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 1 Agora 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 3 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 4 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 5 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 6 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 7 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. 8 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. 9 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. 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