University:
Pennsylvania Western University, California
Course:
STAT 2010 | Statistics for Business
Academic year:

2016
Views:

395

Pages:

36
Author:

Yaretzi Harrison

* Measure of Dispersion * Meaning & Need- * Definition- "Deviation of different values of data from its mean is known as dispersion" There may be several series whose averages may be same but which may differ from each other in many ways. Consider the following data. Batsman A C Test I 100 105 298 Test II 100 100 00 Test III 100 95 02 Total 300 300 300 Mean 100 100 100 Here we see that the average runs of 3 batsman is same. But we can say that A is better than B&C since the figures of A are closer to mean. There for it is important to find some measure which measures this variation & express it in single figure.Such measures are called as measures of dispersion/Variation. * Types of Dispersion- There are following two types of dispersion: 1) Absolute Measures of dispersion 2) Relative Measures of dispersion 1) Absolute Measures of Dispersion- The measure of dispersion which are expressed in terms of the original units of data are called as absolute measures which are used to measure variability. Absolute Measures of Dispersion 1. Range (R) 2. Quartile Deviation (QD) 3. Mean Deviation (MD) about mean, median, mode 4. Standard Deviation (SD) 5. Variance (σ²) Relative Measures of Dispersion "The measure of dispersions which are not expressed in terms of original units are called as relative measures" Which are pure number free from units & are used for comparison. 1. coefficient of Range 2. coeff. quartile deviation 3. coeff. Mean Deviation 4. coeff. Standard Deviation 5. coeff. Variance/Variation * Range (CR) Range is defined as the difference between largest observation & smallest observation of the data. It is denoted by (R). R= Largest obs - smallest obs. R= L-S coeff. of Range = L-S L+S * Note: 1. For Ungrouped data, the largest & smallest value of x are used. 2. For Grouped data, we take lower limit of first class as smallest value & upper limit of last class as largest value. * Example: Calculate Range & its Relative measures for the following data. 20, 21, 25, 22, 30, 32, 37, 65, 47 Here, S=20 L=65 ∴ R=L-S =65-20 ∴ R=45 Now, coeff. of R = L-S / L+S = 45 / 65+20 = 45 / 85 = 0.52 L = 68 S = 62 Range = L-S = 68-62 = 6 Coeff. of Range = L-S / L+S = 6 / 68+62 = 6 / 130 ∴ Coeff. of Range = 0.04 3. The following table gives the age distribution of 50 individuals. Calculate Range & coeff. of Range. Age in Years No. of Persons Class Boundaries 16-20 10 15.5-20.5 21-25 15 20.5-25.5 26-30 17 25.5-30.5 31-35 8 30.5-35.5 L = 35.5 S = 15.5 R = L-S = 35.5 - 15.5 = 20 Coeff. of Range = L-S / L+S = 20 / 35.5 + 15.5 = 20 / 51 = 0.39 Quartile Deviation (QD) = Semi Inter Quartile Range Quartile deviation (QD) is half of difference between upper quartile & lower quartile. QD = (Q₃ - Q₁) / 2 Q₃ = Upper quartile. Q₁ = Lower quartile. coeff. QD = (Q₃ - Q₁) / (Q₃ + Q₁) + Note - The difference (Q₃ - Q₁) is called inter quartile Range. & is also called as semi interquartile range. Examples: Calculate the quartile deviation & its coefficient for following data. 12 12 16 16 18 18 20 20 30 30 40 40 44 46 46 50 50 56 62 62 n=11 Q₁ = size of (n+1)/4th obs. Q₁ = size of (11+1)/4th obs. Q₁ = size of (12)/4th obs. Q₁ = size of (3)rd obs. Q₁ = 16 Q₃ = size of 3(n+1)/4th obs. Q₃ = size of 3(11+1)/4th obs. Q₃ = size of 3(12)/4th obs. Q₃ = size of 3(3)rd obs. Q₃ = size of (9)th obs. Q₃ = 46 QD = (Q₃ - Q₁) / 2 = (46 - 16) / 2 = 30 / 2 = 15 coeff. of QD = (Q₃ - Q₁) / (Q₃ + Q₁) = (46 - 16) / (46 + 16) = 30 / 62 = 0.48 2. Find Range & QD for following data: x Ass.x 125 70 70 100 156 125 100 156 190 173 173 190 250 250 Largest value = 250 Smallest value = 70 Range = L-S = 250-70 = 180 Q₁ = size of (n+1)/4th obs. Q₁ = size of (7+1)/4th obs. Q₁ = size of (8)th obs. Q₁ = size of (2)nd obs. = 100 Q₃ = Size of 3(n+1)/4th obs. Q₃ = size of 3(7+1)/4th obs. Q₃ = size of 3(8)th obs. Q₃ = size of 3(2)nd obs. Q₃ = size of (6)th obs. = 190 QD = (Q₃ - Q₁) / 2 = (190 - 100) / 2 = 90 / 2 = 45 coeff. of QD = (Q₃ - Q₁) / (Q₃ + Q₁) = (190 - 100) / (190 + 100) = 90 / 290 = 0.31 coeff. of QD = 0.31 3. Value f LCF 10 6 6 20 9 15 30 15 30 (79/4) 40 28 58 50 12 70 (79/3*4) 60 6 76 70 3 79 N = 79 Largest value = 70 Smallest value = 10 Range = L-S = 70-10 = 60 R = 60 Q₁ = 1*N/4 = 79/4 = 19.75 Q₁ = 30 Q₃ = 3*N/4 = 3*79/4 = 3*19.75 = 59.25 Q₃ = 50 QD = (Q₃ - Q₁) / 2 = (50 - 30) / 2 = 20 / 2 = 10 QD = 10 coeff. of QD = (Q₃ - Q₁) / (Q₃ + Q₁) = (50 - 30) / (50 + 30) = 20 / 80 = 0.25 coeff. of QD = 0.25 Marks No. of students LCF 20 2 2 18 4 6 16 9 15 14 18 33 (79/4) 12 27 60 10 25 85 (79/3*4) 8 14 99 6 1 N = 100 Largest value = 20 Smallest value = 6 Range = L-S = 20-6 = 14 Range = 14 Q₁ = 1*N/4 = 100/4 = 25 Q₃ = 3*N/4 = 3*100/4 = 3*25 = 75 Q₃ = 10 QD = (Q₃ - Q₁) / 2 = (10 - 14) / 2 = -4 / 2 = -2 QD = -2 coeff. of QD = (Q₃ - Q₁) / (Q₃ + Q₁) = (10 - 14) / (10 + 14) = -4 / 24 = -0.16 coeff. of QD = -0.16 5) Class No. of Persons LCF 20-25 16 24 25-30 8 32 30-35 36 68 (77/14) 35-40 20 88 40-45 15 103 (77/3*4) 45-50 5 120 50-55 5 125 N = 125 Largest ob value=55 Smallest value = 20 Range = L-S = 55-20 = 35 R = 35 Q₁ = 1*N/4 = 125/4 = 31.25 Q₃ = 3*N/4 = 3*125/4 = 3*31.25 = 93.75 Q₃ = 93.75 Q₁ = 1 + (N/4 - cf)*h Q₁ = 30 + (31.25 - 24)*5 Q₁ = 30 + 7.25*5 Q₁ = 30 + 36.25 Q₁ = 30 + 1.81 Q₁ = 31.81