Numerical characteristics of variation series
Arithmetic mean of variation series is the sum of products of all variants on the corresponding frequencies divided on the sum of frequencies:
where xi are variants of a discrete series or middles of intervals of interval variation series; ni are the corresponding to then frequencies; m is the number of intervals or non-repeating variants, wi are relative frequencies of variants or intervals.
Median of variation series is the value of the attribute falling on the middle of ranked series of observations. For a discrete variation series with odd number of members the median is equal to a middle variant and for a series with even number of members – to a half-sum of two middle variants.
Mode of variation series is the variant to which corresponds the greatest frequency.
The elementary parameter of a variation is the variation scope R which is equal to a difference between greatest and least variants of the series: R = xmax – xmin.
Average linear deviation of a variation series is arithmetic mean of absolute quantities of deviations of variants from their arithmetic mean:
Dispersion s2 of a variation series is arithmetic mean of squares of deviations of variants from their arithmetic mean:
Average quadratic deviation s of a variation series is arithmetic value of square root of its dispersion:
Coefficient of variation:
Initial moment of the k-th order of a variation series is
Obviously,
Central moment of the k-th order of a variation series is
Obviously,
Coefficient of asymmetry of a variation series is
Excess (or coefficient of excess) of a variation series is
Glossary
gathering – сбор; gathering data – сбор данных
ordering, systematization – систематизация
processing – обработка; data processing – обработка данных
supervision, observation – наблюдение
revealing – выявление; attribute – признак
occurring – происходящий; wheat – пшеница
farm – хозяйство; change– смена; to rank – ранжировать
frequency – частота; relative frequency – относительная частота
cumulative frequency – накопленная частота
inexpedient – нецелесообразный
arithmetic mean – среднее арифметическое
excess – эксцесс
Exercises for Seminar 12
12.1. There are 100 workers at an enterprise according to the list who have the following categories:
1, 5, 2, 4, 3, 4, 6, 4, 5, 1, 2, 2, 3, 4, 5, 3, 4, 5, 2, 1, 4, 5, 5, 4, 3, 4, 6, 1, 2, 4, 4, 3, 5, 6, 4, 3, 3, 1, 3, 4, 3, 1, 2, 4, 4, 5, 6, 1, 3, 4, 5, 3, 4, 4, 3, 2, 6, 1, 2, 4, 5, 3, 3, 2, 3, 6, 4, 3, 4, 5, 4, 3, 3, 2, 6, 3, 3, 4, 5, 4, 4, 3, 3, 2, 1, 2, 1, 6, 5, 4, 3, 2, 3, 4, 4, 3, 5, 6, 1, 5.
Compose the series of distribution of workers on categories. Find cumulative and relative frequencies. Represent the variation series graphically. Determine the average category of a worker, the modal and median category, the dispersion and the average quadratic deviation.
12.2. Determine the absolute and relative density of distribution of workers of an enterprise on the experience of their work at the given enterprise.
Distribution of workers on the experience of work
| The experience of work, years | Up to 1 | 1-5 | 5-10 | 10-20 | 20-40 | Total |
| Number of workers |
Find the average experience of work, the average quadratic deviation and the coefficient of variation (experience – стаж).
12.3. By oral interrogation the quality of production released by a firm and sold in a shop of this firm was studied. Visitors assessed the quality on a ten-mark scale. The summary data have been received.
Mark estimation of production of the enterprise
| Estimation of quality of production, point | 1-2 | 3-4 | 5-6 | 7-8 | 9-10 |
| Number of cases |
Determine the average score of quality of production, the average quadratic deviation, the coefficient of variation, the parameters of asymmetry and an excess (interrogation – опрос; to release – выпускать).
12.4. According to distribution of students by results of passing examinations determine: the average score of progress of students in each subject and in all subjects; the dispersions of mark of progress in a subject and as a whole in all subjects; the intergroup dispersion. Find the general dispersion of progress by using the rule of addition of dispersions.
Distribution of students of group by results of passing examinations
| Estimation at examination | Number of students who have received an estimation in subjects | |||
(progress – успеваемость).
12.5. Carry out the analysis of the data of annual profit levels of three companies:
| Year | «Cherry Computers» | «Lemon Motors» | «Orange Electronics» |
| 14,2 12,3 -16,2 15,4 17,2 10,3 -6,3 -7,8 3,4 12,2 | -6,2 13,3 -8,4 27,3 28,2 14,5 -2,4 -3,1 15,6 18,2 | 37,5 -10,6 40,3 5,4 6,2 10,2 13,8 11,5 -6,2 27,5 |
Find the average value and the standard deviation of the profit for each of the companies. Compare the received results of their activity for 10 years. What of the companies, in your opinion, is the activity more successful for?
Exercises for Homework 12
12.6. There are the following data on a number of industrial subdivisions on each of 100 agricultural enterprises:
2, 4, 5, 3, 4, 6, 7, 4, 5, 3, 3, 4, 2, 6, 5, 4, 7, 2, 3, 4, 4, 5, 4, 3, 4, 6, 6, 5, 2, 3, 4, 3, 5, 6, 7, 2, 4, 3, 4, 5, 4, 6, 7, 2, 5, 3, 5, 4, 3, 7, 2, 4, 3, 4, 5, 4, 3, 2, 6, 7, 6, 4, 3, 2, 3, 4, 5, 4, 3, 5, 4, 3, 2, 6, 4, 5, 7, 5, 4, 3, 4, 5, 7, 4, 3, 4, 5, 6, 5, 3, 4, 2, 2, 4, 3, 7, 5, 6, 4, 5.
Compose the series of distribution of the agricultural enterprises on number of industrial subdivisions for one economy. Find cumulative and relative frequencies. Represent the variation series graphically. Determine the average number of industrial subdivisions for one economy, the modal and median values of the number of subdivisions, the dispersion and the average quadratic deviation (economy – хозяйство).
12.7. There are the following data on an area of crops of vegetables in the economies of the set of districts.
The area of crops of vegetables on an economy
| District | Number of economies | ||||||
| - | |||||||
| - | |||||||
| - | |||||||
| - | |||||||
| - | |||||||
Give the comparative estimation of variability of the area of crops of vegetables in the economies of two districts (variability – колеблемость; crop – посев).
12.8. Workers of an enterprise are grouped by age.
Distribution of workers of the enterprise by age
| Categories of workers | Age of workers, years | Total number of workers | ||||
| Up to 30 | 30-40 | 40-50 | 50-60 | From above 60 | ||
| Workers | ||||||
| Heads | ||||||
| Specialists | ||||||
| Total number of workers |
Determine: the average age of workers of the enterprise as a whole and on the marked categories; the modal and median values of age of workers on categories and on the enterprise; the dispersion and the average quadratic deviation of age on categories of workers and on the enterprise; the intergroup dispersion of age of workers. Find the general dispersion of age of workers by using the rule of addition of dispersions.
12.9. The administration of a supermarket is interested in the optimal level of stocks of products in a trading hall, and also the monthly average volume of purchases of the goods which are not subjects of daily consumption in a family (for example such as soda). For finding-out of this question the manager of a supermarket within January was registering the frequency of purchases of 100-gramme packages with soda and has collected the following data xi:
4, 4, 9, 3, 3, 1, 2, 0, 4, 2, 3, 5, 7, 10, 6, 5, 7, 3, 2, 9, 8, 1, 4, 6, 5, 4, 2, 1, 0, 8
Construct the variation series, determine its numerical characteristics. Which recommendations would you give to the administration of the supermarket?
L E C T U R E 13