statistics assignment for BBA students, Assignments of Applied Statistics

Download statistics assignment for BBA students and more Assignments Applied Statistics in PDF only on Docsity! UNIVERSITY OF CENTRAL PUNJAB BUSINESS STATISITCS ASSIGNMENT 2 AMEER HAMZA (L1F18BBAM0371) SECTION E SIR ZAHID AHMAD COMPREHENSIVE NOTE ON MEASURE OF DISPERSION Measure of dispersion: It is known as extent to which observations vary about their mean or other average. There are 2 measures of dispersion. An absolute measure of dispersion and relative measure of dispersion. An absolute measure of dispersion is one that measures the dispersion in terms of the same units or in the square of units as the unit of data. Example: Rupees, meters etc. A relative measure of dispersion is one that is expressed in the form ratio or percentage etc. it is useful for comparison of data of different nature. Types of dispersion:  Range  Quartile deviation  Mean deviation  Variance and standard deviation. Range It is difference between maximum value and minimum value of the data given. R=Xm−Xo Where, X m=maximumvalue X o=minimum value Example: R=Xm−Xo R=10−4 R=6 Merits of Range:  Easy to calculate and very simple to understand ¿ Ʃf 4 = 50 4 =12.5  Q 1=44.5+ 5 8 (12.5−5) Q 1=¿ 49.19  Q 3=59.5+ 5 9 (3× 50 4 −34) Q 3=61.4 Q .D= Q 3−Q1 2 Q .D= 61.4−49.19 2 Q .D=6.105 Co-efficient of Quartile Deviation:  Q .D= (Q 3−Q1 ) (Q3+Q 1 ) Q .D= (61.4−49.19 ) (61.4+49.19 ) Q .D=0.1104 Merits of Quartile Deviations:  It is easy to understand and compute.  It is not affected by extreme values.  It is useful measure when it is desired to know the variability in the central half of data. Demerits of Quartile Deviation:  It ignores 50% of data.  It does not show scatter from any particular average.  It is not suitable to algebraic treatment. Mean Deviation It is defined as arithmetic mean of the deviations measured from mean. All deviations are being counted as positive. Mean deviation gives best results when deviations are taken from the median. While computing the value of mean deviation, positive and negative signs are ignored and all values are treated as positive. Deviations can be taken from any of the three averages but in actual practice mean deviation is calculated either from mean or median. The latter is supposed to be better than the former, because the sum of the deviations from the median is less than the sum of deviations from the mean. Formula: Ungrouped data Grouped data Ʃ∨X−X́∨¿ n ¿ Ʃ f∨X− X́∨ ¿ Ʃ f ¿ Find Mean deviation: FOR UNGROUPED DATA 20, 26, 35, 42, 54, 67, 71 x |x -X́ | 20 25 26 19 35 10 42 3 54 9 67 22 71 26 X́ = ƩX n ¿ 315 7 = 45 M.D = Ʃ |X−X́| n = 114 7 = 16.28 FOR GROUPED DATA Class frequency x fx |x -X́ | f |x -X́ | 10-14 9 12 108 -12 108 15-19 10 17 170 -7 70 20-24 17 22 374 -2 34 25-29 10 27 270 3 30 30-34 5 32 160 8 40 35-39 4 37 148 13 52 40-45 5 42 210 18 90 X́= Ʃfx Ʃf = 252.75 Standard Deviation Standard deviation shows the variation of data. If the data is close together, the standard deviation will be small. If the data is spread out, the standard deviation will be large. It is often denoted by the lowercase Greek letter Sigma, σ. Formula: Ungrouped data Grouped data S .D=√ Ʃ ( x− x́ ) 2 n S .D=√ Ʃf (x− x́) 2 Ʃf Take example above of variance question. We just have to put simple square root of on the answer of variance. For Ungrouped data. √ Ʃ ( x− x́ ) 2 n =√ 6665 =√133.2=11.54 For Grouped data. √ Ʃf (x− x́) 2 Ʃf =√ 505520 =√252.75=15.89 Merits of Standard Deviation:  Its value is based on all the observations of a series  It is rigidly defined  It is capable of further algebraic manipulation  It is less affected by the fluctuation of sampling Demerits of Standard Deviation:  It is affected by extreme value like mean deviation.  Standard deviation is not easy to understand.  It is not easy to calculate. Relative dispersions  Co-efficient of Q.D Q3−Q1 Q3+Q1  Co-efficient of M.D M . D X́  Co-efficient of variation S . D X́ MOMENTS: A moment designates the power to which deviations are raised before averaging them. Ungroup data Grouped data m1 = Ʃ (X− X́) n m1 = Ʃf (X−X́ ) Ʃf m2 =Ʃ ¿¿ m2=Ʃf ¿¿ m3 =Ʃ ¿¿ m3=Ʃf ¿¿ m4 =Ʃ ¿¿ m4=Ʃf ¿¿ SKEWNESS Lack of symmetry is called as skewness. Formulas for skewness: 1. sk= m3 2 m2 3 (σby moments) 2. Karl Pearson first formula: sk= Mean−mode standard deviation 3. Karl Pearson second formula: sk= 3(Mean−median) standard deviation 4. Bowley Formula: