II. Mean and indicators of variation

The educational purpose is to solve the problems on calculation of average sizes and parameters of variation, and it consists of the following:

- to study how to calculate the average annual interest rate, modal and median values of the annual interest rate, a dispersion, a standard deviation and indicators of variation of the annual interest rate on the set of banks;

- to learn how to build series of distribution of annual interest rates for an accounting period on the investigated set of banks;

- to give economic interpretation of the received numerical characteristics;

- to formulate the conclusions derived from the received data.

The average size is the generalizing characteristic of set of units on chosen grouping feature, for example, the size of the average annual interest rate calculated for each bank from the investigated set of banks.

In statistics it is necessary to calculate averages by variants which perform grouped averages. In such cases the general average is defined as arithmetic average weighted of group averages, thus in weights are number of units in groups.

The mode is the value of feature which is the most often met in the studied set. For discrete numbers of distribution the mode is a variant with the greatest frequency.

As the median refers to a variant, located in the middle of the ordered set of data, the median divides it into two equal parts.

The median in the discrete variation series of distribution is defined by the sum of the cumulated frequencies which should exceed half of all volume of units of population.

For measurement of a degree variability of the separate values from their average we calculate the basic generalizing parameters of a variation: a dispersion, standard (an average quadratic) deviation and coefficient of variation.

The dispersion is the average arithmetic squares of deviations of separate values from their arithmetic average.

Depending on the initial data the dispersion is calculated under the formula of average arithmetic simple or weighted.

There are the formulae for calculating the dispersion in the general set:

- The simple dispersion (not weighted) in

the general set,

- The weighted dispersion in the general set,

where: μ is the general average, N is the number of units in the general set, k is the number of groups.

There are the formulae for calculating dispersion in the sample:

- The simple dispersion (not weighted) in the

sample.

- The weighed dispersion in the sample,

where: is the sample average, n is the number of units in a sample, k is the number of groups..

The standard deviation represents a root square of a dispersion:

- The standard deviation in the general set

- The standard deviation in the sample.

Unlike the dispersion the standard deviation is an absolute measure of variation and it is expressed in terms of measurement of a varying feature (tenge, tons, percent, etc.)

For comparison of the sizes of the variation of various features, and also for calculation of a degree of variation of the same features in the studied set the relative parameter of a variation – the coefficient of variation, which represents percentage of a standard deviation to average arithmetic, is estimated by the following:

- The coefficient of variation in the general set

- The coefficient of variation in the sample.

With the size of the coefficient of variation it is possible to judge about the degree of variation of the features in the investigated set. The more its size, the more disorder of values of the features around the average meaning, that the set on structure is less homogeneous.

Task # 2. The volume of loans and the weighted mean of annual interest rate on loans from 20 banks for the reporting month were:

Number of a bank	Weighted mean of annual interest rate on loans, % (x)	Volume of loans, million tenge (f)
А
	16,0
	17,3
	21,5
	19,4
	25,5
	18,5
	17,5
	26,2
	18,7
	21,0
	19,6
	28,0
	27,3
	16,5
	24,7
	16,5
	20,7
	20,5
	18,3
	23,3
Total:	-

· Calculate from these 20 banks:

Ø Weighted mean annual interest rate on loans, % ,

Ø Dispersion ,

Ø Standard deviation ,

Ø Coefficient of variation ,

Ø Mode ,

Ø Median ,

· Plot the polygon distribution of the weighted mean annual interest rate on loans for the reporting month for these banks.

· Draw a conclusion.

Solution:

Order and group banks in terms of the weighted mean of annual interest rate on loans:

Weighted mean annual interest rate on loans, x	Volume of loans, f	Cumulated frequencies, S		x-
%	share	mln. tenge	% to the total	mln. tenge	%
16,0	0,160		8,00		8,00	24,00	-0,033	0,001089	0,16335
16,5	0,165		15,89		23,89	49,17	-0,028	0,000784	0,233632
17,3	0,173		7,36		31,25	23,87	-0,020	0,00040	0,0552
17,5	0,175		6,93		38,18	22,75	-0,018	0,000324	0,04212
18,3	0,183		6,93		45,11	23,79	-0,010	0,0001	0,013
18,5	0,185		6,40		51,51	22,20	-0,008	0,000064	0,00768
18,7	0,187		6,29		57,80	22,07	-0,006	0,000036	0,004248
19,4	0,194		6,13		63,93	22,31	0,001	0,000001	0,000115
19,6	0,196		5,07		69,00	18,62	0,003	0,000009	0,000855
20,5	0,205		4,80		73,80	18,45	0,012	0,000144	0,01296
20,7	0,207		4,53		78,33	17,60	0,014	0,000196	0,01666
21,0	0,210		4,43		82,76	17,43	0,017	0,000289	0,023987
21,5	0,215		3,84		86,60	15,48	0,022	0,000484	0,034848
23,3	0,233		2,61		89,21	11,42	0,04	0,0016	0,0784
24,7	0,247		2,51		91,72	11,61	0,054	0,002916	0,137052
25,5	0,255		2,40		94,12	11,48	0,062	0,003844	0,17298
26,2	0,262		2,19		96,31	10,74	0,069	0,004761	0,195201
27,3	0,273		2,08		98,39	10,65	0,08	0,0064	0,2496
28,0	0,280		1,60		99,99	8,40	0,087	0,007569	0,22707
Total:		100,0	-	-	362,04	-	-	1,668958

Rate of Income earned on loan capital

Loans, % = ----------------------------------------------------∙100%

Volume of loans

Weighted mean of annual interest rate on loans, %:

(19,3%)

Dispersion. The standard deviation. Coefficient of variation:

Since the coefficient of variation is less than 33% (the weighted mean of annual interest rate on loans equals to 19,3%), the mean is typical for the given set of banks.

The mode is the variance with the greatest frequency (i.e. in our example, is the interest rate on loans corresponding to the greatest amount of loans). = 16,5%.

The median is the variance, which divides ranked variation series into two equal parts. The number of median is:

or

=18,5%, i.e. half of the loans were issued by the banks at 18,5% or less. Thus, we estimated the weighted mean of annual interest rates on loans (), the mode ( = 16,5%) and the median ( = 18,5%) of the annual interest rates on loans.

When comparing the weighted mean of annual interest rates on loans for all studied banks, the modal and the median values of the interest rate on loans, we see that they differ significantly among themselves, calling into question the typicality of the calculated weighted mean annual interest rate on the set of the investigated 20 banks.

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