Measures of central tendency mean

Measures of central tendency mean

1. ECONOMICS BASIC STATISTICS DR REKHA CHOUDHARY Department of Economics Jai Narain Vyas University, Jodhpur Rajasthan Measures of Centre Tendency: Arithmetic Mean

2. Department of Economics 1.0 Introduction Measures of central tendency are statistical measures which describe the position of a distribution. They are also called statistics of location, and are the complement of statistics of dispersion, which provide information concerning the variance or distribution of observations. In the univariate context, the mean, median and mode are the most commonly used measures of central tendency. Computable values on a distribution that discuss the behavior of the center of a distribution.

3. Department of Economics 1.1 Objectives After going through this unit, you will be able to : Define the term Arithmetic Mean; Explain combined and weighted Arithmetic Mean;   Describe the relation between Mean, Mode and Median;   Describe the calculation of Mean in in individual, discrete and continuous series by different methods; and   Explain the advantages and disadvantages of Mean.  

4. Department of Economics 1.2 Measures of Central Tendency It refers to a single central number or value that condenses the mass data & enables us to give an idea about the whole or entire data. Central tendency is a statistical measure that determines a single value that accurately describes the center of the distribution and represents the entire distribution of scores. The value or the figure which represents the whole series is neither the lowest value in the series nor the highest it lies somewhere between these two extremes. Types: Arithmetic Mean Median Mode Geometric Mean Harmonic Mean

5. Department of Economics 1.3 Arithmetic Mean It is the most commonly used measure of central tendency. It is also called as ‘Average’ .It is defined as additional or summation of all individual observations divided by the total number of observation. Horace Secrist , “The arithmetic mean of a series is the figure obtained by dividing the sum of value of all items by their number” Arithmetic mean is a mathematical average and it is the most popular measures of central tendency. It is frequently referred to as ‘mean’ it is obtained by dividing sum of the values of all observations in a series ( X) by the number of Ʃ items (N) constituting the series. Thus, mean of a set of numbers X1, X2, X3, ……….. Xn denoted by and is defined as 1.3.1 Definition Arithmetic Mean = 𝑆𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑙𝑙 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠

6. Department of Economics 1.3.2 Calculation of Mean 1. Individual Series Direct Method = ƩX N X = Stands for value ̅ X͞ = Stands for Mean ƩX =Stands for summation of values N = Stands for numbers of items Example : Roll No. 1 2 3 4 5 6 7 8 9 10 Total Marks 110 190 160 165 200 190 150 200 165 160 1690 X͞ = ƩX = 1690 = 169 marks N 10

7. Department of Economics 2. Discrete series : Example: Wages (X) Rs . 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 No. of persons (f) 35 40 48 100 125 87 43 22 Wages (X) Rs . No. of persons ( f) Total Wages (f x X) 4.5 35 157.5 5.5 40 220.0 6.5 48 312.0 7.5 100 750.0 8.5 125 1062.5 9.5 87 826.5 10.5 43 451.5 11.5 22 253.0 N= 500 ƩfX = 4033.0 Direct Method X = ƩfX N = 4033 500 Mean Wage =Rs. 8.07

8. Department of Economics 2. Discrete series : One of the values should be taken as assumed average (A) dx = (X- A) Deviation multiplying by their frequency and there after summate the product ( Ʃfdx ) Formula X̅ = A + Ʃfdx N Wages (X) Rs . No. of persons ( f) Deviations from A= 8.5 dx Total Wages (f x dx) 4.5 35 -4 -140 5.5 40 -3 -120 6.5 48 -2 -96 7.5 100 -1 -100 8.5 125 9.5 87 1 87 10.5 43 2 86 11.5 22 3 66 N= 500 Ʃfdx = -217 Short-cut Method X̅ = A + Ʃfdx N = 8.5 + (-217) 500 = 8.066 X̅ Arithmetic mean ͞ A Assumed mean Ʃfdx Product of deviations from assumed mean and f N Total no. of items

9. Department of Economics 3. Continuous series Direct method 1. In the continuous series, the arithmetic average is determined in the same manner as in case of discrete series except that central or mid-value of class groups are assumed as ‘X’ . Example : Height (in ft.) (X) Frequency (f) 0-7 26 7-14 31 14-21 35 21-28 42 28-35 82 35-42 71 42-49 54 49-56 19 360 Height (in ft.) Mid-value (X) Frequency (f) f x X 0-7 3.5 26 91.0 7-14 10.5 31 325.5 14-21 17.5 35 612.5 21-28 24.5 42 1029.0 28-35 31.5 82 2583.0 35-42 38.5 71 2733.5 42-49 45.5 54 2457.0 49-56 52.5 19 997.5 360 (N) ƩfX 10829.0 X = ƩfX = 10829 N 360 Mean height = 30.08 ft

10. Department of Economics Short cut method Height (in ft.) Mid-value (X) Frequency (f) Deviation from A =31.5 dx f x dx 0-7 3.5 26 -28 -728 7-14 10.5 31 -21 -651 14-21 17.5 35 -14 -490 21-28 24.5 42 -7 -294 28-35 31.5 82 35-42 38.5 71 7 497 42-49 45.5 54 14 756 49-56 52.5 19 21 399 Total 360 (N) Ʃfdx = -511 X̅ =A + Ʃfdx N = 31.5 + (-511) 360 = 30.08 ft.

11. Department of Economics Step Deviation method 1.One value, usually the middle value is assumed as an (A) 2. d́ = X – A i 3. Find out Ʃf d́ 4. Formula X̅ =A + Ʃf d́ x i N X̅ =A + Ʃf d́ x i N = 31.5 + (-73) x 7 360 = 30.08 ft. Height (in ft.) Mid-value (X) Frequency (f) Deviation from A =31.5 d́ = X – A i f x d́ 0-7 3.5 26 -4 -104 7-14 10.5 31 -3 -93 14-21 17.5 35 -2 -70 21-28 24.5 42 -1 -42 28-35 31.5 82 35-42 38.5 71 1 71 42-49 45.5 54 2 108 49-56 52.5 19 3 57 Total 360 (N) Ʃf d ́ = -73

12. Department of Economics 1.4 Combined Arithmetic Mean If mean of two or more components of a group are given separately along with the number of items, the combined mean of the whole group can be ascertained Formula Combined Mean X̅ = X̅1N1 + X̅2N2 + X̅3N3 …………….+ X̅nNn N1 +N2 +N3…………….Nn X̅1 , X̅2,X̅3 are the arithmetic mean of different components N1, N2, N3 are the number of items of different components

13. Department of Economics 1.5 Weighted Arithmetic Mean Example: X̅ = 3 x 3.75 + 4 x 3.5 + 2 x 4 3 + 4 + 2 Credit (Weight) Grade 3.00 3.75 4.00 3.5 2.00 4 Sometimes we associate with the numbers X1, X2, X3,……..XN with certain weighting factors w1, w2, w3,……..wN depending on the importance of that number X̅ = 𝑤1 𝑋1 + 𝑤2 𝑋2 + 𝑤3 𝑋3 + …….. + 𝑤𝑛𝑋𝑁 = Ʃ 𝑤𝑋 N 𝑤1 + 𝑤2 + 𝑤3 + …….. + 𝑤N Ʃ𝑤 Weighted mean = 3.69

14. Department of Economics 1.6. Advantages and Disadvantages of Mean Advantages of Mean: • It is easy to understand & simple calculate. • It is based on all the values. • It is rigidly defined . • It is easy to understand the arithmetic average even if some of the details of the data are lacking. • It is not based on the position in the series. Disadvantages of Mean: • It is affected by extreme values. • It cannot be calculated for open end classes. • It cannot be located graphically • It gives misleading conclusions. • It has upward bias.

15. Department of Economics 1.7 Relation between Mean , Mode and Median  Mode = 3 Median – 2 Mean Symmetrical distribution Asymmetrical distribution Symmetrical distribution The observations are equally distributed. The values of mean, median and mode are always equal. i.e. Mean = Median = Mode Asymmetrical distribution The observations are not equally distributed. Two possibilities are there: Positively Skewed Negatively Skewed

16. Department of Economics 1.8 Let us Sum up In conclusion, a measure of central tendency is a measure that tells us where the middle of a bunch of data lies. The term average is used frequently in everyday life to express an amount that is typical for a group of people or things. ... Averages are  useful  because they: summaries a large amount of data into a single value; and. indicate that there is some variability around this single value within the original data . The  mean  is often  used  in research, academics and in sports. When you watch a cricket match and you see the player's batting average, that number represents the total number of hits divided by the number of times at bat. In other words, that number is the  mean . So, mean is also used in daily life.

17. Department of Economics 1.9 Unit End Questions     1. Calculate Arithmetic Mean from the following data . Income (in Rs ) No. of persons 50-100 15 50-150 29 50-200 46 50-250 75 50- 300 90 2. Write the advantages and disadvantages of arithmetic mean? 3. Write the formula of Combined arithmetic mean ?

18. Department of Economics 1.10 Suggested Readings Asthana H.S, and Bhushan , B.(2007) Statistics for Social Sciences (with SPSS Applications). Prentice Hall of India B.L.Aggrawal (2009). Basic Statistics . New Age International Publisher, Delhi. Gupta, S.C.(1990) Fundamentals of Statistics . Himalaya Publishing House, Mumbai Elhance , D.N: Fundamental of Statistics Singhal , M.L: Elements of Statistics Nagar, A.L. and Das, R.K.: Basic Statistics Croxton Cowden: Applied General Statistics Nagar, K.N.: Sankhyiki ke mool tatva Gupta, BN : Sankhyiki Balasubramanian  , P., &  Baladhandayutham , A.  (2011).Research methodology in library science. (pp. 164- 170). New Delhi: Deep & Deep Publications. 

19. Department of Economics THANKS