed.

Central moment of the third order

.

Kurtosis indicator
,

where is the central moment of the fourth order;
is the standard deviation to the fourth power.

Average Skewness Error

.

Average kurtosis error

.

Pearson goodness-of-fit test

,

where – empirical frequency; is the theoretical frequency.

Theoretical frequencies

,

where – interval value; t =
.

Kolmogorov's criterion

,

where D is the maximum value of the difference between the accumulated empirical and theoretical frequencies.

TASKS

Tasks for independent solution

3.15. The distribution of construction firms by investment volume is characterized by the following data:

Determine the characteristics of the distribution: a) average; b) fashion;

c) standard deviation; d) coefficient of variation and asymmetry. Draw conclusions about the nature of the distribution of construction firms.

3.16. The distribution of city families by the number of children is characterized by the following data:

Determine the coefficients of skewness and kurtosis using the central moments of the first four orders. Draw conclusions about the nature of the distribution of families.

3.17. According to the task 3.16 determine the characteristics of the distribution: a) average; b) fashion; c) root-mean-square distribution; d) Pearson's coefficient of variation and asymmetry. Draw conclusions about the nature of the distribution of trade.

3.18. The distribution of commercial banks by assets is characterized by the following data:

Determine the indicators of asymmetry and kurtosis of the distribution of commercial banks by asset size. Draw your own conclusions.

3.19. Distribution of the number of unemployed by age groups in the Nth region for 2000-2003 characterized by the following data:

Determine Pearson's goodness-of-fit test (χ 2) and check the closeness of the empirical and theoretical distributions of the number of unemployed for 2000.

3.20. The part is processed in the workshop on a semi-automatic lathe. As of January 25, the following data was received on the size of machined parts (in deviations from the nominal):

To characterize the state of the technological process, it is required to check the correspondence of the empirical distribution to the normal distribution law, using K. Pearson's goodness-of-fit test.

3.21. The following data are available on the amount of overhaul run for ZIL-133 vehicles:

Give a graphical representation of the series in the form of a histogram and cumulate. Using graphic images, determine the numerical values ​​of the mode and median. Determine the asymmetry index. Formulate conclusions.

Exercise 1. The distribution of trading firms by the volume of monthly turnover is characterized by the following data:

• cumulative (cumulative) frequencySi(frequencySD) characterizes the volume of the population with the values ​​of the options not exceedingXi.

S1= f1, S2= f1+ f2, S3= f1+ f2+ f3;

Task 2. The distribution of city families by the number of children is characterized by the following data:

Tasks for independent work

Exercise 1. The following data are available on the structure of production equipment in the industry of the Russian Federation in 2009.

Task 2. In 2009, the Russian Federation had the following distribution of unemployed men by age groups. Calculate the average variation. Describe the presented population.

Task 3. In 2009, the Russian Federation had the following distribution of unemployed women by age groups. Calculate the average variation. Describe the presented population.

Task 4. Based on the sample survey of household budgets, the following distribution of the population of Moscow by the level of average monthly per capita income in 2011 was obtained. Calculate the average variation. Describe the presented population.

Task 5. Operating credit institutions in the Russian Federation at the beginning of 2011 by the amount of registered authorized capital. Calculate the average variation. Describe the presented population.

Task 6. Distribution of agricultural enterprises by the size of sown areas. Calculate the average variation. Describe the presented population.

Task 7. The distribution of the mileage of a trading company van. Calculate the average variation. Describe the presented population.

Task 8. When studying consumer demand in the shoe departments of the shopping complex, the following data were obtained on the distribution of shoes sold by size. Calculate the average variation. Describe the presented population.

Task 9. The following data are available on the distribution of the population of the Russian Federation by the level of average monthly per capita income in 2011. Calculate the average variation. Describe the presented population.

Task 10. The following data are available on the distribution of the population of the Russian Federation by the level of average monthly per capita income in 2010. Calculate the average variation. Describe the presented population.

Task 11. Grouping of acting credit organizations in the Russian Federation at the beginning of 2010 in terms of registered authorized capital. Calculate the average variation. Describe the presented population.

Task 12. Based on the results of the winter examination session, the following distribution of scores was obtained. Calculate the average variation. Describe the presented population.

Task 13. Distribution of the unemployed by the duration of the break in work (men). Calculate the average variation. Describe the presented population.

Task 14. The fixed assets of the enterprises of the city of the production sector are characterized by the following data:

Calculate the average variation. Describe the presented population.
Task 15. The fixed assets of the city's non-manufacturing enterprises are characterized by the following data:

Task 16. The distribution of the number of words in the telegram of post office A is characterized by the following data:

Task 17. The distribution of the number of words in the telegram of post office B is characterized by the following data:

Calculate the average variation. Characterize the presented population .
Task 18. The distribution of juvenile delinquency in one of the regions for 2003 is characterized by the following data:

Task 19. The distribution of commercial banks by assets is characterized by the following data:

Calculate the average variation. Describe the presented population.
Task 20. The distribution of commercial banks by the amount of credit investments is characterized by the following data:

Calculate the average variation. Describe the presented population.
Task 21.

Task 22. According to the State Statistics Committee of the Russian Federation, the number of people employed in the economy by age in 2009 was distributed as follows:

Task 23. The distribution of construction firms by investment volume is characterized by the following data:

Task 24. For October 2009, the average accrued wage workers by age groups is characterized by the following data: