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 |
8 |
12 |
16 |
14 |
10 |
60 |
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
3
8
36
89
45
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 |
|||
1 |
2 |
3 |
4 |
|
2 |
2 |
1 |
4 |
3 |
3 |
6 |
10 |
8 |
8 |
4 |
10 |
8 |
9 |
9 |
5 |
7 |
6 |
4 |
5 |
(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» |
1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 |
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?