







































AGORA INTERNATIONAL JOURNAL OF ECONOMICAL SCIENCES, 
AIJES, ISSN 2067-3310, E - ISSN 2067 – 7669, VOL. 12 (2018) 

 

The Use of Relative Statistical Indicators, of Central Trend Indicators and 

Their Interpretation 

Cristina Ioana Antik1 

1Agora University of Oradea, Tineretului Square Str., No. 8, Oradea, Romania 

E-mail antikcristina@yahoo.com  

Abstract  

Statistics is one of the economic evidence form in charge with the study of socioeconomic staple phenomenons, 

quantitatively speaking, in close connection with their qualitative side, in concrete conditions of space and time. 

The present article uses the importance of statistical indicators in the economic analysis of an economic entity, 

showing the calculation of theoretical and practical relative indicators, and the central trend indicators.  

Keywords: statistical indicators, relative indicators, relative sizes, interpretation of statistical data, arithmetic 

mean, harmonic mean, geometric mean, mean square value, statistical interpretation, economic analysis. 

1 Introduction  

Relative indicators are expressed in relative quantities and are used to analyze the evolution of 

economic phenomena over time. 

It is calculated as the ratio between the size it compares and the size taken as a basis for 

comparison. 

This ratio is a coefficient, which can be multiplied by: 

a. 100 and the result is called a percentage; 

b. 1000 and the result is called a promil. 

The types of relative sizes are [2]: 

- Structure 

- Of the dynamics 

- Coordination 

- Intensity 

- Of the plan (provisions) 

2 Types of Relative Sizes 

2.1 Structure related sizes 

It is calculated as the ratio between the size (in absolute amount) of a group and the size of the 

whole community [1].  

Calculation formula:  

o The size of a group  

o The size of the community  

They are also called specific weights or weights, ie how much a group represents in the 

community.  

Obs: The sum of the group weights = 100% 

 

Example:  

mailto:antikcristina@yahoo.com


2 
 

Table 1. The structure of clients at a company with tourist activity in 2018 (conventional data) 
Categories of clients Number of clients Weight % 

Internal clients  1.200 30 

External clients  2.800 70 

Total 4.000 100 

Calculation of shares: 

                              Internal clients no.              1.200   

a.Internal clients = ---------------------- x 100 = ------- x 100 = 30% 

                              Total clients no.                  4.000 

                              External clients no.            2.800  

b.External clients = ---------------------- x 100 = ------ x 100 = 70%  

                               Total clients no.                 4.000  

Interpretation: the number of external customers represents over 2/3 of the total number of 

customers (70%). 

2.2 Dynamics related sizes 

It is calculated as the ratio between the size of the indicator in the current period (1) and the 

size of the indicator in the base period [3]. 

Calculation formula: 

 The size of the indicator in the current period (1) 

 The size of the indicator in the base period (0) 

 

They can be expressed in percentages (%) and promils. The relative quantities of the dynamics 

express the evolution in time of the economic phenomena (increase or decrease) [4]. 

Example: 

Table 2. The dynamics of the customer structure in 2018 compared to 2017 at a company with 

tourist activity 

Categories of 

clients 

No. of clients  

year 2017 

Base period (0) 

No. of clients  

year 2018 

Current period (1) 

Weight (%) 

2017 2018 

Internal clients 1.200 1.760 30 40 

External clients 2.800 2.640 70 60 

Total 4.000 4.400 100 100 

Calculation of customer shares in 2018: 

                              1.760  

a.Internal clients = ------ x 100 = 40%   

                              4.400  

                               2.640   

b.External clients =  ------ x 100 = 60%  

                               4.400 

 

Dynamics indicator: 

                              1.760 

c.Internal clients = ----- x 100 = 146,66%  

                              1.200  

                              2.640 

d.External clients = ------ x 100 = 94,28%  

                              2.800   

                           4.400  

e.Total clients = ------- x 100 = 110% 

                           4.000   



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Interpretation: The total number of tourists at the analyzed travel agency increased in 2018 

compared to 2017 by 10% or 110% or 1.1 times. The situation by categories of customers is as 

follows: in the case of domestic tourists, the number of domestic customers increased in 2018 

compared to 2018 by 46.66%; in the case of foreign tourists, the number of foreign customers 

decreased in 2018 compared to 2017 by 5.72%. 

2.3 Coordination related sizes 

They are calculated as the ratio between the values of two groups, phenomena (communities), 

different territorial units [5]. 

Calculation formula: 

The size of a group (communities), territorial unit to be compared (XA) 

-------------------------------------------------------------------------------------------- 

The size of a group (communities), territorial unit base of comparison (XB) 

OR 

XB 

XA 

Example: Comparison of wheat production at two farms (data are conventional for 2018): 

farm A (1) the production was 30,000,000 lei 

farm B (2) the production was 24,000,000 lei 

 

                                Wheat production at farm 1             30.000.000 lei 

a. Variant 1 = ---------------------------------- x 100 = -------- x 100 

                                 Wheat production at farm 2              12 mil   

    Result = 125% 

 

Interpretation: 

Wheat production on farm 1 was 25% higher than wheat production on farm 2 in 2018. 

 

                               Wheat production at farm 2              24.000.000 lei 

b. Variant 2 = ---------------------------------- x 100 = -------- x 100 

                                Wheat production at farm 1              15 mil   

    Result = 80% 

 

Interpretation: Wheat production on farm 2 was 20% lower than wheat production on farm 1 

in 2018. 

2.4 Intensity related sizes 

They are calculated as the ratio between the value of two related primary characteristics and 

express how much of the value of the compared characteristic returns per characteristic unit - 

basis of comparison. 

Example:  

Table 3. Average production per worker (W) at two companies in 2018 with the same object 

of activity (textile garments) 
Commercial 

company 

Production 

value 

No. of 

workers 

Average production per 

worker 

A 200.000 1.000 100 

B 240.000 1.000 120 

                                                      WB               120     

             Size related to intensiy = ----- x 100 = ---- x 100 = 1,2  

                                                      WA               100  



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Interpretation: The average productivity per worker is 1.2 times higher in company B compared 

to company A, in 2018 

2.5 Provisions related sizes 

It is calculated as relative quantities of: 

                        Planned volume in the current period        

a.Plan tasks = -----------------------------------------------x100 

                         Achieved volume in the base period 

This index can be: 

Sub-unitary, when a reduction is planned compared to the achievements of the previous period 

(expenses, costs); 

Supra-unitary, when an increase is planned compared to the plan load (production, sales) 

                                            Achieved volume in the current period  

b.Other plan achievements = --------------------------------------------- 

                                              Planned volume in the current period 

This index can be:  

- Sub-unitary, when the achievements of the current period are lower than the provisions 

(reduction);  

- Supra-unitary, when the achievements of the current period are higher than the provisions 

(increase). 

Example:  

Table 4. The volume of goods sales at the SC FRESH FRUCT SRL store in 2018 
Commercial company Sales volume 

Actual sales 

2017 

Provisions 

2018 

Actual sales 

2018 

SC FRESH FRUCT SRL 2.000.000 2.400.000 2.640.000 

→ The relative sizes of the plan task for 2018 

Provisions 2018              2.400.000 

--------------------- x 100 = ------------ x 100 = 120% 

Achievements 2018         2.000.000 

Interpretation:  The company has forecast (scheduled) for 2018 an increase in sales volume by 

20% (growth rate). 

→ The relative sizes of the 2018 plan implementation 

Achievements 2018             2.640.000  

------------------------ x 100 = ----------- x 100 = 110% 

Provisions 2018                   2.400.000     

Interpretation:  The company registered an increase in sales compared to the annual provisions 

by 10% in 2018. The central trend indicators used in the statistical analysis of mass socio-

economic phenomena are studied using: 

- Average indices  

- Position indices 

3 Average Sizes (Indices) 

These average sizes are grouped by the nature of the characteristic into the following categories: 

-Arithmetic mean 

- Harmonic average 

- Square mean 

- Geometric average 

- Chronological average 



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Each average is calculated as: 

• Simple average: when the number of characteristic variants is equal to the number of observed 

units; 

• Weighted average: when the same variant is found in several observation units. 

3.1 The arithmetic mean 

It can only be calculated for quantitative variables. The arithmetic mean can be simple and 

weighted. 

3.1.1 Simple arithmetic mean 

It is calculated as the ratio between the sum of the values (levels) of the characteristics and 

their number. 

 
where: 

n = number of variants 

Xi = variants of the characteristics 

X = statistical characteristic 

Example: A student obtained the following grades in the discipline "Statistics" in the school 

year 2017/2018: 4,5,6,7,8,9 

The average student at the end of the school year is: 

          4 + 5 + 6 + 7 + 8 + 9 

Ms = ------------------------- = 39 / 6 = 6,50 

                       6 

3.1.2 Weighted arithmetic mean 

 
Where: 

n = frequencies  

i=  number of characteristic variants 

 

Example: The situation of students in a class according to their average in mathematics for a 

semester is as follows:  

Table 5. The situation of students in a class according to their average 
Math average (x) No. of students (n) xn 

5 7 35 

6 7 42 

7 5 35 

8 6 48 

9 5 45 

Total 30 205 

The general average of the students in this class in mathematics is calculated as follows: 

           205 

Mp = ----- = 6,83 

            30 

Example: Calculate the weighted arithmetic averages for a series of intervals: the distribution 

of students in a class of students at the end of a school year, according to the number of 

absences. 

 

n

x
x

i
=




=
i

ii

n

nx
x



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Table 6.  Distribution of students in a class by number of absences 
Groups of students by 

number of absences 

No. of students 

(f) 

The center (middle) 

of the range (x) 

xf 

0-10 18 5 90 

11-20 4 15 60 

21-30 3 25 75 

31-40 3 35 105 

41-50 2 45 90 

TOTAL 30 - 420 

Ma = Σxf / Σf = 420 / 30 = 14 

Interpretation: The average number of absences per student is 14, it is between the first interval 

(with the highest number of students) and the second interval. 

Example:  Application for calculating the average for the alternative characteristic with answer 

variants: yes - no. At the end of a school year, in a class, the situation of students' promotion is 

as follows: out of 30 students, 24 are promoted and 6 are not promoted. 

                                              No. of promoted students 

No. of promoted students (%) = --------------------------= 24/30 = 80% 

                                              Total no. of students 

                                               No. of failed students 

No. of failed students (%) = ----------------------- = 6/30 = 20% 

                                                Total no. of students 

3.2 The harmonic mean  

It is calculated as the inverse size of the arithmetic mean from the inverse values of the terms. 

a) Simple:  𝑋h=
𝑛

∑
1

𝑥
∙𝑓

 , where  

n= number of series terms; 

x= characteristic variants. 

b) Weighted:  𝑋h=
∑ 𝑓

∑
1

𝑥
∙𝑓

, where 

f= frequencies (weights) 

x= characteristic variants. 

This average is used in different situations: 

- Calculation of the average harvest per "hectare" on a farm; 

- Calculating the average salary of a company, when the salary fund for different departments 

and the average salaries of employees are known. 

Application:  

Table 7. Calculation of the average salary at S.C. FRESH FRUCT S.R.L. 

Compartments 
Average salary / 

employee (𝑺) 

Salary fund / lei 

(FS) 

- Sales 

- Administrative 

2.200 

2.500 

110.000 

80.000 

TOTAL S.C. FRESH FRUCT SRL - 190.000 (𝚺𝑭𝑺) 

The number of employees is at the sales department 50, and at administrative 32. The average 

salary on the company in 2018 was 2,317.06 lei. Average salary per company is calculated: 

𝑆=
Σ𝐹𝑆

Σ𝑁
=

Σ𝐹𝑆

Σ
1

𝑆1
∙𝐹𝑆

=
190.000

50+32
=

190.000

82
=2.317,06 lei 

The number of employees is calculated taking into account the average salary by compartments: 

𝑁𝑠=
𝐹𝑆

𝑆
=

1

𝑆
·FS 

 



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In the "sales" section: 

𝑁𝑎𝑛𝑔=
1

2.200
·110.000 = 

1.100

11
= 100 

In the "administrative" section: 

𝑁𝑎𝑛𝑔=
1

2.500
·80.000 = 

8000

125
= 64 

3.3 The square mean 

It is calculated by extracting the square root from the arithmetic mean of the squared terms of 

the series. 

It is used to calculate the variation of the characteristic (mean square deviation) or when it is 

indicated to give importance to higher levels in a series of statistics [6]. 

Calculation mode: 

a) For simple series:  

𝑋p= √
∑ 𝑋2

𝑛
 

b) For the frequency series: 

𝑋p= √
∑ 𝑋2𝑓

∑ 𝑓
 

Application: Calculation of the average number of products made by a worker, who works 

according to a company with production activity. 

Table 8.  Average number of products made by a worker 
Workers No. of products (X) 

1 

2 

3 

4 

12 

18 

10 

20 

TOTAL 60 (∑ 𝑿) 

If simple arithmetic mean is used, 

𝑋a= 
∑ 𝑋

∑ 𝑛
= 

60

4
 =15, where:  

n = no. of variants workers 

x = characteristic variants 

If simple quadratic arithmetic mean is used, 

𝑋p= √
∑ 𝑋2

∑ 𝑛
= √

122+182+102+202

4
=√

968

4
=√242~ 15,56 

So, the average number of products made by a worker calculated according to the arithmetic 

average is 15, and according to the square average is 15.56. 

3.4 The geometric mean 

It is based on the product relationship between the series terms and is used when the lower 

terms are given importance in the absolute value [7]. It is mainly used for the calculation of 

averages in the case of relative quantities of dynamics, respectively for establishing the average 

growth rate. The geometric mean is the value, which by replacing the terms of the series and 

the effect of their product, the resulting value is equal to the product of the terms, meaning:  

𝑥1 ∙ 𝑥2·𝑥3 ∙ …  ·𝑥𝑛 =  Π𝑥, 

where: 𝛱𝑥 = the product of the series terms. 

Geometric mean: 

𝑋𝑔= √Π𝑥
𝑛

 

This average is used if there is a negative or zero term in the statistical series. 



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3.5 The chronologic mean 

It is used in the case of the statistical series of dynamics (chronological) for the calculation of 

the average level. 

𝑋𝑐𝑟=

𝑥1
2

+𝑥2+𝑥3+ …+
𝑥𝑛
2

𝑛−1
 

where: x = characteristic variants; 

           n = number of terms. 

Calculation mode: 

- Make the sum between half the value of the first term and the last term, with the full value of 

the other terms of the series;  

- Divide the total amount obtained by the number of series terms minus 1. 

a) Simple:  

It is used in the case of the dynamic series of moments with equal intervals, usually for the 

statistical calculation of the average stock (goods, materials) 

 

Application:  

Table 9. The situation of the stock of goods at a store for the second quarter of 2018 
Period Goods stock (lei) Average stock 

April 

May 

June 

July 

10.000 

10.400 

11.600 

10.800 

The average stock of goods in each month of the 

second quarter was 7,133.33 lei. 

Average level of stock of goods  = 

10.000

2
+1

10.400

2
+

11.600

2
+

10.800

2

3
= 

5.000+5.200+5.800+5.400

3
= =

 
21.400

3
=7.133,33 lei 

b) Weighted in the case of dynamic series with unequal intervals. 

𝑋𝑐𝑟= 

𝑥1+𝑥2
2

·𝑓1+ 
𝑥2+𝑥3

2
·𝑓2+⋯+

𝑥𝑛−1+𝑥𝑛
2

·𝑓𝑛

𝑓1+𝑓2+⋯+𝑓𝑛
 

 

Applicațion: 

Table 10. The situation of raw material stocks at a company with production activity in 2018 
Period Good stock (lei) (x)  

January 

March 

July 

September 

December 

16.000   (x1) 

13.000   (x2) 

10.000   (x3) 

15.000   (x4) 

12.000   (x5) 

Time intervals (in 

months) are: 

𝑓1 = 2 

𝑓2 = 4 

𝑓3 = 2 

𝑓4 = 3 

 

Average annual stock =  

= 

16.000+13.000

2
∙2+

13.000+10.000

2
∙4+

10.000+15.000

2
∙2+

15.000+12.000

2
∙3

2+4+2+3
=

29.500+46.000+25.000+40.500

11
= 

141.000

11
= 

12.818,18 lei 

The average annual stock of raw materials for 2018 was 12,818.18 lei. 

4 Position Sizes (Indicators) 

They are used for distribution series. 

 

 

 



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4.1 The median 

Represents the central value of a statistical series ordered ascending or descending, which 

divides the terms of the series into equal parts: half of the terms are smaller than the median 

and half are larger than the median. 

If the series has an odd number of terms, the median represents the value of the central term, 

which is calculated by the relation      
𝑛+1

2
,  𝑛 = serial term number. 

Example:  

Statistical series 3,7,8,9,4,2,5, n = 7  

Place of the median = (7 + 1) / 2 = 8/2 = 4  

The median is 9, ie the fourth term. 

If the series has an even number of terms, the value of the median is calculated as the 

arithmetic mean of the two central terms. 

 

Example: 

Statistical series in ascending order: 6,8,9,12,15,16  

The median = 
9+12

2
 =

21

2
 =10,5 

The series has an odd number of terms, the series is ordered ascending: 5,5,6,7,7; the series 

has 5 terms, therefore the formula is used: (n+1) / 2 

The place of the median is 
5+1

2
 =3 

The median is 6, occupying the 3rd place (middle). 

 

Application: 

The series has an even number of terms (4). 

The series is ordered in ascending order: 10,12,18,20. 

Place of median between term 2 and 3: 

The median = 
12+18

2
 =

30

2
 =15 

4.2 The module 

Represents the characteristic level that has the highest frequency. Calculated only for frequency 

distribution statistical series. In table no. 1, the maximum frequency is 7, so the modulus, ie the 

corresponding value is 5 and 6 (averages of 5 and 6). In table no. 4, the highest frequency is 20, 

so the module corresponds to worker no. 4, which made this number of products (20). 

5 Conclusions 

The importance of the statistical calculation lies not only in the calculation of the arithmetic 

ratio of these indicators, but in the examples and their interpretation, from which the top-level 

management (managers) can come out to conclusions and arguments in order to correct the 

business evolution. The theoretical and practical presentation of these statistical indicators is 

addressed to practicing accountants, economists, students enrolled in an economic profile, 

departments of economic analysis within companies, as well as those interested in the 

introduction of the calculation methods and the interpretation of statistical indicators. 

REFERENCES  

[1] Anghelache, C., Statistică teoretică și economică – teorie și aplicații, Editura Economică, 2004 

[2] Dorin, V., Stan, F., Statistică și analiză economică,Editura CD Press, București, 2007; 

[3] Ișfănescu, A, colectiv, Analiza economico-financiară, Editura ASE, București, 2002; 



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[4] Isaic, M.A., colectiv, Statistică – teorie și aplicații, Editura Cison, București, 2006; 

[5] Meșter, I. Simuț, R., Statistică economică – aplicații practice, Editura Universității din Oradea, 2009; 

[6] Meșter, I., Silaghi, S., Statistică economică – aplicații practice, Editura Universității din Oradea, 2008; 

[7] Wagner, P., colectiv, Statistică, Editura Didactică și Pedagogică, București, 1995.   


