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# Statistics-Measures of dispersions

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### Statistics-Measures of dispersions

1. 1. MEASURES OF DISPERSIONS STATISTICS
2. 2. MEASURES OF DISPERSIONS • A quantity that measures the variability among the data, or how the data one dispersed about the average, known as Measures of dispersion, scatter, or variations.
3. 3. 2. Common Measures of Dispersion • The main measures of dispersion 1. Range 2. Mean deviation or the average deviation 3. The variance & the standard deviation
4. 4. 1. RANGE • It is the difference between the largest and the smallest observation in a set of data. • Range = xm – xo • Its relative measure known as coefficient of dispersion. • Coefficient of dispersion = • It is used in daily temperature recording stick prices rate • It ignores all the information available in middle of data. • It might give a misleading picture of the spread of data. om om xx xx + −
5. 5. 1. RANGE • Example: 1. Find the range in the following data. 31,26,15,43,19,10,12,37 Range = xm – xo 33 = 43 – 10 2. Find the range in the following F.D. (Ungrouped) 5 = 8 – 3 Range 5 = 8 – 3 3. Find the range in the following data. Range = 60 – 10 = 50 X 3 4 5 6 7 8 f 5 8 12 10 4 2 X 10 - 20 20 - 30 30- 40 40 – 50 50 - 60 f 5 8 12 10 4
6. 6. MEAN (OR AVERAGE) DEVIATION • It is defined as the “Arithmetic mean of the absolute deviation measured either from the mean or median. • or for ungroup. • or for grouped. n xx DM ∑ − =.. N xxf∑ − = N medianx∑ − N medianxf∑ − =
7. 7. MEAN (OR AVERAGE) DEVIATION • Example: 1. Calculate mean deviation from the FD (Ungrouped Data). MD (x) = 33.6 / 20 = 1.68 X f f.x I x – 4.9 I f I x - 4.9 I 2 3 6 2.9 8.7 4 9 36 0.9 8.1 6 5 30 1.1 5.5 8 2 16 3.1 6.2 10 1 10 5.1 5.1 Total Σf =20 Σf.x =98 Σ f I x - 4.9 I = 33.6
8. 8. MEAN (OR AVERAGE) DEVIATION • Exp: Calculate mean deviation from the FD (Grouped Data). MD (x) = 33.6 / 20 = 1.68 M.D = 23.72 / 14 = 1.69 X f Class Mark ( x ) f.x I x – 6.57 I f I x – 6.57 I 2 – 4 2 3 6 3.57 7.14 4 - 6 3 5 15 1.57 4.71 6 – 8 6 7 42 0.43 2.58 8 – 10 2 9 18 2.43 4.86 10 – 12 1 11 11 4.43 4.43 Total Σf =14 Σ f.x =92 Σ f I x – 6.57 I = 23.72 =92/14=6.57ẋ
9. 9. • It is an absolute measure. • It is relative measure is coefficient of M.D. • Coefficient of M.D. = • It is based on all the observed values. MEAN (OR AVERAGE) DEVIATION median DM or mean DM ....
10. 10. • EXAMPLES MEAN (OR AVERAGE) DEVIATION
11. 11. THE VARIANCE AND STANDARD DEVIATION • It is defined as “The mean of the squares of deviations of all the observation from their mean.” It’s square root is called “standard deviation”. • Usually it is denoted by (for population of statistics) S2 (for sample) • = for ungrouped 2 σ 2 σ n xx∑ − 2 )(
12. 12. • = for grouped • It is an absolute measure; • It is relative measure is coefficient of variation. • • Shortcut method N xxf∑ − 2 )(2 σ 100. ×= µ σ VC 100 .. .. ×= x DS VC 22 2         −= ∑∑ N x N x σ 22 2 .         −= ∑∑ N fx N xf σ THE VARIANCE AND STANDARD DEVIATION
13. 13. • EXAMPLES THE VARIANCE AND STANDARD DEVIATION
14. 14. VARIANCE AND STANDARD DEVIATION• Example: 1. Calculate Variance and SD from the FD (Ungrouped Data). Using Short cut method var = (564 / 20) - (98 / 20) ^ 2 = 28.2 – 24.01 = 4.09 Sd = √ σ^2 = √ 4.09 = 2.02 X f f.x X^2 f.x^2 2 3 6 4 12 4 9 36 16 144 6 5 30 36 180 8 2 16 64 128 10 1 10 100 100 Total Σf =20 Σf.x = 98 Σ f.x^2=564 22 2 .         −= ∑∑ N fx N xf σ
15. 15. VARIANCE AND STANDARD DEVIATION • Exp: Calculate Variance and Standard deviation from the FD (Grouped Data). Using Short cut method: var = (670 /14) - (92 / 14) ^ 2 = 47.85 – 43.18 = 4.67 Sd = √ σ^2 = √ 4.67 = 2.16 X f Class Mark ( x ) f.x x^2 f.x^2 2 – 4 2 3 6 9 18 4 - 6 3 5 15 25 75 6 – 8 6 7 42 49 294 8 – 10 2 9 18 81 162 10 – 12 1 11 11 121 121 Total Σf =14 Σ f.x =92 Σ f.x^2 =670 22 2 .         −= ∑∑ N fx N xf σ
16. 16. 16 Relative Measures ofRelative Measures of DispersionDispersion  Coefficient of Range  Coefficient of Quartile Deviation  Coefficient of Mean Deviation  Coefficient of Variation (CV) 01:38 PM
17. 17. 17 Relative Measures of VariationRelative Measures of Variation Largest Smallest Largest Smallest Coefficient of Range X X X X − = + 3 1 3 1 Coefficient of Quartile Deviation Q Q Q Q − = + Coefficient of Mean Deviation MD Mean = 01:38 PM
18. 18. Coefficient of Variation (CV)Coefficient of Variation (CV) Can be used to compare two or more sets of data measured in different units or same units but different average size. 01:38 PM 18 100% X S CV ⋅        =
19. 19. 19 Use of Coefficient of VariationUse of Coefficient of Variation Stock A: Average price last year = \$50 Standard deviation = \$5 Stock B: Average price last year = \$100 Standard deviation = \$5 but stock B is less variable relative to its price 10%100% \$50 \$5 100% X S CVA =⋅=⋅        = 5%100% \$100 \$5 100% X S CVB =⋅=⋅        = Both stocks have the same standard deviation 01:38 PM
20. 20. 20 Five Number SummaryFive Number Summary The five number summary of a data set consists of the minimum value, the first quartile, the second quartile, the third quartile and the maximum value written in that order: Min, Q1, Q2, Q3, Max. From the three quartiles we can obtain a measure of central tendency (the median, Q2) and measures of variation of the two middle quarters of the distribution, Q2-Q1 for the second quarter and Q3-Q2 for the third quarter. 01:38 PM
21. 21. 21 The weekly TV viewing times (in hours). 25 41 27 32 43 66 35 31 15 5 34 26 32 38 16 30 38 30 20 21 The array of the above data is given below: 5 15 16 20 21 25 26 27 30 30 31 32 32 34 35 37 38 41 43 66 Five Number SummaryFive Number Summary 01:38 PM
22. 22. 22 Hrs22.021}-0.25{2521obs.}5th-obs.0.25{6thobs.5th;Q1ofVALUE obs.5.25thdatain theobs.th 4 1)1(20 ;Q1ofLOCATION =+=+ = + Five Number SummaryFive Number Summary Hrs30.530}-0.50{3103obs.}10th-obs.0.50{11thobs.th10;Q2ofVALUE obs.th50.10datain theobs.th 4 1)2(20 ;2QofLOCATION =+=+ = + Minimum value=5.0 Maximum value=66.0 Hrs36.535}-0.75{3735obs}15th-obs{16th75.0obs15th;3QofVALUE obs.15.75thdatain theobs.th 4 1)3(20 ;3QofLOCATION =+=+ = + 01:38 PM
23. 23. 23 Box and Whisker DiagramBox and Whisker Diagram A box and whisker diagram or box-plot is a graphical mean for displaying the five number summary of a set of data. In a box-plot the first quartile is placed at the lower hinge and the third quartile is placed at the upper hinge. The median is placed in between these two hinges. The two lines emanating from the box are called whiskers. The box and whisker diagram was introduced by Professor Jhon W. Tukey. 01:38 PM
24. 24. 24 Construction of Box-PlotConstruction of Box-Plot 1. Start the box from Q1 and end at Q3 2. Within the box draw a line to represent Q2 3. Draw lower whisker to Min. Value up to Q1 4. Draw upper Whisker from Q3 up to Max. Value Q1 Q3 Q2 01:38 PM Max Value Min Value
25. 25. 25 Construction of Box-PlotConstruction of Box-Plot 1. Q1=22.0 Q3=36.5 2. Q2=30.5 3. Minimum Value=5.0 4. Maximum Value=66.0 70 60 50 40 30 20 10 0 01:38 PM
26. 26. 26 Interpretation of Box-PlotInterpretation of Box-Plot 70 60 50 40 30 20 10 0 Box-Whisker Plot is useful to identify •Maximum and Minimum Values in the data •Median of the data •IQR=Q3-Q1, Lengthy box indicates more variability in the data •Shape of the data From Position of line within box Line At the center of the box----Symmetrical Line above center of the box----Negatively skewed Line below center of the box----Positively Skewed •Detection of Outliers in the data 01:38 PM
27. 27. 27 OutliersOutliers An outlier is the values that falls well outside the overall pattern of the data. It might be • the result of a measurement or recording error, • a member from a different population, • simply an unusual extreme value. An extreme value needs not to be an outliers; it might, instead, be an indication of skewness. 01:38 PM
28. 28. 28 Inner and Outer FencesInner and Outer Fences If Q1=22.0 Q2=30.5 Q3=36.5 ( ) ( )   =+= =−= 25.58IQR1.5QFenceInnerUpper 25.0IQR1.5QFenceInnerLower :FencesInner 3 1 ( ) ( )   =+= −=−= 0.80IQR3QFenceOuterUpper 5.21IQR3QFenceOuterLower :FencesOuter 3 1 01:38 PM Lower Inner Fence 22-1.5(36.5-22) = 0.25 Upper Inner Fence 36+1.5(36.5-22) = 58.25 Lower Outer Fence 22-3(36.5-22) = -21.5 Upper Outer Fence 36+3(36.5-22) = 80.0
29. 29. 29 Identification of the OutliersIdentification of the Outliers 1. The values that lie within inner fences are normal values 2. The values that lie outside inner fences but inside outer fences are possible/suspected/mild outliers 3. The values that lie outside outer fences are sure outliers 80 70 60 50 40 30 20 10 0 Plot each suspected outliers with an asterisk and each sure outliers with an hollow dot. * Only 66 is a mild outlier 01:38 PM
30. 30. 30 Box plots are especially suitable for comparing two or more data sets. In such a situation the box plots are constructed on the same scale. Uses of Box and Whisker DiagramUses of Box and Whisker Diagram Male Female 01:38 PM
31. 31. Standardized VariableStandardized Variable A variable that has mean “0” and Variance “1” is called standardized variable Values of standardized variable are called standard scores Values of standard variable i.e standard scores are unit-less Construction VariableofDeviationStandard VariableofMeanVariable Z − = 01:38 PM 31
32. 32. X Z 3 25 -1.3624 1.85611.8561 6 4 -0.5450 0.29700.2970 11 9 0.81741 0.66820.6682 12 16 1.0899 1.18791.1879 32 54 0 4.009 5.13 4 54 8 4 32 2 == === ∑ xS n X X 2 )( XX − 67.3 8− = − = X Sx XX Z 1 4 009.4 0 2 ≅= == ∑ zS n Z Z 2 )( ZZ − Variable Z has mean “0” and variance “1” so Z is a standard variable. Standard Score at X=11 is 8174.0 67.3 811 = − = − = Sx XX Z 01:38 PM Standardized VariableStandardized Variable
33. 33. 33 The industry in which sales rep Mr. Atif works has mean annual sales=\$2,500 standard deviation=\$500. The industry in which sales rep Mr. Asad works has mean annual sales=\$4,800 standard deviation=\$600. Last year Mr. Atif’s sales were \$4,000 and Mr. Asad’s sales were \$6,000. Performance evaluation by z-scoresPerformance evaluation by z-scores Which of the representatives would you hire if you have one sales position to fill? 01:38 PM
34. 34. 34 Performance evaluation by z-scoresPerformance evaluation by z-scores 3 500 500,2000,4 = − = − = B B BB B Z S XX Z Sales rep. Atif XB= \$2,500 SΒ= \$500 XB= \$4,000 Sales rep. Asad XP =\$4,800 SP = \$600 XP= \$6,000 2 600 800,4000,6 = − = − = P P PP P Z S XX Z Mr. Atif is the best choice 01:38 PM
35. 35. 35 valuesof68%aboutcontains1SX ± The Empirical RuleThe Empirical Rule X 68% 1SX ± valuesof99.7%aboutcontains3SX ± valuesof95%aboutcontains2SX ± 95% X 2S± X 3S± 99.7% 01:38 PM
36. 36. Chebysev’s TheoremChebysev’s Theorem
37. 37. 37 A distribution in which the values equidistant from the centre have equal frequencies is defined to be symmetrical and any departure from symmetry is called skewness. 1. Length of Right Tail = Length of Left Tail 2. Mean = Median = Mode 3. Sk=0 a) Sk=(Mean-Mode)/SD b) Sk=(Q3-2Q2+Q1)/(Q3-Q1) 01:38 PM Measures of Skewness
38. 38. 38 A distribution is positively skewed, if the observations tend to concentrate more at the lower end of the possible values of the variable than the upper end. A positively skewed frequency curve has a longer tail on the right hand side 1. Length of Right Tail > Length of Left Tail 2. Mean > Median > Mode 3. SK>0 a) Sk=(Mean-Mode)/SD b) Sk=(Q3-2Q2+Q1)/(Q3-Q1) MeasuresMeasures ofof SkewnessSkewness 01:38 PM
39. 39. 39 A distribution is negatively skewed, if the observations tend to concentrate more at the upper end of the possible values of the variable than the lower end. A negatively skewed frequency curve has a longer tail on the left side. 1. Length of Right Tail < Length of Left Tail 2. Mean < Median < Mode 3. SK< 0 a) Sk=(Mean-Mode)/SD b) Sk=(Q3-2Q2+Q1)/(Q3-Q1) 01:38 PM Measures of Skewness
40. 40. 01:38 PM 40 The Kurtosis is the degree of peakedness or flatness of a unimodal (single humped) distribution, • When the values of a variable are highly concentrated around the mode, the peak of the curve becomes relatively high; the curve is Leptokurtic. • When the values of a variable have low concentration around the mode, the peak of the curve becomes relatively flat;curve is Platykurtic. • A curve, which is neither very peaked nor very flat-toped, it is taken as a basis for comparison, is called Mesokurtic/Normal. Measures of Kurtosis
41. 41. 4101:38 PM Measures of Kurtosis
42. 42. 42 Measures of Kurtosis 1. If Coefficient of Kurtosis > 3 ----------------- Leptokurtic. 2. If Coefficient of Kurtosis = 3 ----------------- Mesokurtic. 3. If Coefficient of Kurtosis < 3 ----------------- is Platykurtic. ( ) ( ) 4 22 n X-X Coefficient of Kurtosis= X-X    ∑ ∑ 01:38 PM
43. 43. Moments about origin C.I f x f.x f. x^2 f.x^3 f.x^4 2.5 2.9 2 2.7 5.4 14.58 39.366 106.2882 3 3.4 7 3.2 22.4 71.68 229.376 734.0032 3.5 3.9 17 3.7 62.9 232.73 861.101 3186.074 4 4.4 25 4.2 105 441 1852.2 7779.24 4.5 4.9 20 4.7 94 441.8 2076.46 9759.362 5 5.4 12 5.2 62.4 324.48 1687.296 8773.939 5.5 5.9 9 5.7 51.3 292.41 1666.737 9500.401 6 6.4 8 6.2 49.6 307.52 1906.624 11821.07 Total (Σ) 100 453 2126.2 10319.16 51660.38 01:38 PM 43
44. 44. Moments about origin 01:38 PM 44
45. 45. Regression X Price Y (Quantity Demanded) x*y x^2 15 440 6600 225 20 430 8600 400 25 450 11250 625 30 370 11100 900 40 340 13600 1600 50 370 18500 2500 Σx=180 Σy=2400 Σx.y=69650 Σx^2=6250 01:38 PM 45
46. 46. 01:38 PM 46