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A Six Sigma Analysis of Mobile Data Usage

The document presents a six sigma analysis of mobile data usage from March 2015 to March 2016, focusing on various mobile carriers and their data plans. It includes statistical evaluations such as Monte Carlo simulations, process capability assessments, and hypothesis tests, revealing significant overage charges among customers of major carriers. The analysis also addresses the appropriateness of data limits across different plans and offers insights into monthly and daily data usage trends.

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A Six Sigma Analysis of Mobile Data Usage 2016 WCQI Session W10 Brandon Theiss, PE Brandon.Theiss@gmail.com
Motivation Is my current mobile data plan with Republic Wireless Optimal Given my data usage?
Learning Objectives • Apply the Six Sigma Methodology to Non Traditional Applications • Utilize Monte Carlo simulations to make predictions • Utilize Non Parametric Hypothesis testing • Utilize Process Capability to determine specification limitations for non-normal data
4 Major Mobile Phone Carriers
Plans Offered By Verizon 20% of Verizon customers charged overages in past year*
Plans Offered By AT&T 28% of AT&T customers charged overages in past year*
Plans Offered By T-Mobile 12% of T-Mobile customers charged overages in past year*
5% of Sprint customers charged overages in past year* Plans Offered By Sprint
Plans Offered By Republic Wireless
121110987654321 12000 10000 8000 6000 4000 2000 0 Bill Number TotalUsage 1000 2000 3000 5000 6000 10000 12000 3095.80 1911.60 3203.80 2674.30 3224.90 4517.40 4846.80 5905.40 3039.10 3784.20 4612.40 4254.40 Chart of Total Data Usage The Data Set Data was collected from March 23, 2015 Through March 24, 2016
121110987654321 $130 $120 $110 $100 $90 $80 $70 $60 $50 $40 Bill Number BilledAmmount Verizon (1GB) ATT (2GB) T-Mobile (2GB) Sprint (1GB) Republic (2GB) Variable Time Series Plot of Small Verizon, ATT, T-Mobile, Sprint, Republic Comparison of Carriers Small Data Plans Data Speed Potentially Decreased
121110987654321 $120 $110 $100 $90 $80 $70 $60 $50 Bill Number BilledAmmount Verizon (3GB) ATT (2GB) T-Mobile (2GB) Sprint (3GB) Republic (3GB) Variable Time Series Plot of MediumVerizon, ATT, T-Mobile, Sprint, Republic Comparison of Carriers Medium Data Plans Data Speed Potentially Decreased
121110987654321 $90 $85 $80 $75 $70 $65 Bill Number BilledAmmount Verizon (6GB) ATT (5GB) T-Mobile (6GB) Sprint (6GB) Republic (5GB) Variable Time Series Plot of Large Verizon, ATT, T-Mobile, Sprint, Republic Comparison of Carriers Large Data Plans
Comparison of Carriers X-Large Data Plans 121110987654321 140 120 100 80 60 40 20 0 Index Data Verizon (12GB) ATT (15GB) T-Mobile (10GB) Sprint (12GB) Republic (Not Offered) Variable Time Series Plot of XL Verizon, ATT, T-Mobile, Sprint, Republic
ATT (15G B) Verizon (12G B) Verizon (1G B) Sprint (1G B) ATT (2G B) Republic (5G B) Verizon (3G B) Sprint (12G B) T-M obile (10G B) Verizon (6G B) ATT (5G B) Sprint (3G B) Sprint (6G B) T-M obile (6G B) Republic (3G B) T-M obile (2G B) Republic (2G B) $ 1,600.00 $ 1,400.00 $ 1,200.00 $ 1,000.00 $ 800.00 $ 600.00 $ 400.00 $ 200.00 $ 0.00 Plan Annual Chart of Annual Cost How Much Would Each Plan have cost for the Year?
1st Quartile 3053.3 Median 3504.6 3rd Quartile 4588.6 Maximum 5905.4 3052.5 4459.2 3054.0 4587.4 784.2 1879.6 A-Squared 0.25 P-Value 0.687 Mean 3755.8 StDev 1107.0 Variance 1225527.2 Skewness 0.314666 Kurtosis -0.123559 N 12 Minimum 1911.6 Anderson-Darling Normality Test 95% Confidence Interval for Mean 95% Confidence Interval for Median 95% Confidence Interval for StDev 60005000400030002000 Median Mean 4500425040003750350032503000 95% Confidence Intervals Summary Report for Total Monthly Usage A First Statistical Approach (monthly data)
800070006000500040003000200010000 99 95 80 50 20 5 1 Total Usage Percent Goodness of Fit Test Normal AD = 0.248 P-Value = 0.687 Probability Plot for Total Usage Normal - 95% CI Is The Data Normally Distributed?
121110987654321 6000 4000 2000 Observation IndividualValue _ X=3756 UCL=6424 LCL=1088 121110987654321 3000 2000 1000 0 Observation MovingRange __ MR=1003 UCL=3278 LCL=0 I-MR Chart of Total Monthly Usage Is The Data Is Statistical Control?
600050004000300020001000 LSL * Target * USL 1000 Sample Mean 3755.84 Sample N 12 StDev(Overall) 1107.04 StDev(Within) 889.313 Process Data Pp * PPL * PPU -0.83 Ppk -0.83 Cpm * Cp * CPL * CPU -1.03 Cpk -1.03 Potential (Within) Capability Overall Capability % < LSL * * * % > USL 100.00 99.36 99.90 % Total 100.00 99.36 99.90 Observed Expected Overall Expected Within Performance USL Overall Within Process Capability Report for Total Usage (1GB) Plan Annual Cost ATT (2GB) $ 1,065.00 Sprint (1GB) $ 1,065.00 Is a 1GB (1,000MB) Limit Appropriate?
60005000400030002000 LSL * Target * USL 2000 Sample Mean 3755.84 Sample N 12 StDev(Overall) 1107.04 StDev(Within) 889.313 Process Data Pp * PPL * PPU -0.53 Ppk -0.53 Cpm * Cp * CPL * CPU -0.66 Cpk -0.66 Potential (Within) Capability Overall Capability % < LSL * * * % > USL 91.67 94.36 97.58 % Total 91.67 94.36 97.58 Observed Expected Overall Expected Within Performance USL Overall Within Process Capability Report for Total Usage (2GB) Is a 2GB (2,000MB) Limit Appropriate? Plan Annual Cost Republic (2GB) $ 480.00 T-Mobile (2GB) $ 600.00 ATT(2GB) $ 1,065.00
60005000400030002000 LSL * Target * USL 3000 Sample Mean 3755.84 Sample N 12 StDev(Overall) 1107.04 StDev(Within) 889.313 Process Data Pp * PPL * PPU -0.23 Ppk -0.23 Cpm * Cp * CPL * CPU -0.28 Cpk -0.28 Potential (Within) Capability Overall Capability % < LSL * * * % > USL 83.33 75.26 80.23 % Total 83.33 75.26 80.23 Observed Expected Overall Expected Within Performance USL Overall Within Process Capability Report for Total Usage (3GB) Plan Annual Cost Republic (3GB) $ 660.00 Sprint (3GB) $ 840.00 Verizon (3GB) $ 1,020.00 Is a 3GB (3,000MB) Limit Appropriate?
60005000400030002000 LSL * Target * USL 5000 Sample Mean 3755.84 Sample N 12 StDev(Overall) 1107.04 StDev(Within) 889.313 Process Data Pp * PPL * PPU 0.37 Ppk 0.37 Cpm * Cp * CPL * CPU 0.47 Cpk 0.47 Potential (Within) Capability Overall Capability % < LSL * * * % > USL 8.33 13.05 8.09 % Total 8.33 13.05 8.09 Observed Expected Overall Expected Within Performance USL Overall Within Process Capability Report for Total Usage (5GB) Plan Annual Cost ATT (5GB) $ 915.00 Republic (5GB) $ 1,020.00 ATT (5GB) $ 1,500.00 Is a 5GB (5,000MB) Limit Appropriate?
60005000400030002000 LSL * Target * USL 6000 Sample Mean 3755.84 Sample N 12 StDev(Overall) 1107.04 StDev(Within) 889.313 Process Data Pp * PPL * PPU 0.68 Ppk 0.68 Cpm * Cp * CPL * CPU 0.84 Cpk 0.84 Potential (Within) Capability Overall Capability % < LSL * * * % > USL 0.00 2.13 0.58 % Total 0.00 2.13 0.58 Observed Expected Overall Expected Within Performance USL Overall Within Process Capability Report for Total Usage (6GB) Plan Annual Cost T-Mobile (6GB) $ 780.00 Sprint (6GB) $ 780.00 Verizon (6GB) $ 960.00 Is a 6GB (6,000MB) Limit Appropriate?
900075006000450030001500 LSL * Target * USL 10000 Sample Mean 3755.84 Sample N 12 StDev(Overall) 1107.04 StDev(Within) 889.313 Process Data Pp * PPL * PPU 1.88 Ppk 1.88 Cpm * Cp * CPL * CPU 2.34 Cpk 2.34 Potential (Within) Capability Overall Capability % < LSL * * * % > USL 0.00 0.00 0.00 % Total 0.00 0.00 0.00 Observed Expected Overall Expected Within Performance USL Overall Within Process Capability Report for Total Usage (10GB) Plan Annual Cost T-Mobile (10GB) $ 960.00 Is a 10GB (10,000MB) Limit Appropriate? ~6 Sigma !
1200010500900075006000450030001500 LSL * Target * USL 12000 Sample Mean 3755.84 Sample N 12 StDev(Overall) 1107.04 StDev(Within) 889.313 Process Data Pp * PPL * PPU 2.48 Ppk 2.48 Cpm * Cp * CPL * CPU 3.09 Cpk 3.09 Potential (Within) Capability Overall Capability % < LSL * * * % > USL 0.00 0.00 0.00 % Total 0.00 0.00 0.00 Observed Expected Overall Expected Within Performance USL Overall Within Process Capability Report for Total Usage (12GB) Plan Annual Cost Sprint (12GB) $ 960.00 Verizon (12GB) $ 1200.00 Is a 12GB (12,000MB) Limit Appropriate? Greater than 6 Sigma!
2/19/2016 1/13/2016 12/7/2015 10/31/2015 9/24/2015 8/18/2015 7/12/2015 6/5/2015 4/29/2015 3/24/2015 1200 1000 800 600 400 200 0 Date DataUsage Time Series Plot of Data Usage A Second Statistical Approach (daily data)
1st Quartile 69.13 Median 96.70 3rd Quartile 138.00 Maximum 1100.00 112.59 133.69 88.25 102.97 95.71 110.67 A-Squared 27.78 P-Value <0.005 Mean 123.14 StDev 102.64 Variance 10535.65 Skewness 3.9407 Kurtosis 26.1682 N 366 Minimum 0.00 Anderson-Darling Normality Test 95% Confidence Interval for Mean 95% Confidence Interval for Median 95% Confidence Interval for StDev 10008006004002000 Median Mean 14013012011010090 95% Confidence Intervals Summary Report for Data Usage Descriptive Statistics On Daily Usage
12008004000 99.9 99 90 50 10 1 0.1 Data Usage Percent 10000 1000 100101 0.1 0.01 0.001 0.0001 99.9 99 90 50 10 1 0.1 Data Usage Percent 100010010 99.9 99 90 50 10 1 0.1 Data Usage - Threshold Percent 20-2-4 99.9 99 90 50 10 1 0.1 Data Usage Percent 3-Parameter Loglogistic AD = 1.975 P-Value = * Johnson Transformation AD = 0.171 P-Value = 0.932 Goodness of Fit Test Logistic AD = 13.251 P-Value < 0.005 Loglogistic AD = 9.501 P-Value < 0.005 After Johnson transformation Probability Plot for Data Usage Logistic - 95% CI Loglogistic - 95% CI 3-Parameter Loglogistic - 95% CI Normal - 95% CI If The Data Is Not Normal What Approximates The Data?
12008004000 99.9 99 90 50 10 1 0.1 N 366 AD 27.776 P-Value <0.005 Percent 20-2-4 99.9 99 90 50 10 1 0.1 N 366 AD 0.171 P-Value 0.932 Percent 1.21.00.80.60.40.2 0.8 0.6 0.4 0.2 0.0 Z Value P-ValueforADtest 0.38 Ref P P-Value for Best Fit: 0.931848 Z for Best Fit: 0.38 Best Transformation Type: SU Transformation function equals -0.996951+ 0.885314 × Asinh( ( X - 59.1002 ) / 25.8392 ) Probability Plot for Original Data Probability Plot for Transformed Data Select a Transformation (P-Value = 0.005 means ≤ 0.005) Johnson Transformation for Data Usage The Johnson Transformation of the Data
111098754321 3.0 1.5 0.0 -1.5 -3.0 Billing Cycle IndividualValue _ X=-0.003 UCL=2.430 LCL=-2.436 111098754321 4 3 2 1 0 Billing Cycle MovingRange __ MR=0.915 UCL=2.989 LCL=0 1 1 1 1 1 1 1 1 1 1 I-MR Chart of Transformed Data Usage Is the Data In Statistical Control?
121110987654321 1200 1000 800 600 400 200 0 Billing Cycle DataUsage 106.752 61.6645 103.34889.1433104.029 150.58156.348 190.497 101.303122.071 153.747137.239 Boxplot of Data Usage A Third Statistical Approach
10005000 99.9 99 90 50 10 1 0.1 Residual Percent 20015010050 1000 750 500 250 0 Fitted Value Residual 9007506004503001500 120 90 60 30 0 Residual Frequency 350300250200150100501 1000 750 500 250 0 Observation Order Residual Normal Probability Plot Versus Fits Histogram Versus Order Residual Plots for Data Usage Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Billing Cycle 11 429109 39010 4.04 0.000 Error 354 3416405 9651 Total 365 3845514 Model Summary S R-sq R-sq(adj) R-sq(pred) 98.2388 11.16% 8.40% 4.99% Method Null hypothesis All means are equal Alternative hypothesis At least one mean is different Significance level α = 0.05 Equal variances were assumed for the analysis. Factor Information Factor Levels Values Billing Cycle 12 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 One-way ANOVA: Data Usage versus Billing Cycle Is There Statistically Significant Difference Between The Months?
But ANOVA Requires The Data to be Normal
Kruskal-Wallis Test: Data Usage versus Billing Cycle Kruskal-Wallis Test on Data Usage Billing Cycle N Median Ave Rank Z 1 31 108.80 217.0 1.84 2 30 130.50 249.8 3.58 3 31 88.40 187.4 0.21 4 30 85.60 160.9 -1.22 5 31 137.90 265.5 4.51 6 31 129.40 234.7 2.82 7 30 88.15 182.3 -0.07 8 31 93.80 187.9 0.24 9 30 75.70 135.9 -2.57 10 31 75.00 148.9 -1.90 11 31 62.50 86.3 -5.35 12 29 73.20 142.6 -2.17 Overall 366 183.5 H = 82.19 DF = 11 P = 0.000 H = 82.19 DF = 11 P = 0.000 (adjusted for ties) A First Non-Parametric Approach
20 10 0 1050 900 750 600 450 300 150 0 1050 900 750 600 450 300 150 0 20 10 0 1050 900 750 600 450 300 1500 20 10 0 1050 900 750 600 450 300 1500 1 Data Usage Frequency 2 3 4 5 6 7 8 9 10 11 12 Histogram of Data Usage Panel variable: Billing Cycle Kruskal-Wallis Test Requires The Distributions To Have Similar Shapes
Mood Median Test: Data Usage versus Billing Cycle Mood median test for Data Usage Chi-Square = 70.53 DF = 11 P = 0.000 Billing Individual 95.0% CIs Cycle N≤ N> Median Q3-Q1 --+---------+---------+--------- +---- 1 10 21 109 68 (*--) 2 4 26 131 45 (-*---) 3 17 14 88 59 (-*-----) 4 19 11 86 46 (-*-) 5 5 26 138 156 (-----*---------------) 6 8 23 129 81 (----*----) 7 16 14 88 78 (--*-----) 8 17 14 94 44 (-*-) 9 21 9 76 44 (-*--) 10 22 9 75 46 (-*-) 11 26 5 63 36 (-*-) 12 18 11 73 83 (--*-----) --+---------+---------+--------- +---- 60 120 180 240 Overall median = 97 A Second Non-Parametric Approach
A Fourth Statistical Approach
SaturdayFridayThursdayWednesdayTuesdayMondaySunday 1200 1000 800 600 400 200 0 Day Of Week DataUsage 124.35117.36125.612116.7687.934 127.687163.094 Boxplot of Data Usage A Fifth Statistical Approach (by days of the week)
20 10 0 10509007506004503001500 10509007506004503001500 20 10 0 10509007506004503001500 20 10 0 Sunday Data Usage Frequency Monday Tuesday Wednesday Thursday Friday Saturday Histogram of Data Usage Panel variable: Day Of Week What Do The Distributions Of Each Day Look Like?
SUNDAY
1st Quartile 74.32 Median 120.15 3rd Quartile 197.35 Maximum 1100.00 115.83 210.36 84.71 152.41 142.27 210.53 A-Squared 4.65 P-Value <0.005 Mean 163.09 StDev 169.77 Variance 28821.13 Skewness 3.7420 Kurtosis 18.3156 N 52 Minimum 0.00 Anderson-Darling Normality Test 95% Confidence Interval for Mean 95% Confidence Interval for Median 95% Confidence Interval for StDev 9607204802400 Median Mean 225200175150125100 95% Confidence Intervals Summary Report for Data Usage Sunday Descriptive Statistics
Weibull AD = 0.728 P-Value = 0.053 3-Parameter Weibull AD = 0.355 P-Value = 0.475 Goodness of Fit Test Exponential AD = 4.176 P-Value < 0.003 2-Parameter Exponential AD = 1.614 P-Value = 0.017 1000100101 99.9 90 50 10 1 Data Usage Percent 1000100101 99.9 90 50 10 1 Data Usage - Threshold Percent 100010010 99.9 90 50 10 1 Data Usage Percent 1000100101 99.9 90 50 10 1 Data Usage - Threshold Percent Probability Plot for Data Usage Exponential - 95% CI 2-Parameter Exponential - 95% CI Weibull - 95% CI 3-Parameter Weibull - 95% CI What Distribution Models Sunday?
9607204802400 40 30 20 10 0 Shape # 1.369 Scale # 125.2 Thresh # 26.73 N 50 Data Usage Frequency Histogram of Data Usage 3-Parameter Weibull # This estimated historical parameter is used in the calculations. A 3-Parameter Weibull Models Sunday Data Red Bars indicate outliers that were excluded from parameter determination
MONDAY
1st Quartile 67.92 Median 89.85 3rd Quartile 134.92 Maximum 619.30 94.34 161.03 78.71 108.15 100.37 148.52 A-Squared 5.23 P-Value <0.005 Mean 127.69 StDev 119.76 Variance 14342.75 Skewness 2.55234 Kurtosis 7.19780 N 52 Minimum 0.00 Anderson-Darling Normality Test 95% Confidence Interval for Mean 95% Confidence Interval for Median 95% Confidence Interval for StDev 6004803602401200 Median Mean 16014012010080 95% Confidence Intervals Summary Report for Data Usage Monday Descriptive Statistics
10001001010.10.010.0010.0001 90 50 10 1 Data Usage Percent 1000100101 90 50 10 1 Data Usage - Threshold Percent 10001001010.10.010.0010.0001 90 50 10 1 Data Usage Percent 10010 90 50 10 1 Data Usage - Threshold Percent Weibull AD = 2.383 P-Value < 0.010 3-Parameter Weibull AD = 0.398 P-Value = 0.342 Goodness of Fit Test Exponential AD = 6.080 P-Value < 0.003 2-Parameter Exponential AD = 6.124 P-Value < 0.010 Probability Plot for Data Usage Exponential - 95% CI 2-Parameter Exponential - 95% CI Weibull - 95% CI 3-Parameter Weibull - 95% CI What Distribution Models Monday?
600480360240120 35 30 25 20 15 10 5 0 Shape # 1.916 Scale # 74.12 Thresh # 29.30 N 48 Data Usage Frequency Histogram of Data Usage 3-Parameter Weibull # This estimated historical parameter is used in the calculations. A 3-Parameter Weibull Models Monday Data Red Bars indicate outliers that were excluded from parameter determination
TUESDAY
1st Quartile 61.250 Median 81.400 3rd Quartile 105.600 Maximum 289.700 75.526 100.342 72.217 89.345 37.785 55.699 A-Squared 1.76 P-Value <0.005 Mean 87.934 StDev 45.017 Variance 2026.544 Skewness 2.02797 Kurtosis 7.44336 N 53 Minimum 0.000 Anderson-Darling Normality Test 95% Confidence Interval for Mean 95% Confidence Interval for Median 95% Confidence Interval for StDev 300240180120600 Median Mean 100959085807570 95% Confidence Intervals Summary Report for Data Usage Tuesday Descriptive Statistics
Weibull AD = 0.382 P-Value > 0.250 3-Parameter Weibull AD = 0.203 P-Value > 0.500 Goodness of Fit Test Exponential AD = 10.303 P-Value < 0.003 2-Parameter Exponential AD = 3.239 P-Value < 0.010 1000100101 99.9 90 50 10 1 Data Usage Percent 10001001010.1 99.9 90 50 10 1 Data Usage - Threshold Percent 10010 99.9 90 50 10 1 Data Usage Percent 10010 99.9 90 50 10 1 Data Usage - Threshold Percent Probability Plot for Data Usage Exponential - 95% CI 2-Parameter Exponential - 95% CI Weibull - 95% CI 3-Parameter Weibull - 95% CI What Distribution Models Tuesday?
30024018012060 25 20 15 10 5 0 Shape # 1.882 Scale # 57.02 Thresh # 34.60 N 51 Data Usage Frequency Histogram of Data Usage 3-Parameter Weibull # This estimated historical parameter is used in the calculations. A 3-Parameter Weibull Models Tuesday Data Red Bars indicate outliers that were excluded from parameter determination
WEDNESDAY
1st Quartile 69.00 Median 97.10 3rd Quartile 154.00 Maximum 321.50 97.32 136.20 77.27 113.79 59.20 87.27 A-Squared 2.07 P-Value <0.005 Mean 116.76 StDev 70.53 Variance 4974.95 Skewness 1.10549 Kurtosis 0.67508 N 53 Minimum 0.00 Anderson-Darling Normality Test 95% Confidence Interval for Mean 95% Confidence Interval for Median 95% Confidence Interval for StDev 320240160800 Median Mean 1401301201101009080 95% Confidence Intervals Summary Report for Data Usage Wednesday Descriptive Statistics
Weibull AD = 1.186 P-Value < 0.010 3-Parameter Weibull AD = 0.618 P-Value = 0.113 Goodness of Fit Test Exponential AD = 5.427 P-Value < 0.003 2-Parameter Exponential AD = 2.310 P-Value < 0.010 1000100101 99.9 90 50 10 1 Data Usage Percent 1000100101 99.9 90 50 10 1 Data Usage - Threshold Percent 100010010 99.9 90 50 10 1 Data Usage Percent 1000100101 99.9 90 50 10 1 Data Usage - Threshold Percent Probability Plot for Data Usage Exponential - 95% CI 2-Parameter Exponential - 95% CI Weibull - 95% CI 3-Parameter Weibull - 95% CI What Distribution Models Wednesday?
3202802402001601208040 20 15 10 5 0 Shape 1.430 Scale 104.1 Thresh 24.66 N 52 Data Usage Frequency Histogram of Data Usage 3-Parameter Weibull A 3-Parameter Weibull Models Wednesday Data
THURSDAY
1st Quartile 74.90 Median 102.00 3rd Quartile 163.27 Maximum 449.50 105.25 145.97 81.45 134.75 61.29 90.69 A-Squared 1.99 P-Value <0.005 Mean 125.61 StDev 73.13 Variance 5347.80 Skewness 2.03570 Kurtosis 6.42894 N 52 Minimum 42.10 Anderson-Darling Normality Test 95% Confidence Interval for Mean 95% Confidence Interval for Median 95% Confidence Interval for StDev 400300200100 Median Mean 1401301201101009080 95% Confidence Intervals Summary Report for Data Usage Thursday Descriptive Statistics
Weibull AD = 0.904 P-Value = 0.020 3-Parameter Weibull AD = 0.324 P-Value > 0.500 Goodness of Fit Test Exponential AD = 6.944 P-Value < 0.003 2-Parameter Exponential AD = 1.454 P-Value = 0.025 1000100101 99.9 90 50 10 1 Data Usage Percent 1000100101 99.9 90 50 10 1 Data Usage - Threshold Percent 100010010 99.9 90 50 10 1 Data Usage Percent 1000100101 99.9 90 50 10 1 Data Usage - Threshold Percent Probability Plot for Data Usage Exponential - 95% CI 2-Parameter Exponential - 95% CI Weibull - 95% CI 3-Parameter Weibull - 95% CI What Distribution Models Thursday?
400300200100 25 20 15 10 5 0 Shape # 1.364 Scale # 85.54 Thresh # 40.89 N 52 Data Usage Frequency Histogram of Data Usage 3-Parameter Weibull # This estimated historical parameter is used in the calculations. A 3-Parameter Weibull Models Thursday Data Red Bars indicate outliers that were excluded from parameter determination
FRIDAY
1st Quartile 67.70 Median 100.95 3rd Quartile 122.95 Maximum 435.30 94.58 140.14 84.26 105.70 68.58 101.49 A-Squared 4.30 P-Value <0.005 Mean 117.36 StDev 81.84 Variance 6697.42 Skewness 2.21566 Kurtosis 5.35910 N 52 Minimum 10.70 Anderson-Darling Normality Test 95% Confidence Interval for Mean 95% Confidence Interval for Median 95% Confidence Interval for StDev 4003002001000 Median Mean 1401301201101009080 95% Confidence Intervals Summary Report for Data Usage Friday Descriptive Statistics
Weibull AD = 0.607 P-Value = 0.111 3-Parameter Weibull AD = 0.392 P-Value = 0.404 Goodness of Fit Test Exponential AD = 9.088 P-Value < 0.003 2-Parameter Exponential AD = 2.477 P-Value < 0.010 1000100101 90 50 10 1 Data Usage Percent 10001001010.1 90 50 10 1 Data Usage - Threshold Percent 10010 90 50 10 1 Data Usage Percent 10010 90 50 10 1 Data Usage - Threshold Percent Probability Plot for Data Usage Exponential - 95% CI 2-Parameter Exponential - 95% CI Weibull - 95% CI 3-Parameter Weibull - 95% CI What Distribution Models Friday?
400300200100 35 30 25 20 15 10 5 0 Shape # 1.670 Scale # 61.32 Thresh # 39.17 N 51 Data Usage Frequency Histogram of Data Usage 3-Parameter Weibull # This estimated historical parameter is used in the calculations. A 3-Parameter Weibull Models Friday Data Red Bars indicate outliers that were excluded from parameter determination
SATURDAY
1st Quartile 69.73 Median 101.85 3rd Quartile 137.40 Maximum 597.70 96.46 152.24 82.46 121.30 83.94 124.22 A-Squared 4.52 P-Value <0.005 Mean 124.35 StDev 100.17 Variance 10033.47 Skewness 2.79744 Kurtosis 9.97571 N 52 Minimum 0.00 Anderson-Darling Normality Test 95% Confidence Interval for Mean 95% Confidence Interval for Median 95% Confidence Interval for StDev 6004803602401200 Median Mean 16014012010080 95% Confidence Intervals Summary Report for Data Usage Saturday Descriptive Statistics
Weibull AD = 1.262 P-Value < 0.010 3-Parameter Weibull AD = 0.441 P-Value = 0.310 Goodness of Fit Test Exponential AD = 7.494 P-Value < 0.003 2-Parameter Exponential AD = 1.317 P-Value = 0.037 1000100101 90 50 10 1 Data Usage Percent 1000100101 90 50 10 1 Data Usage - Threshold Percent 10010 90 50 10 1 Data Usage Percent 1000100101 90 50 10 1 Data Usage - Threshold Percent Probability Plot for Data Usage Exponential - 95% CI 2-Parameter Exponential - 95% CI Weibull - 95% CI 3-Parameter Weibull - 95% CI What Distribution Models Saturday?
600480360240120 35 30 25 20 15 10 5 0 Shape # 1.246 Scale # 69.33 Thresh # 44.41 N 50 Data Usage Frequency Histogram of Data Usage 3-Parameter Weibull # This estimated historical parameter is used in the calculations. A 3-Parameter Weibull Models Saturday Data Red Bars indicate outliers that were excluded from parameter determination
THE SIMULATION
The Simulation Equation Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sunday Monday Tuesday Wednesday Thursday Tuesday Wednesday Thursday Friday Saturday + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +=Bill 1 Sunday Monday Tuesday Wednesday Thursday Friday Saturday Total Bill1 4 4 5 5 5 4 4 31 Bill2 4 4 4 4 4 5 5 30 Bill3 5 5 5 4 4 4 4 31 Bill4 4 4 4 5 5 4 4 30 Bill5 5 4 4 4 4 5 5 31 Bill6 4 5 5 5 4 4 4 31 Bill7 4 4 4 4 5 5 4 30 Bill8 5 5 4 4 4 4 5 31 Bill9 4 4 5 5 4 4 4 30 Bill10 4 4 4 4 5 5 5 31 Bill11 5 5 5 4 4 4 4 31 Bill12 4 4 4 5 4 4 4 29
The Simulation Parameters
The Simulation Results
The Simulation Results
The Simulation Results
ASSESSING CAPABILITY FROM SIMULATION RESULTS
Is a 1GB (1,000MB) Limit Appropriate?
Is a 2GB (2,000MB) Limit Appropriate?
Is a 3GB (3,000MB) Limit Appropriate?
Is a 4GB (4,000MB) Limit Appropriate?
Is a 5GB (5,000MB) Limit Appropriate?
Is a 6GB (6,000MB) Limit Appropriate?
Is a 10GB (10,000MB) Limit Appropriate?
Is a 12GB (12,000MB) Limit Appropriate?
Data Usage <1 1-2 2-3 3-4 4-5 5-6 >6 Expected Monthly Charge0.000% 0.330% 29.190% 52.890% 15.850% 1.650% 0.090% Sprint (1GB) $ 40.00 $ 0.05 $ 4.38 $ 7.93 $ 2.38 $ 0.25 $ 0.01 $ 55.00 Sprint (3GB) $ 50.00 $ 7.93 $ 2.38 $ 0.25 $ 0.01 $ 60.57 Sprint (6GB) $ 65.00 $ 65.00 VZ (1Gb) $ 50.00 $ 0.05 $ 4.38 $ 7.93 $ 2.38 $ 0.25 $ 0.01 $ 65.00 ATT (2GB) $ 55.00 $ 4.38 $ 7.93 $ 2.38 $ 0.25 $ 0.01 $ 69.95 ATT (5GB) $ 75.00 $ 0.25 $ 0.01 $ 75.26 VZ (3GB) $ 65.00 $ 7.93 $ 2.38 $ 0.25 $ 0.01 $ 75.57 Sprint (12GB) $ 80.00 $ 80.00 VZ (6GB) $ 80.00 $ 0.01 $ 80.01 VZ(12GB) $ 100.00 $ 100.00 ATT (15GB) $ 125.00 $ 125.00 Plan Selection Based on Simulation
Measured Simulation Ppk % Ppk % 1GB -0.83 99.36% -1.22 100% 2GB -0.53 94.36% -0.7047 99.67 3GB -0.23 75.26% -0.1984 70.48% 5GB 0.37 13.05% 0.81 1.74% 6GB 0.68 2.13% 1.31 0.09% 10GB 1.88 0.00% 3.33 0.00% 12GB 2.48 0.00% 4.35 0.00% Comparison of Simulated and Measured Capability
Conclusion • Mobile Phone Data usage can be analyzed using: – Descriptive Statistics – Run Charts – Probability Plots – Control Chart – Process Capability • Non-Normal Data requires different hypothesis test including: – Kruskal-Wallis – Mood Median • A Stochastic Simulation Model can be created by: – Determining a distribution that characterized each factor – Specifying a mathematical relationship between the factors • A Process Capability on simulated data can be used to determine specification limits
Questions? Contact Information: Brandon R. Theiss, PE Rutgers School of Law- Camden Brandon.Theiss@Rutgers.edu

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A Six Sigma Analysis of Mobile Data Usage

  • 1.
    A Six SigmaAnalysis of Mobile Data Usage 2016 WCQI Session W10 Brandon Theiss, PE Brandon.Theiss@gmail.com
  • 2.
    Motivation Is my currentmobile data plan with Republic Wireless Optimal Given my data usage?
  • 3.
    Learning Objectives • Applythe Six Sigma Methodology to Non Traditional Applications • Utilize Monte Carlo simulations to make predictions • Utilize Non Parametric Hypothesis testing • Utilize Process Capability to determine specification limitations for non-normal data
  • 4.
    4 Major MobilePhone Carriers
  • 5.
    Plans Offered ByVerizon 20% of Verizon customers charged overages in past year*
  • 6.
    Plans Offered ByAT&T 28% of AT&T customers charged overages in past year*
  • 7.
    Plans Offered ByT-Mobile 12% of T-Mobile customers charged overages in past year*
  • 8.
    5% of Sprintcustomers charged overages in past year* Plans Offered By Sprint
  • 9.
    Plans Offered ByRepublic Wireless
  • 10.
  • 11.
    121110987654321 $130 $120 $110 $100 $90 $80 $70 $60 $50 $40 Bill Number BilledAmmount Verizon (1GB) ATT(2GB) T-Mobile (2GB) Sprint (1GB) Republic (2GB) Variable Time Series Plot of Small Verizon, ATT, T-Mobile, Sprint, Republic Comparison of Carriers Small Data Plans Data Speed Potentially Decreased
  • 12.
    121110987654321 $120 $110 $100 $90 $80 $70 $60 $50 Bill Number BilledAmmount Verizon (3GB) ATT(2GB) T-Mobile (2GB) Sprint (3GB) Republic (3GB) Variable Time Series Plot of MediumVerizon, ATT, T-Mobile, Sprint, Republic Comparison of Carriers Medium Data Plans Data Speed Potentially Decreased
  • 13.
    121110987654321 $90 $85 $80 $75 $70 $65 Bill Number BilledAmmount Verizon (6GB) ATT(5GB) T-Mobile (6GB) Sprint (6GB) Republic (5GB) Variable Time Series Plot of Large Verizon, ATT, T-Mobile, Sprint, Republic Comparison of Carriers Large Data Plans
  • 14.
    Comparison of CarriersX-Large Data Plans 121110987654321 140 120 100 80 60 40 20 0 Index Data Verizon (12GB) ATT (15GB) T-Mobile (10GB) Sprint (12GB) Republic (Not Offered) Variable Time Series Plot of XL Verizon, ATT, T-Mobile, Sprint, Republic
  • 15.
    ATT (15G B) Verizon (12G B) Verizon (1G B) Sprint (1G B) ATT (2G B) Republic (5G B) Verizon (3G B) Sprint (12G B) T-M obile (10G B) Verizon (6G B) ATT (5G B) Sprint(3G B) Sprint (6G B) T-M obile (6G B) Republic (3G B) T-M obile (2G B) Republic (2G B) $ 1,600.00 $ 1,400.00 $ 1,200.00 $ 1,000.00 $ 800.00 $ 600.00 $ 400.00 $ 200.00 $ 0.00 Plan Annual Chart of Annual Cost How Much Would Each Plan have cost for the Year?
  • 16.
    1st Quartile 3053.3 Median3504.6 3rd Quartile 4588.6 Maximum 5905.4 3052.5 4459.2 3054.0 4587.4 784.2 1879.6 A-Squared 0.25 P-Value 0.687 Mean 3755.8 StDev 1107.0 Variance 1225527.2 Skewness 0.314666 Kurtosis -0.123559 N 12 Minimum 1911.6 Anderson-Darling Normality Test 95% Confidence Interval for Mean 95% Confidence Interval for Median 95% Confidence Interval for StDev 60005000400030002000 Median Mean 4500425040003750350032503000 95% Confidence Intervals Summary Report for Total Monthly Usage A First Statistical Approach (monthly data)
  • 17.
    800070006000500040003000200010000 99 95 80 50 20 5 1 Total Usage Percent Goodness ofFit Test Normal AD = 0.248 P-Value = 0.687 Probability Plot for Total Usage Normal - 95% CI Is The Data Normally Distributed?
  • 18.
  • 19.
    600050004000300020001000 LSL * Target * USL1000 Sample Mean 3755.84 Sample N 12 StDev(Overall) 1107.04 StDev(Within) 889.313 Process Data Pp * PPL * PPU -0.83 Ppk -0.83 Cpm * Cp * CPL * CPU -1.03 Cpk -1.03 Potential (Within) Capability Overall Capability % < LSL * * * % > USL 100.00 99.36 99.90 % Total 100.00 99.36 99.90 Observed Expected Overall Expected Within Performance USL Overall Within Process Capability Report for Total Usage (1GB) Plan Annual Cost ATT (2GB) $ 1,065.00 Sprint (1GB) $ 1,065.00 Is a 1GB (1,000MB) Limit Appropriate?
  • 20.
    60005000400030002000 LSL * Target * USL2000 Sample Mean 3755.84 Sample N 12 StDev(Overall) 1107.04 StDev(Within) 889.313 Process Data Pp * PPL * PPU -0.53 Ppk -0.53 Cpm * Cp * CPL * CPU -0.66 Cpk -0.66 Potential (Within) Capability Overall Capability % < LSL * * * % > USL 91.67 94.36 97.58 % Total 91.67 94.36 97.58 Observed Expected Overall Expected Within Performance USL Overall Within Process Capability Report for Total Usage (2GB) Is a 2GB (2,000MB) Limit Appropriate? Plan Annual Cost Republic (2GB) $ 480.00 T-Mobile (2GB) $ 600.00 ATT(2GB) $ 1,065.00
  • 21.
    60005000400030002000 LSL * Target * USL3000 Sample Mean 3755.84 Sample N 12 StDev(Overall) 1107.04 StDev(Within) 889.313 Process Data Pp * PPL * PPU -0.23 Ppk -0.23 Cpm * Cp * CPL * CPU -0.28 Cpk -0.28 Potential (Within) Capability Overall Capability % < LSL * * * % > USL 83.33 75.26 80.23 % Total 83.33 75.26 80.23 Observed Expected Overall Expected Within Performance USL Overall Within Process Capability Report for Total Usage (3GB) Plan Annual Cost Republic (3GB) $ 660.00 Sprint (3GB) $ 840.00 Verizon (3GB) $ 1,020.00 Is a 3GB (3,000MB) Limit Appropriate?
  • 22.
    60005000400030002000 LSL * Target * USL5000 Sample Mean 3755.84 Sample N 12 StDev(Overall) 1107.04 StDev(Within) 889.313 Process Data Pp * PPL * PPU 0.37 Ppk 0.37 Cpm * Cp * CPL * CPU 0.47 Cpk 0.47 Potential (Within) Capability Overall Capability % < LSL * * * % > USL 8.33 13.05 8.09 % Total 8.33 13.05 8.09 Observed Expected Overall Expected Within Performance USL Overall Within Process Capability Report for Total Usage (5GB) Plan Annual Cost ATT (5GB) $ 915.00 Republic (5GB) $ 1,020.00 ATT (5GB) $ 1,500.00 Is a 5GB (5,000MB) Limit Appropriate?
  • 23.
    60005000400030002000 LSL * Target * USL6000 Sample Mean 3755.84 Sample N 12 StDev(Overall) 1107.04 StDev(Within) 889.313 Process Data Pp * PPL * PPU 0.68 Ppk 0.68 Cpm * Cp * CPL * CPU 0.84 Cpk 0.84 Potential (Within) Capability Overall Capability % < LSL * * * % > USL 0.00 2.13 0.58 % Total 0.00 2.13 0.58 Observed Expected Overall Expected Within Performance USL Overall Within Process Capability Report for Total Usage (6GB) Plan Annual Cost T-Mobile (6GB) $ 780.00 Sprint (6GB) $ 780.00 Verizon (6GB) $ 960.00 Is a 6GB (6,000MB) Limit Appropriate?
  • 24.
    900075006000450030001500 LSL * Target * USL10000 Sample Mean 3755.84 Sample N 12 StDev(Overall) 1107.04 StDev(Within) 889.313 Process Data Pp * PPL * PPU 1.88 Ppk 1.88 Cpm * Cp * CPL * CPU 2.34 Cpk 2.34 Potential (Within) Capability Overall Capability % < LSL * * * % > USL 0.00 0.00 0.00 % Total 0.00 0.00 0.00 Observed Expected Overall Expected Within Performance USL Overall Within Process Capability Report for Total Usage (10GB) Plan Annual Cost T-Mobile (10GB) $ 960.00 Is a 10GB (10,000MB) Limit Appropriate? ~6 Sigma !
  • 25.
    1200010500900075006000450030001500 LSL * Target * USL12000 Sample Mean 3755.84 Sample N 12 StDev(Overall) 1107.04 StDev(Within) 889.313 Process Data Pp * PPL * PPU 2.48 Ppk 2.48 Cpm * Cp * CPL * CPU 3.09 Cpk 3.09 Potential (Within) Capability Overall Capability % < LSL * * * % > USL 0.00 0.00 0.00 % Total 0.00 0.00 0.00 Observed Expected Overall Expected Within Performance USL Overall Within Process Capability Report for Total Usage (12GB) Plan Annual Cost Sprint (12GB) $ 960.00 Verizon (12GB) $ 1200.00 Is a 12GB (12,000MB) Limit Appropriate? Greater than 6 Sigma!
  • 26.
  • 27.
    1st Quartile 69.13 Median96.70 3rd Quartile 138.00 Maximum 1100.00 112.59 133.69 88.25 102.97 95.71 110.67 A-Squared 27.78 P-Value <0.005 Mean 123.14 StDev 102.64 Variance 10535.65 Skewness 3.9407 Kurtosis 26.1682 N 366 Minimum 0.00 Anderson-Darling Normality Test 95% Confidence Interval for Mean 95% Confidence Interval for Median 95% Confidence Interval for StDev 10008006004002000 Median Mean 14013012011010090 95% Confidence Intervals Summary Report for Data Usage Descriptive Statistics On Daily Usage
  • 28.
    12008004000 99.9 99 90 50 10 1 0.1 Data Usage Percent 10000 1000 100101 0.1 0.01 0.001 0.0001 99.9 99 90 50 10 1 0.1 Data Usage Percent 100010010 99.9 99 90 50 10 1 0.1 DataUsage - Threshold Percent 20-2-4 99.9 99 90 50 10 1 0.1 Data Usage Percent 3-Parameter Loglogistic AD = 1.975 P-Value = * Johnson Transformation AD = 0.171 P-Value = 0.932 Goodness of Fit Test Logistic AD = 13.251 P-Value < 0.005 Loglogistic AD = 9.501 P-Value < 0.005 After Johnson transformation Probability Plot for Data Usage Logistic - 95% CI Loglogistic - 95% CI 3-Parameter Loglogistic - 95% CI Normal - 95% CI If The Data Is Not Normal What Approximates The Data?
  • 29.
    12008004000 99.9 99 90 50 10 1 0.1 N 366 AD 27.776 P-Value<0.005 Percent 20-2-4 99.9 99 90 50 10 1 0.1 N 366 AD 0.171 P-Value 0.932 Percent 1.21.00.80.60.40.2 0.8 0.6 0.4 0.2 0.0 Z Value P-ValueforADtest 0.38 Ref P P-Value for Best Fit: 0.931848 Z for Best Fit: 0.38 Best Transformation Type: SU Transformation function equals -0.996951+ 0.885314 × Asinh( ( X - 59.1002 ) / 25.8392 ) Probability Plot for Original Data Probability Plot for Transformed Data Select a Transformation (P-Value = 0.005 means ≤ 0.005) Johnson Transformation for Data Usage The Johnson Transformation of the Data
  • 30.
  • 31.
  • 32.
    10005000 99.9 99 90 50 10 1 0.1 Residual Percent 20015010050 1000 750 500 250 0 Fitted Value Residual 9007506004503001500 120 90 60 30 0 Residual Frequency 350300250200150100501 1000 750 500 250 0 Observation Order Residual NormalProbability Plot Versus Fits Histogram Versus Order Residual Plots for Data Usage Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Billing Cycle 11 429109 39010 4.04 0.000 Error 354 3416405 9651 Total 365 3845514 Model Summary S R-sq R-sq(adj) R-sq(pred) 98.2388 11.16% 8.40% 4.99% Method Null hypothesis All means are equal Alternative hypothesis At least one mean is different Significance level α = 0.05 Equal variances were assumed for the analysis. Factor Information Factor Levels Values Billing Cycle 12 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 One-way ANOVA: Data Usage versus Billing Cycle Is There Statistically Significant Difference Between The Months?
  • 33.
    But ANOVA RequiresThe Data to be Normal
  • 34.
    Kruskal-Wallis Test: DataUsage versus Billing Cycle Kruskal-Wallis Test on Data Usage Billing Cycle N Median Ave Rank Z 1 31 108.80 217.0 1.84 2 30 130.50 249.8 3.58 3 31 88.40 187.4 0.21 4 30 85.60 160.9 -1.22 5 31 137.90 265.5 4.51 6 31 129.40 234.7 2.82 7 30 88.15 182.3 -0.07 8 31 93.80 187.9 0.24 9 30 75.70 135.9 -2.57 10 31 75.00 148.9 -1.90 11 31 62.50 86.3 -5.35 12 29 73.20 142.6 -2.17 Overall 366 183.5 H = 82.19 DF = 11 P = 0.000 H = 82.19 DF = 11 P = 0.000 (adjusted for ties) A First Non-Parametric Approach
  • 35.
    20 10 0 1050 900 750 600 450 300 150 0 1050 900 750 600 450 300 150 0 20 10 0 1050 900 750 600 450 300 1500 20 10 0 1050 900 750 600 450 300 1500 1 Data Usage Frequency 23 4 5 6 7 8 9 10 11 12 Histogram of Data Usage Panel variable: Billing Cycle Kruskal-Wallis Test Requires The Distributions To Have Similar Shapes
  • 36.
    Mood Median Test:Data Usage versus Billing Cycle Mood median test for Data Usage Chi-Square = 70.53 DF = 11 P = 0.000 Billing Individual 95.0% CIs Cycle N≤ N> Median Q3-Q1 --+---------+---------+--------- +---- 1 10 21 109 68 (*--) 2 4 26 131 45 (-*---) 3 17 14 88 59 (-*-----) 4 19 11 86 46 (-*-) 5 5 26 138 156 (-----*---------------) 6 8 23 129 81 (----*----) 7 16 14 88 78 (--*-----) 8 17 14 94 44 (-*-) 9 21 9 76 44 (-*--) 10 22 9 75 46 (-*-) 11 26 5 63 36 (-*-) 12 18 11 73 83 (--*-----) --+---------+---------+--------- +---- 60 120 180 240 Overall median = 97 A Second Non-Parametric Approach
  • 37.
  • 38.
  • 39.
    20 10 0 10509007506004503001500 10509007506004503001500 20 10 0 10509007506004503001500 20 10 0 Sunday Data Usage Frequency Monday Tuesday WednesdayThursday Friday Saturday Histogram of Data Usage Panel variable: Day Of Week What Do The Distributions Of Each Day Look Like?
  • 40.
  • 41.
    1st Quartile 74.32 Median120.15 3rd Quartile 197.35 Maximum 1100.00 115.83 210.36 84.71 152.41 142.27 210.53 A-Squared 4.65 P-Value <0.005 Mean 163.09 StDev 169.77 Variance 28821.13 Skewness 3.7420 Kurtosis 18.3156 N 52 Minimum 0.00 Anderson-Darling Normality Test 95% Confidence Interval for Mean 95% Confidence Interval for Median 95% Confidence Interval for StDev 9607204802400 Median Mean 225200175150125100 95% Confidence Intervals Summary Report for Data Usage Sunday Descriptive Statistics
  • 42.
    Weibull AD = 0.728 P-Value= 0.053 3-Parameter Weibull AD = 0.355 P-Value = 0.475 Goodness of Fit Test Exponential AD = 4.176 P-Value < 0.003 2-Parameter Exponential AD = 1.614 P-Value = 0.017 1000100101 99.9 90 50 10 1 Data Usage Percent 1000100101 99.9 90 50 10 1 Data Usage - Threshold Percent 100010010 99.9 90 50 10 1 Data Usage Percent 1000100101 99.9 90 50 10 1 Data Usage - Threshold Percent Probability Plot for Data Usage Exponential - 95% CI 2-Parameter Exponential - 95% CI Weibull - 95% CI 3-Parameter Weibull - 95% CI What Distribution Models Sunday?
  • 43.
    9607204802400 40 30 20 10 0 Shape # 1.369 Scale# 125.2 Thresh # 26.73 N 50 Data Usage Frequency Histogram of Data Usage 3-Parameter Weibull # This estimated historical parameter is used in the calculations. A 3-Parameter Weibull Models Sunday Data Red Bars indicate outliers that were excluded from parameter determination
  • 44.
  • 45.
    1st Quartile 67.92 Median89.85 3rd Quartile 134.92 Maximum 619.30 94.34 161.03 78.71 108.15 100.37 148.52 A-Squared 5.23 P-Value <0.005 Mean 127.69 StDev 119.76 Variance 14342.75 Skewness 2.55234 Kurtosis 7.19780 N 52 Minimum 0.00 Anderson-Darling Normality Test 95% Confidence Interval for Mean 95% Confidence Interval for Median 95% Confidence Interval for StDev 6004803602401200 Median Mean 16014012010080 95% Confidence Intervals Summary Report for Data Usage Monday Descriptive Statistics
  • 46.
    10001001010.10.010.0010.0001 90 50 10 1 Data Usage Percent 1000100101 90 50 10 1 Data Usage- Threshold Percent 10001001010.10.010.0010.0001 90 50 10 1 Data Usage Percent 10010 90 50 10 1 Data Usage - Threshold Percent Weibull AD = 2.383 P-Value < 0.010 3-Parameter Weibull AD = 0.398 P-Value = 0.342 Goodness of Fit Test Exponential AD = 6.080 P-Value < 0.003 2-Parameter Exponential AD = 6.124 P-Value < 0.010 Probability Plot for Data Usage Exponential - 95% CI 2-Parameter Exponential - 95% CI Weibull - 95% CI 3-Parameter Weibull - 95% CI What Distribution Models Monday?
  • 47.
    600480360240120 35 30 25 20 15 10 5 0 Shape # 1.916 Scale# 74.12 Thresh # 29.30 N 48 Data Usage Frequency Histogram of Data Usage 3-Parameter Weibull # This estimated historical parameter is used in the calculations. A 3-Parameter Weibull Models Monday Data Red Bars indicate outliers that were excluded from parameter determination
  • 48.
  • 49.
    1st Quartile 61.250 Median81.400 3rd Quartile 105.600 Maximum 289.700 75.526 100.342 72.217 89.345 37.785 55.699 A-Squared 1.76 P-Value <0.005 Mean 87.934 StDev 45.017 Variance 2026.544 Skewness 2.02797 Kurtosis 7.44336 N 53 Minimum 0.000 Anderson-Darling Normality Test 95% Confidence Interval for Mean 95% Confidence Interval for Median 95% Confidence Interval for StDev 300240180120600 Median Mean 100959085807570 95% Confidence Intervals Summary Report for Data Usage Tuesday Descriptive Statistics
  • 50.
    Weibull AD = 0.382 P-Value> 0.250 3-Parameter Weibull AD = 0.203 P-Value > 0.500 Goodness of Fit Test Exponential AD = 10.303 P-Value < 0.003 2-Parameter Exponential AD = 3.239 P-Value < 0.010 1000100101 99.9 90 50 10 1 Data Usage Percent 10001001010.1 99.9 90 50 10 1 Data Usage - Threshold Percent 10010 99.9 90 50 10 1 Data Usage Percent 10010 99.9 90 50 10 1 Data Usage - Threshold Percent Probability Plot for Data Usage Exponential - 95% CI 2-Parameter Exponential - 95% CI Weibull - 95% CI 3-Parameter Weibull - 95% CI What Distribution Models Tuesday?
  • 51.
    30024018012060 25 20 15 10 5 0 Shape # 1.882 Scale# 57.02 Thresh # 34.60 N 51 Data Usage Frequency Histogram of Data Usage 3-Parameter Weibull # This estimated historical parameter is used in the calculations. A 3-Parameter Weibull Models Tuesday Data Red Bars indicate outliers that were excluded from parameter determination
  • 52.
  • 53.
    1st Quartile 69.00 Median97.10 3rd Quartile 154.00 Maximum 321.50 97.32 136.20 77.27 113.79 59.20 87.27 A-Squared 2.07 P-Value <0.005 Mean 116.76 StDev 70.53 Variance 4974.95 Skewness 1.10549 Kurtosis 0.67508 N 53 Minimum 0.00 Anderson-Darling Normality Test 95% Confidence Interval for Mean 95% Confidence Interval for Median 95% Confidence Interval for StDev 320240160800 Median Mean 1401301201101009080 95% Confidence Intervals Summary Report for Data Usage Wednesday Descriptive Statistics
  • 54.
    Weibull AD = 1.186 P-Value< 0.010 3-Parameter Weibull AD = 0.618 P-Value = 0.113 Goodness of Fit Test Exponential AD = 5.427 P-Value < 0.003 2-Parameter Exponential AD = 2.310 P-Value < 0.010 1000100101 99.9 90 50 10 1 Data Usage Percent 1000100101 99.9 90 50 10 1 Data Usage - Threshold Percent 100010010 99.9 90 50 10 1 Data Usage Percent 1000100101 99.9 90 50 10 1 Data Usage - Threshold Percent Probability Plot for Data Usage Exponential - 95% CI 2-Parameter Exponential - 95% CI Weibull - 95% CI 3-Parameter Weibull - 95% CI What Distribution Models Wednesday?
  • 55.
    3202802402001601208040 20 15 10 5 0 Shape 1.430 Scale 104.1 Thresh24.66 N 52 Data Usage Frequency Histogram of Data Usage 3-Parameter Weibull A 3-Parameter Weibull Models Wednesday Data
  • 56.
  • 57.
    1st Quartile 74.90 Median102.00 3rd Quartile 163.27 Maximum 449.50 105.25 145.97 81.45 134.75 61.29 90.69 A-Squared 1.99 P-Value <0.005 Mean 125.61 StDev 73.13 Variance 5347.80 Skewness 2.03570 Kurtosis 6.42894 N 52 Minimum 42.10 Anderson-Darling Normality Test 95% Confidence Interval for Mean 95% Confidence Interval for Median 95% Confidence Interval for StDev 400300200100 Median Mean 1401301201101009080 95% Confidence Intervals Summary Report for Data Usage Thursday Descriptive Statistics
  • 58.
    Weibull AD = 0.904 P-Value= 0.020 3-Parameter Weibull AD = 0.324 P-Value > 0.500 Goodness of Fit Test Exponential AD = 6.944 P-Value < 0.003 2-Parameter Exponential AD = 1.454 P-Value = 0.025 1000100101 99.9 90 50 10 1 Data Usage Percent 1000100101 99.9 90 50 10 1 Data Usage - Threshold Percent 100010010 99.9 90 50 10 1 Data Usage Percent 1000100101 99.9 90 50 10 1 Data Usage - Threshold Percent Probability Plot for Data Usage Exponential - 95% CI 2-Parameter Exponential - 95% CI Weibull - 95% CI 3-Parameter Weibull - 95% CI What Distribution Models Thursday?
  • 59.
    400300200100 25 20 15 10 5 0 Shape # 1.364 Scale# 85.54 Thresh # 40.89 N 52 Data Usage Frequency Histogram of Data Usage 3-Parameter Weibull # This estimated historical parameter is used in the calculations. A 3-Parameter Weibull Models Thursday Data Red Bars indicate outliers that were excluded from parameter determination
  • 60.
  • 61.
    1st Quartile 67.70 Median100.95 3rd Quartile 122.95 Maximum 435.30 94.58 140.14 84.26 105.70 68.58 101.49 A-Squared 4.30 P-Value <0.005 Mean 117.36 StDev 81.84 Variance 6697.42 Skewness 2.21566 Kurtosis 5.35910 N 52 Minimum 10.70 Anderson-Darling Normality Test 95% Confidence Interval for Mean 95% Confidence Interval for Median 95% Confidence Interval for StDev 4003002001000 Median Mean 1401301201101009080 95% Confidence Intervals Summary Report for Data Usage Friday Descriptive Statistics
  • 62.
    Weibull AD = 0.607 P-Value= 0.111 3-Parameter Weibull AD = 0.392 P-Value = 0.404 Goodness of Fit Test Exponential AD = 9.088 P-Value < 0.003 2-Parameter Exponential AD = 2.477 P-Value < 0.010 1000100101 90 50 10 1 Data Usage Percent 10001001010.1 90 50 10 1 Data Usage - Threshold Percent 10010 90 50 10 1 Data Usage Percent 10010 90 50 10 1 Data Usage - Threshold Percent Probability Plot for Data Usage Exponential - 95% CI 2-Parameter Exponential - 95% CI Weibull - 95% CI 3-Parameter Weibull - 95% CI What Distribution Models Friday?
  • 63.
    400300200100 35 30 25 20 15 10 5 0 Shape # 1.670 Scale# 61.32 Thresh # 39.17 N 51 Data Usage Frequency Histogram of Data Usage 3-Parameter Weibull # This estimated historical parameter is used in the calculations. A 3-Parameter Weibull Models Friday Data Red Bars indicate outliers that were excluded from parameter determination
  • 64.
  • 65.
    1st Quartile 69.73 Median101.85 3rd Quartile 137.40 Maximum 597.70 96.46 152.24 82.46 121.30 83.94 124.22 A-Squared 4.52 P-Value <0.005 Mean 124.35 StDev 100.17 Variance 10033.47 Skewness 2.79744 Kurtosis 9.97571 N 52 Minimum 0.00 Anderson-Darling Normality Test 95% Confidence Interval for Mean 95% Confidence Interval for Median 95% Confidence Interval for StDev 6004803602401200 Median Mean 16014012010080 95% Confidence Intervals Summary Report for Data Usage Saturday Descriptive Statistics
  • 66.
    Weibull AD = 1.262 P-Value< 0.010 3-Parameter Weibull AD = 0.441 P-Value = 0.310 Goodness of Fit Test Exponential AD = 7.494 P-Value < 0.003 2-Parameter Exponential AD = 1.317 P-Value = 0.037 1000100101 90 50 10 1 Data Usage Percent 1000100101 90 50 10 1 Data Usage - Threshold Percent 10010 90 50 10 1 Data Usage Percent 1000100101 90 50 10 1 Data Usage - Threshold Percent Probability Plot for Data Usage Exponential - 95% CI 2-Parameter Exponential - 95% CI Weibull - 95% CI 3-Parameter Weibull - 95% CI What Distribution Models Saturday?
  • 67.
    600480360240120 35 30 25 20 15 10 5 0 Shape # 1.246 Scale# 69.33 Thresh # 44.41 N 50 Data Usage Frequency Histogram of Data Usage 3-Parameter Weibull # This estimated historical parameter is used in the calculations. A 3-Parameter Weibull Models Saturday Data Red Bars indicate outliers that were excluded from parameter determination
  • 68.
  • 69.
    The Simulation Equation SundayMonday Tuesday Wednesday Thursday Friday Saturday Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sunday Monday Tuesday Wednesday Thursday Tuesday Wednesday Thursday Friday Saturday + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +=Bill 1 Sunday Monday Tuesday Wednesday Thursday Friday Saturday Total Bill1 4 4 5 5 5 4 4 31 Bill2 4 4 4 4 4 5 5 30 Bill3 5 5 5 4 4 4 4 31 Bill4 4 4 4 5 5 4 4 30 Bill5 5 4 4 4 4 5 5 31 Bill6 4 5 5 5 4 4 4 31 Bill7 4 4 4 4 5 5 4 30 Bill8 5 5 4 4 4 4 5 31 Bill9 4 4 5 5 4 4 4 30 Bill10 4 4 4 4 5 5 5 31 Bill11 5 5 5 4 4 4 4 31 Bill12 4 4 4 5 4 4 4 29
  • 70.
  • 71.
  • 72.
  • 73.
  • 74.
  • 75.
    Is a 1GB(1,000MB) Limit Appropriate?
  • 76.
    Is a 2GB(2,000MB) Limit Appropriate?
  • 77.
    Is a 3GB(3,000MB) Limit Appropriate?
  • 78.
    Is a 4GB(4,000MB) Limit Appropriate?
  • 79.
    Is a 5GB(5,000MB) Limit Appropriate?
  • 80.
    Is a 6GB(6,000MB) Limit Appropriate?
  • 81.
    Is a 10GB(10,000MB) Limit Appropriate?
  • 82.
    Is a 12GB(12,000MB) Limit Appropriate?
  • 83.
    Data Usage <11-2 2-3 3-4 4-5 5-6 >6 Expected Monthly Charge0.000% 0.330% 29.190% 52.890% 15.850% 1.650% 0.090% Sprint (1GB) $ 40.00 $ 0.05 $ 4.38 $ 7.93 $ 2.38 $ 0.25 $ 0.01 $ 55.00 Sprint (3GB) $ 50.00 $ 7.93 $ 2.38 $ 0.25 $ 0.01 $ 60.57 Sprint (6GB) $ 65.00 $ 65.00 VZ (1Gb) $ 50.00 $ 0.05 $ 4.38 $ 7.93 $ 2.38 $ 0.25 $ 0.01 $ 65.00 ATT (2GB) $ 55.00 $ 4.38 $ 7.93 $ 2.38 $ 0.25 $ 0.01 $ 69.95 ATT (5GB) $ 75.00 $ 0.25 $ 0.01 $ 75.26 VZ (3GB) $ 65.00 $ 7.93 $ 2.38 $ 0.25 $ 0.01 $ 75.57 Sprint (12GB) $ 80.00 $ 80.00 VZ (6GB) $ 80.00 $ 0.01 $ 80.01 VZ(12GB) $ 100.00 $ 100.00 ATT (15GB) $ 125.00 $ 125.00 Plan Selection Based on Simulation
  • 84.
    Measured Simulation Ppk %Ppk % 1GB -0.83 99.36% -1.22 100% 2GB -0.53 94.36% -0.7047 99.67 3GB -0.23 75.26% -0.1984 70.48% 5GB 0.37 13.05% 0.81 1.74% 6GB 0.68 2.13% 1.31 0.09% 10GB 1.88 0.00% 3.33 0.00% 12GB 2.48 0.00% 4.35 0.00% Comparison of Simulated and Measured Capability
  • 85.
    Conclusion • Mobile PhoneData usage can be analyzed using: – Descriptive Statistics – Run Charts – Probability Plots – Control Chart – Process Capability • Non-Normal Data requires different hypothesis test including: – Kruskal-Wallis – Mood Median • A Stochastic Simulation Model can be created by: – Determining a distribution that characterized each factor – Specifying a mathematical relationship between the factors • A Process Capability on simulated data can be used to determine specification limits
  • 86.
    Questions? Contact Information: Brandon R.Theiss, PE Rutgers School of Law- Camden Brandon.Theiss@Rutgers.edu

Editor's Notes

  • #6 http://money.cnn.com/2016/01/14/technology/overage-charges/
  • #7 http://money.cnn.com/2016/01/14/technology/overage-charges/
  • #8 http://money.cnn.com/2016/01/14/technology/overage-charges/
  • #9 http://money.cnn.com/2016/01/14/technology/overage-charges/