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QMS 102 Measures of Variability The local Bank has been monitoring the daily average waiting times for its customers. The manager believed that there would be an improvement in the waiting times by having 1 permanent clerk and 2 floating clerks instead of the current 2 permanent clerks. The floating clerks would do other tasks except when the ques became excessive and then would switch an open other wickets. Last week the manager implemented the new system. Old Policy: 4, 6, 12, 15, 18, 4, 4, 4, 14, 21, 0, 7, 8, 9, 18, 6, 6, 12, 22, 25 New Policy: 1, 3, 9, 14, 16 Question: Were waiting times more variable using the "Old Policy" or the "New Policy"? QMS 102 Measures of Variability Since the shapes of the distributions can be expected to be different, we will not be able to answer the question be comparing the distributions of the 2 processes visually. QMS 102 Measures of Variability Old Policy: New Policy: 4, 6, 12, 15, 18, 4, 4, 4, 14, 21, 0, 7, 8, 9, 18, 6, 6, 12, 22, 25 1, 3, 9, 14, 16 A more precise comparison can be accomplished by choosing a criteria to assign a single number variability measure to each group! We will then compare the measure to answer the question. ie Criteria: Use the highest value to represent the variability of the group. Since 25>16 , old procedure is more variable. What is wrong with this criteria ? QMS 102 Measures of Variability Old Policy: New Policy: 4, 6, 12, 15, 18, 4, 4, 4, 14, 21, 0, 7, 8, 9, 18, 6, 6, 12, 22, 25 1, 3, 9, 14, 16 A more precise comparison can be accomplished by choosing a criteria to assign a single variability measure to each group! We will then compare the variability measures to answer the question. ie Criteria: Use the highest value to represent the variability of the group. Since 25>16 , old procedure is more variable. What is wrong with this criteria ? Excessive influenced of analysts choices of sample sizes {A bigger sample will have more significant extremes} Insensitive to the existence and magnitude of most values. {Note how adding or moving values usually has no effect} QMS 102 Measures of Variability Range xmax- xmin , distance on the real line required to hold the data Interquartile Range Q3 - Q1 , distance on the real line required to hold the middle 50% {Mean Absolute Deviation} Population Variance Population Standard Deviation Population Coefficient of Variation Others: Sample Variance Sample Standard Deviation Sample Coefficient of Variation QMS 102 Measures of Variability Old Policy: New Policy: 4, 6, 12, 15, 18, 4, 4, 4, 14, 21, 0, 7, 8, 9, 18, 6, 6, 12, 22, 25 1, 3, 9, 14, 16 Range Old Policy = Largest - Smallest = 25 - 0 = 25 min Casio [Me nu][List] List5 "4,6 ,12,....” [Menu][Stat][Calc] [Set] Xlist:“List1” Xfreq “1” [Exit][1-Var] xmin =0 xmax= 25 Range New Policy = ? 16-1=15 QMS 102 Measures of Variability Interquartile Range Old Policy: New Policy: IQR Old Policy 4, 6, 12, 15, 18, 4, 4, 4, 14, 21, 0, 7, 8, 9, 18, 6, 6, 12, 22, 25 1, 3, 9, 14, 16 = Q3 - Q1 = 16.5 - 5 = 11.5 min Casio [Me nu][List] List5 "4,6 ,12,....” [Menu][Stat][Calc] [Set] Xlist:“List1” Xfreq “1” [Exit][1-Var] Q1= 5 Q3= 16 .5 [Menu][Run][Var][Stat][Grph] [Q3] - [Q1][EXE] 11 .5 IQR New Policy = ? 15-2=13 QMS 102 Measures of Variability Old Policy: New Policy: Mean Absolute Deviation 4, 6, 12, 15, 18, 4, 4, 4, 14, 21, 0, 7, 8, 9, 18, 6, 6, 12, 22, 25 1, 3, 9, 14, 16 Mean Absolute Deviation Old Policy = MAD Old Policy = Mean Absolute Deviation New Policy = 5.28 (WILL NOT BE ON TESTS) QMS 102 Measures of Variability Example Data A: 1, -1, 1, -1 Which Population is More Variable? B: 2, 0, -2, 0 QMS 102 Measures of Variability Example continued: Data A: 1, -1, 1, -1 Mean of 0, all “deviations” are 1. MADA = ( |1-0| + |1-0| + |1-0| + |1-0| ) / 4 = 1 Data B: 2, 0, -2, 0 Mean of 0, 2 deviations of 0, 2 - deviations of 2. MADB = ( |2-0| + |0-0| + |2-0| + |0-0| ) / 4 = 1 MAD is not sensitivity enough to the data at the "edges".{large deviation}! How can we emphasize the edge values? Squaring deviations does the job! .1 1 10 Some deviations |---|-----|------------------|----------------------------------------------------------------------.01 1 Squared deviations 100 ||--------|----------------------------------------------------------------------------------/\/\-----| QMS 102 Measures of Variability Old Policy: New Policy: 4, 6, 12, 15, 18, 4, 4, 4, 14, 21, 0, 7, 8, 9, 18, 6, 6, 12, 22, 25 1, 3, 9, 14, 16 Population Variance Old Policy = ( minutes squared Population Variance New Policy = 5.892 = 34.64 Population Variance )2 = 6.93 2= 48.09 Old Policy Casio [Menu][List] List5 "4,6 ,12,....” [Menu][Stat][Calc] [Set] Xlist:“List1” Xfreq “1” [Exit][1-Var] x n= 6.93 [Menu][Run][Var][Stat][X] [x n][x2][EXE] 48.09 2 New Policy = ? QMS 102 Measures of Variability Population Standard Deviation Casio [Me nu][List] List5 "4,6 ,12,....” [Menu][Stat][Calc] [Set] Xlist:“List1” Xfreq “1” [Exit][1-Var] x n= 6.93 This estimation is not usually this accurate. Population Standard Deviation New Policy = ? 5.89 QMS 102 Population Variance / Standard Deviation Demonstration of the edge effect! Assume that a set of data has a Mean of 0 and the set includes 1 and 10. Move 1 ---> 2 this increases the numerator of the variance calculation by ? Instead , move 10 --->11 How does the numerator change? Variance and standard deviation are much more sensitive to values far from the mean! QMS102 Measures of Variability Which one should we use? Criteria Properties Range Insensitive to the existence and magnitude of most values. Based only on location of the two extremes. Easy to calculate. IQR Robust. Totally insensitive to magnitude of the outer 50%. {MAD} Equally sensitive to the existence and magnitude of all values. Easy to understand.} Variance Much more sensitive to the existence and magnitude of extreme values than central values. Calculation supported. Unusual units. Standard Deviation Same as Variance but nice units.