Measures of dispersion range qd md

Measures of dispersion: Range, Quartile Deviation and Mean Deviation

1. DR REKHA CHOUDHARY Department of Economics Jai Narain Vyas University, Jodhpur Rajasthan Measures of Dispersion: Range, Quartile Deviation, Mean Deviation ECONOMICS BASIC STATISTICS

2. Introduction Dispersion measures the extent to which the items vary from some central value. It may be noted that the measures of dispersion or variation measure only the degree but not the direction of the variation. The measures of dispersion are also called averages of the second order because these are based on the deviations of the different values from the mean or other measures of central tendency which are called averages of the first order. Department of Economics 2

3. Objectives After going through this unit, you will be able to:   To understand the objectives of Dispersion; Types of Dispersion; Define Range , Quartile deviation and Mean deviation ; Merits and Demerits of Range , Quartile deviation and Mean deviation Department of Economics 3

4. Dispersion refers to the variations of the items among themselves / around an average. Greater the variation amongst different items of a series, the more will be the dispersion . As per Bowley , “Dispersion is a measure of the variation of the items”. In the words of Spelgel , “The degree to which numerical data tend to spread about an average values is called the variation or dispersion of data” D ispersion Definition

5. To determine the reliability of an average To compare the variability of two or more series For facilitating the use of other statistical measures To get information about the composition of the series To help in controlling variability O bjectives of Measuring D ispersion

6. Easy to understand Simple to calculate Uniquely defined Based on all observations Not affected by extreme observations Capable of further algebraic treatment   P roperties of a Good M easure of Dispersion

7. Absolute: Expressed in the same units in which data is expressed Ex: Rupees, Kgs, Ltr , Km etc. Relative: In the form of ratio or percentage, so is independent of units It is also called Coefficient of Dispersion Measures Of Dispersion Measures Of Dispersion

8. Range Interquartile Range Quartile Deviation Mean Deviation Standard Deviation Coefficient of Variation Lorenz Curve Methods Of Measuring Dispersion

9. It is the simplest measures of dispersion It is defined as the difference between the largest and smallest values in the series Formula R = Range R = L – S , 3 5 7 9 10 12 L = Largest Value, Min Range Max S = Smallest Value Coefficient of Range = 𝐿 −𝑆 𝐿 +𝑆 Range (R) Definition Range = 12- 3 = 9

10. Example : Find the range & Coefficient of Range for the following data :  Class Frequency 1-5 2 6-10 8 11-15 15 16-20 35 21-25 20 26-30 10 In case of inclusive series, first it should be converted into exclusive series, in the example the lowest limit is 0.5 and the highest limit 30.5 Range R =L –S = 30.5-0.5 or 30 = 30 Coefficient of Range C. Of R . = L –S = 30.5-0.5 = 30 L+ S = 30.5+0.5 31 C Of R = 0.97

11. Merits Simple to understand Easy to calculate Widely used in statistical quality control Demerits Can’t be calculated in open ended distributions Not based on all the observations Affected by sampling fluctuations Affected by extreme values Merits and Demerits of Range

12. Interquartile Range is the difference between the upper quartile (Q3) and the lower quartile ( Q1). It covers dispersion of middle 50% of the items of the series Formula Interquartile Range = Q3 – Q1 Definition Quartile Deviation is half of the interquartile range. It is also called Semi Interquartile Range Formula Quartile Deviation = 𝑄3 −𝑄1 2 Coefficient of Quartile Deviation: It is the relative measure of quartile deviation. Coefficient of Q.D. = 𝑄3 −𝑄 1 𝑄 3 +𝑄1 Interquartile Range & Quartile Deviation Definition

13. Example: Find quartile deviation and coefficient of quartile deviation : Central Size 1 2 3 4 5 6 7 8 9 10 Frequency 2 9 11 14 20 24 20 16 5 2 Central Size Class Interval Frequency Cum. f 1 0.5-1.5 2 2 2 1.5-2.5 9 11 3 2.5-3.5 11 22 4 3.5-4.5 14 36 5 4.5-5.5 20 56 6 5.5-6.5 24 80 7 6.5-7.5 20 100 8 7.5-8.5 16 116 9 8.5-9.5 5 121 10 9.5-10.5 2 123 N= 123 Q₁ =Size of N th item 4 Q₁ =Size of 123 th item=30.75 th item 4 Q ₁ =3.5 + 1 (30.75-22) =4.125 14 Q ₃ =Size of 3 N th item 4 Q₃ =Size of 3x123 th item=92.25 h item 4 Q ₃ =6.5 + 1 (92.25-80) =7.1125 20 C of Q.D= Q₃ -Q ₁ = 0.266 Q₃ +Q ₁

14. Merits Simple to understand Minimum effect of extreme values Dispersion of middle part Demerits Formation of quartiles of two series cannot be studied by this method Not based on all the values of variable Affected by sampling fluctuations It is not suitable for further algebraic treatment Merits and Demerits of Quartile Deviation

15. Mean Deviation is the arithmetic average of deviations of all the values taken from a measure of central tendency (Mean, Mode or Median) of the series. In taking deviations of values, algebraic signs + and – are not taken into consideration. . M.D. from Median = Σ | 𝑋 −𝑀| or Σ | 𝑑ₘ | 𝑁 𝑁 M.D . from Mean = Σ | 𝑋 −𝑋̅| or Σ | 𝑑ₓ | 𝑁 𝑁 Coefficient of M.D . ₘ = 𝑀.𝐷 .ₘ 𝑀𝑒𝑑𝑖𝑎𝑛 Coefficient of M.D . ₓ = 𝑀.𝐷. ₓ 𝑀𝑒𝑎𝑛 Mean Deviation (M.D.) Definition

16. Example : Calculate M.D. from Mean & Median & coefficient of Mean Deviation from the following data : Individual series Direct Method Find out average Take deviations of given values from median( any other average) ignoring algebraic signs |d| There deviations are aggregated Ʃ |d | Apply formula: δₘ = Ʃ | d ₘ | ; δ ₓ = Ʃ | d ₓ | N N Weight Deviation from M (50) Ʃ |d ₘ Ignoring signs Deviations from X (52) Ʃ |d ₓ | 45 5 7 47 3 5 47 3 5 49 1 3 50 2 53 3 1 58 8 6 59 9 7 60 10 8 468 42 44 Ʃ X Ʃ |d ₘ Ʃ |d ₓ | MD from Median Median =size of (N + 1) th item 2 = 5 th item = 50 δₘ = Ʃ |d ₘ | = 42 or 4.67 N 9 C of δ ₘ = δ ₘ or 4.67 =.0934 M 50 MD from Mean _ X = Ʃ X = 468 = 52 N 9 δ ₓ = Ʃ |d ₓ | = 44 or 4.89 N 9 C of δ ₓ = δ ₓ or 4.89 =. 0940 X 52

17. Example : Calculate M.D. from Mean & Median & coefficient of Mean Deviation from the following data : Discrete series Direct Method Find out Mean Take deviations of given values from median( any other average) ignoring algebraic signs |d| There deviations are aggregated Ʃ |d |, multiplying by respective frequencies Apply formula: δₘ = Ʃf | d ₘ | ; δ ₓ = Ʃ f|d ₓ | N N Size 4 6 8 10 12 14 16 Frequency 2 4 5 3 2 1 4

18. Department of Economics Size frequency C.f f x X Deviation from median Deviation from mean (9.71) Median of signs Total deviation Ignoring + and - Total deviation X F C.f fxX |d ₘ | f x |d ₘ | |d ₓ | fx |d ₓ | 4 2 2 8 4 8 5.71 11.42 6 4 6 24 2 8 3.71 14.84 8 5 11 40 1.71 8.55 10 3 14 30 2 6 0.29 0.87 12 2 16 24 4 8 2.29 4.58 14 1 17 14 6 6 4.29 4.29 16 4 21 64 8 32 6.29 25.16 Total 21 204 68 69.71 N Ʃ fx Ʃf |d ₘ | Ʃ f|d ₓ| MD from Median Median =size of (N + 1) th item 2 = 11 th item = 8 δₘ = Ʃ f|d ₘ | = 68 or 3.24 N 21 C of δ ₘ = δ ₘ or 3.24 = 0.405 M 8 MD from Mean _ X = ƩfX = 204 = 9.71 N 21 δ ₓ = Ʃf |d ₓ | = 69.21 or 3.32 N 21 C of δ ₓ = δ ₓ or 3.32 =0.342 X 9.71 18

19. Department of Economics Example : Calculate M.D. from Mean & Median & coefficient of Mean Deviation from the following data : Continous series Direct Method Find the mid value Find out Mean Take deviations of given values from median( any other average) ignoring algebraic signs |d| There deviations are aggregated Ʃ |d |, multiplying by respective frequencies Apply formula: δₘ = Ʃf | d ₘ | ; δ ₓ = Ʃ f|d ₓ | N N Marks 5-15 15-25 25-35 35-45 45-55 55-65 65-75 75-85 85-95 Total Frequency 3 8 15 20 25 10 9 6 4 100 19

20. Department of Economics MD from Mean _ X = ƩfX = 4760 = 47.6 N 100 δ ₓ = Ʃf |d ₓ | = 1499.2 or 14.99 N 100 C of δ ₓ = δ ₓ or 14.99 = 0.314 X 47.6 MD from Median Median =size of (N ) th item 2 = 50 th item = (45-55) M = l + i (m –c) = 45 + 10 (50-46) f 25 = 46.6 δₘ = Ʃ f|d ₘ | = 1507.2 or 15.07 N 100 C of δₘ = δₘ or 15.07 = 0.323 M 46.6 Mid-point frequency C.f f x X Deviation from median (46.6) Deviation from mean (47.6) Median of signs Total deviation Ignoring + and - Total deviation X F C.f fxX |d ₘ | f x |d ₘ | |d ₓ | fx |d ₓ | 10 3 3 30 36.6 109.8 37.6 112.8 20 8 11 160 26.6 212.8 27.6 220.8 30 15 26 450 16.6 249.0 17.6 264.0 40 20 46 800 6.6 132.0 7.6 152.0 50 25 71 1250 3.4 85.0 2.4 60.0 60 10 81 600 13.4 134.0 12.4 124.0 70 9 90 630 23.4 210.6 22.4 201.6 80 6 96 480 33.4 200.4 32.4 194.4 90 4 100 360 43.4 173.6 42.4 169.6 Total 100 4760 1507.2 1499.2 N Ʃ fx Ʃf |d ₘ | Ʃ f|d ₓ| 20

21. Merits Simple to understand Easy to compute Less effected by extreme items Useful in fields like Economics, Commerce etc. Comparisons about formation of different series can be easily made as deviations are taken from a central value Demerits Ignoring ‘±’ signs are not appropriate Not accurate for Mode Difficult to calculate if value of Mean or Median comes in fractions Not capable of further algebraic treatment Not used in statistical conclusions. Merits and Demerits of Mean Deviation

22. Unit End Questions Find out Range of the following values- 20,8,10,0,-20,10,4 2. Calculate coefficient of Quartile deviation from the following – Class 0-10 10-20 20-30 30-40 40-50 f 4 15 28 16 7 3 . Calculate coefficient of Mean deviation from the following – Class 0-10 0-20 0-30 0-40 0-50 f 12 13 28 29 50

23. Required Readings References https ://www.google.com/url?sa=i&url=http%3A%2F%2Fmakemeanalyst.com%2Fexplore-your-data-range-interquartile-range-and-box-plot%2F&psig=AOvVaw3hXiW_vSzIxwJXOf_OLgNw&ust=1598435199938000&source=images&cd=vfe&ved=0CAIQjRxqFwoTCMDU8LGJtusCFQAAAAAdAAAAABAD https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.colourbox.com%2Fimage%2Fpen-and-calculator-on-the-financial-newspaper-image-2257139&psig=AOvVaw0juutpJyjRq7qYZMbMZF8O&ust=1598436637417000&source=images&cd=vfe&ved=0CAIQjRxqFwoTCIDimtKOtusCFQAAAAAdAAAAABAD 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

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