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1. 1. NewGate India Hyderbad, Andhra Pradesh- 500038 Website: www.newgate.in Email: contact@newgate.in Slideshare URL : http://www.slideshare.net/newgateindia Business Statistics Descriptive Statistical Analysis ForLocation of students of PGP Jan-09 students 1
2. 2. CONTENTS:I. Questions: ………………………………………………….3II. Solution: Calculation1. Data Collection………………………………………………32. Data Classification to intervals………………………………53. DATA GENERATED through MS EXCEL…………………………64. Sum, Count & Range………………………………………...65. Mean…………………………………………………………76. Variance & Standard Deviation…………………………….7 6.1 Standard Deviation………………………………7 6.2 Variance…………………………………………77. Median………………………………………………..……..78. Quartile……………………………………………………...89. Inter Quartile Range…………………………………………910. Upper limit & Lower Limit……………………………1011. Mode……………………………………………………1112. Skewness……………………………………………….11III. Analysis:13. Central tendencies & Dispersion……………………....12 13.1 Central tendency………………………………...12 13.2 Dispersion……………………………………….13 13.3 Coefficient of variation…………………………..1414. Mode…………………………………………………...1415. Box & Whisker’s Plot………………………...……..…15 15.1 Outliers…………….……………………………15 15.2 Evidence of skewness………………………..…16 Appendix – 1……………………………………………17 Appendix – 2……………………………………………18 Appendix – 3……………………………………………19 Bibliography……………………………………………..20 2
3. 3. 1. __________________________________________________________________________ I.QUESTION __________________________________________________________________________ Collect information on any variable for a group of 30 members. Write a report summarizing those data, including the following activities. a. Calculate appropriate measures of central tendency and dispersion. b. Do these data have a mode? c. Draw a box-and-whisker plot. Are there any outliers? Does the plot show any evidence of skewness? __________________________________________________________________________ II. Solution: Calculation __________________________________________________________________________ 1.DATA COLLECTION: A survey was done for a sample of 30 students from PGP JAN 2010 batch of Alliance Business school. Survey was done on the basis of the places to which each one of them belonged. The distance of each of the places were calculated with reference to Bangalore. SAMPLE : 30 Students POPULATION : 52 Students NOTE: The distance of students from Bangalore were recorded as 0 Km as all distance were measured from Bangalore. The scale value for all 30 students were collected and quoted as follows. Sources to calculate distance : http://maps.google.co.in/ 3
4. 4. COLLECTED DATA TABLE 1 Distance inS.No Names Location Km 1 Abhigna Vijaywada 658 2 Adil Bangalore 0 3 Akshay jaipur 1855 4 Anupriya Lucknow 1879 5 Bhavya Kodagui 206 6 Chandreep Kolkata 1929 7 Deepak Bokaro 1981 8 Girish Bangalore 0 9 Irfan kolkata 192910 Kiran Alleppy 44111 Kreetika Bangalore 012 Laxman Jamshedpur 142013 Mohit Delhi 207914 Mridul Patna 202215 Neha Shimla 244416 Pradeeep Mangalore 29817 Prajuktri Kolkata 192918 Prashant Chennai 32719 Priyank Agra 187920 Rama Chennai 32721 Rishab Delhi 207922 Ritesh Jamshedpur 142023 Ritu Agartala 330424 satish Chennai 32725 Shruti Bangalore 026 Soumya kanpur 178327 Sunam Rourkela 170828 Tanay Bhopal 146629 yeshwant Guntur 61930 Yetin Chennai 327 4
5. 5. 2. DATA GENERATED through MS EXCELData analysis was done for descriptive statistics through MS-EXCEL. All the above 30data were taken into account while producing the below report. MS-EXCEL REPORT 1 Mean 1221.2 Median 1443 Mode 0 Standard Deviation 915.7679423 Sample Variance 838630.9241 Sample Variance 838630.9241 Skewness 0.117333504 Range 3304 Minimum 0 Maximum 3304 Sum 36636 Count 30 Largest(1) 3304 Smallest(1) 0 Confidence Level(95.0%) 341.9533643 Q1(First Quartile) 327 Q2(Second Quartile) 1443 Q3(Third Quartile) 1929 IQR 1602 Upper Limit 4332 Lower Limit -2076 Lower Limit -2076 5
6. 6. 3. DATA CLASSIFIED TO CLASS INTERVAL To have better understanding of the behavior of the large number of sample data. We have categorized the collected data to ordinal values within the class intervals. CLASSIFIED CLASS INTERVAL TABLE : 2 X ( in Km) f 0-500 10 500-1000 2 1000-1500 3 1500-2000 9 2000-2500 5 2500-3000 0 3000-3500 14. SUM , COUNT & RANGEARRANGEMENT ( SORTING OF DATA) :First fall of all variable data were arranged from top to bottom in their increasing order Xmid X ( in km) ( in Km) f Xmid X f 0-500 250 10 2500 500-1000 750 2 1500 1000-1500 1250 3 3750 1500-2000 1750 9 15750 2000-2500 2250 5 11250 2500-3000 2750 0 0 3000-3500 3250 1 3250 Total 30 38000 SUM : 38000 COUNT : n = 30 6
7. 7. Range : Range = Maximum value - Minimum Value Max = 3304 Min = 0 Range = 3304_________________________________________________________________________________________ 5. MEAN: X = Σ xi ( i = 1,2,3………n ) n X= { ( 250 x 10) + ( 750 x 2) + ( 1250 x 3) + ( 1750 x 9) + ( 250 x 10) + ( 2250 x 5) + ( 2750 x 0) + ( 3250 x 1) }/30 = 38000 / 30 X = 1266.66 ________________________________________________________________________________________ 6. VARIANCE & STANDARD DEVIATION: 6.1 Standard deviation S = Σ|x–x| n–1 S= [ | ( 250 – 1266.66 ) | x 10 + | ( 750 – 1266.66 ) | x 2 + | ( 1250 – 1266.66 ) | x 3 + | ( 1750 – 1266.66 ) | x 9 + | ( 250 – 1266.66 ) | x 10 + | ( 2250 – 1266.66 )| x 5 + | ( 2750 – 1266.66 )| x 0 + | ( 3250 – 1266.66 )| x 1 ] /30 S = 759.31 6.2 VARIANCE : S = √ Variance Variance = S X S  Variance = 576552.2 7
8. 8. _________________________________________________________________________________________ 7. MEDIAN: Median = L + N/2 – C X i f  L: Lower limit of the class interval  N : Number of observations  ,f : Frequency of particular observation  , i: Width of the class interval  C : Cumulative frequency of previous observation N = 30 N/2 =15 N =15 C = 12 Cumulative f= 3 x f i=500 f 0-500 10 10 L=1000 500-1000 2 12 1000-1500 3 15 Q2 ( Median) 1500-2000 9 24 = 1000 + [ { (15-12) /3 } X 500 ] 2000-2500 5 29 2500-3000 0 29 Q2 = 1500 3000-3500 1 30 Total 30 30_________________________________________________________________________________________ 8
9. 9. 8. QUARTILE: Quartiles Q1, Q2, Q3 are the percentile values dividing the whole samples of data into 4 four equal quadrant.  Q1 = first Quartile or 25th Percentile  Q2( Median) = Second Quartile or 50th Percentile  Q3 = Third Quartile or 75th Percentile Q1 = L + N/4 – C X i f N = 30 N/4 =7.5 N =15 Cumulative C=0 x f f= 10 f i=500 0-500 10 10 L=0 500-1000 2 121000-1500 3 15 Q1 ( Median)1500-2000 9 242000-2500 5 29 = 0 + [ { (7.5-0) /10 } X 500 ]2500-3000 0 293000-3500 1 30 Q1 = 375 Total 30 30 9
10. 10. Q3 = L + 3N/4 – C X i f N = 30 3N/4 = 22.5 N =15 Cumulative C = 15 x f f f= 9 0-500 10 10 i=500 500-1000 2 12 L=1500 1000-1500 3 15 1500-2000 9 24 Q1 ( Median) 2000-2500 5 29 = 1500 + [{ (22.5-15) /9 } X 500 ] 2500-3000 0 29 3000-3500 1 30 Q3 = 1916.66 Total 30 30 Q1 = 375 Q2 = 1500 Q3 = 1916.66 _________________________________________________________________________________________ 9. INTER QUARTILE RANGE ( IQR) IQR = Q3 – Q1 IQR = 19166.66 -376  IQR = 541.66__________________________________________________________________________ 10
11. 11. 10. UPPER LIMIT & LOWER LIMIT Upper Limit U = Q3 + 1.5 IQR Lower Limit L = Q1 - 1.5 IQR U = 1916.66 + 1.5 x 541.66 = 4228.49 L = 375 – 1.5 x 541.66 = -1937.49__________________________________________________________________________ 11. MODE: Mode is the maximum frequency of particular data for a given variable. x Xmid f 0-500 250 10 Frequency is maximum at interval 500-1000 750 2 0-500 that is 10 times 1000-1500 1250 3 1500-2000 1750 9 2000-2500 2250 5 Mode = 250 2500-3000 2750 0 3000-3500 3250 1 Total 30__________________________________________________________________________ 12. Skweness: Skp = {3 ( Mean – Median )} / S Skp = [ 3 ( 1266.66 -1500)] / 759.31 Skp = - 0.9219__________________________________________________________________________ 11
12. 12. III. ANALYSIS: .13. Appropriate measures of central tendency and dispersion 13.1 Central Tendency: Mean,Median,Q1,Q3 Mean 1266.66 Km Median 1500 Km 1500 1400 1300 1200 1100 Mean Median  Observation:  It can inferred that students of PGP Jan-10 batch travel an average distance of 1266.66 Km to come to Alliance Business School.  50 % of the observations lies above 1500 km Q1 ( First Quartile) 375 Km Q3 ( Third Quartile) 1916.66 Km 2500 2000 1500 1000 500 0 Q1 Q2 Q3 12
13. 13.  Observation:  25 % of observation lies below 375 Km  50% of observation lies between 375 Km to 1916.66 Km  25% of observation lies above 1916.66 Km13.2 Dispersion: Standard Deviation, Sample Variance,Range,IQR,Coefficient Standard Deviation 915.7679423 Km Sample Variance 838630.9241 Km Observation:  The central tendency predicted to summarize the whole sample can deviate/differ from its mean at by an average value of 915.78 ( more or less ) Mean + Deviation = 2182.44 Km Mean – Deviation = 350.88 Km  It will deviate/differ over a range of 2 X 915.78 = 1831.56 Km  It will deviate/differ at an average from lower most value 350.88 Km to higher most value 350.88 Km IQR Range Standard Deviation 0 500 1000 1500 2000 2500 3000 3500 Range 3304 Km IQR 1602 Km Observation:  The student’s hometown are spread up over a coverage of 3304 Km  50 % of the observations lies in the coverage area of 1602 Km 13
14. 14. 13.3 Coefficient of variation: Coefficient of variation = Standard Deviation/Mean X 100 = 915.7679423/1266.66 = 72.23 %_________________________________________________________________________________________ 14.Maximum Observed data ( Mode)  Observation:  The maximum observed interval were 0 Km to 500km  Mode is its mid value (0 + 500) / 2 = 250 Km f Total 3000-3500 2500-3000 Axis Title 2000-2500 1500-2000 1000-1500 f 500-1000 0-500 0 5 10 15 20 25 30 35 Axis Title_________________________________________________________________________________________ 14
15. 15. 15. Box-and-whisker plot Q1 375 327 Km Q2 1500 1443 Km Q3 1916.66 1929 Km IQR 1541.66 1602 Km Upper Limit 4228.49 4332 Km Lower Limit -1937.49 -2076 Km 15
16. 16. 15.1 Outliers : UpperLimit,Lower Limit Upper Limit 4228.49 Km Lower Limit -1937.49 Km  Max value : 3304 Km < upper limit:4228.49 Km  Min Value : 0 Km < lower limit : -1937.49 Km Hence, No outliers  Observation:  All samples data should be within 4228.49 Km to escape being outliers.  The maximum sample date is of 3304 Km < 4228.49 Km.So there is no outliers 5000 4000 3000 2000 1000 0 Upper Limit Lower Limit -1000 -2000 -3000 15.2 .Evidence Of Skweness Skewness -0.9219  Observation:  It is negatively skewed as mean < median  The skewness is -0.9219  In the box plot the median do not lie exactly in between Q1 and Q2.________________________________________________________________________________________ 16
17. 17. APPENDIX – 1 COLLECTED DATA TABLE 1 Distance inS.No Names Location Km 1 Abhigna Vijaywada 658 2 Adil Bangalore 0 3 Akshay jaipur 1855 4 Anupriya Lucknow 1879 5 Bhavya Kodagui 206 6 Chandreep Kolkata 1929 7 Deepak Bokaro 1981 8 Girish Bangalore 0 9 Irfan kolkata 192910 Kiran Alleppy 44111 Kreetika Bangalore 012 Laxman Jamshedpur 142013 Mohit Delhi 207914 Mridul Patna 202215 Neha Shimla 244416 Pradeeep Mangalore 29817 Prajuktri Kolkata 192918 Prashant Chennai 32719 Priyank Agra 187920 Rama Chennai 32721 Rishab Delhi 207922 Ritesh Jamshedpur 142023 Ritu Agartala 330424 satish Chennai 32725 Shruti Bangalore 026 Soumya kanpur 178327 Sunam Rourkela 170828 Tanay Bhopal 146629 yeshwant Guntur 61930 Yetin Chennai 327 17
18. 18. APPENDIX – 2 CLASSIFIED CLASS INTERVAL TABLE : 2 XmidX (in Km) f cumulative f Xmid X f ( in Km) 0-500 250 10 10 2500500-1000 750 2 12 15001000-1500 1250 3 15 37501500-2000 1750 9 24 157502000-2500 2250 5 29 112502500-3000 2750 0 29 03000-3500 3250 1 30 3250 Total 30 30 38000 MS-EXCEL Vs CALCULATED REPORT Simulated By Calculated Value Ms-excel Mean 1266.66 1221.2 Mode 250 0 Median 1500 1443 Standard 759.31 915.7679423 Deviation Sample variance 576552.2 838630.9241 Skewness -0.9219 0.117333504 Range 3304 3304 Minimum 0 0 Maximum 3304 3304 Sum 38000 36636 Count 30 30 Largest(1) 3304 3304 Smallest(1) 0 0 Q1 375 327 Q2 1500 1443 Q3 1916.66 1929 IQR 1541.66 1602 Upper Limit 4228.49 4332 Lower Limit -1937.49 -2076 18
19. 19. APPENDIX – 3 MS-EXCEL Vs CALCULATED REPORT3500300025002000 calculated value1500 Ms-Excel value1000 500 0 Mean Median Standard Range Deviation 5000 4000 3000 2000 Axis Title Calculated data 1000 MS-Excel Value 0 Q1 Q2 Q3 IQR Upper Lower Limit Limit -1000 -2000 -3000 Axis Title 19
20. 20. BIBLIOGRAPHY Statistics of Business and Economics( Anderson,Sweeney,Williams,Cenage Learning,9th Edition) www.wikipdeia.org www.stats4u.com www.maps.google.co.in 20