a six sigma analysis of mobile data usage...a six sigma analysis of mobile data usage minitab...

A Six Sigma Analysis of Mobile Data

Usage

Minitab Insights Conference

September 13, 2016

Brandon Theiss, PE, [email protected]

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

The Data Set

Data was collected from March 23, 2015

Through March 24, 2016

Comparison of Carriers Small Data Plans

Data Speed Potentially Decreased

Comparison of Carriers Medium Data Plans

Data Speed Potentially Decreased

Comparison of Carriers Large Data Plans

Comparison of Carriers X-Large Data Plans

ATT (1

5GB)

Verizon

(12G

B)

Veri zon

(1GB)

Sprin

t (1G

B)

ATT (2

GB)

Republic

(5GB)

Verizon

(3GB)

Sprin

t (12

GB)

T-M

obile (1

0GB)

Verizon (6

GB)

ATT (5

GB)

Sprin

t (3G

B)

Sprin

t (6G

B)

T-M

obile (6

GB)

Repub

lic (3

GB)

T-M

obile (2

GB)

Republic

(2GB)

$ 1,600.00

$ 1,400.00

$ 1,200.00

$ 1,000.00

$ 800.00

$ 600.00

$ 400.00

$ 200.00

$ 0.00

Plan

An

nu

al

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)

Is The Data Normally Distributed?

121110987654321

6000

4000

2000

Observation

Ind

ivid

ua

l Va

lue

_

X=3756

UCL=6424

LCL=1088

121110987654321

3000

2000

1000

0

Observation

Mo

vin

g R

an

ge

__

MR=1003

UCL=3278

LCL=0

I-MR Chart of Total Monthly Usage

Is The Data Is Statistical Control?

Plan Annual Cost

ATT (2GB) $ 1,065.00

Sprint (1GB) $ 1,065.00

Is a 1GB (1,000MB) Limit Appropriate?


Plan Annual Cost

Republic (2GB) $ 480.00

T-Mobile (2GB) $ 600.00

ATT(2GB) $ 1,065.00

Plan Annual Cost

Republic (3GB) $ 660.00

Sprint (3GB) $ 840.00

Verizon (3GB) $ 1,020.00


Plan Annual Cost

ATT (5GB) $ 915.00

Republic (5GB) $ 1,020.00

ATT (5GB)

$ 1,500.00


Plan Annual Cost

T-Mobile (6GB) $ 780.00

Sprint (6GB) $ 780.00

Verizon (6GB) $ 960.00


Plan Annual Cost

T-Mobile (10GB) $ 960.00


~6 Sigma !

Plan Annual Cost

Sprint (12GB) $ 960.00

Verizon (12GB) $ 1200.00


Greater than 6 Sigma!

2/19/

2016

1/13

/201

6

12/7

/201

5

10/3

1/201

5

9/24

/201

5

8/18

/201

5

7/12/

2015

6/5/

2015

4/29

/201

5

3/24/

2015

1200

1000

800

600

400

200

0

Date

Data

Usa

ge

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





10008006004002000

Median

Mean

14013012011010090


Summary Report for Data Usage

Descriptive Statistics On Daily Usage

If The Data Is Not Normal What Approximates The Data?

The Johnson Transformation of the Data

111098754321

3.0

1.5

0.0

-1.5

-3.0

Billing Cycle

Ind

ivid

ua

l Va

lue

_X=-0.003

UCL=2.430

LCL=-2.436

111098754321

4

3

2

1

0

Billing Cycle

Mo

vin

g R

an

ge

__MR=0.915

UCL=2.989

LCL=0

1

11

11

11

11

1

I-MR Chart of Transformed Data Usage

Is the Data In Statistical Control?

121110987654321

1200

1000

800

600

400

200

0

Billing Cycle

Data

Usa

ge

106.75261.6645

103.34889.1433104.029150.58156.348

190.497

101.303122.071153.747137.239

Boxplot of Data Usage

A Third Statistical Approach

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

105090

075

060

045030

015

00

20

10

0

105090

075

060

045030

015

00

1

Data Usage

Fre

qu

en

cy

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

Data

Usa

ge

124.35117.36125.612116.7687.934127.687163.094

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

Fre

qu

en

cy

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





9607204802400

Median

Mean

225200175150125100



Sunday Descriptive Statistics

What Distribution Models Sunday?

9607204802400

40

30

20

10

0

Shape # 1.369

Scale # 125.2

Thresh # 26.73

N 50

Data Usage

Fre

qu

en

cy

Histogram of Data Usage3-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





6004803602401200

Median

Mean

16014012010080



Monday Descriptive Statistics

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

Fre

qu

en

cy



A 3-Parameter Weibull Models Monday Data


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





300240180120600

Median

Mean

100959085807570



Tuesday Descriptive Statistics

What Distribution Models Tuesday?

30024018012060

25

20

15

10

5

0

Shape # 1.882

Scale # 57.02

Thresh # 34.60

N 51

Data Usage

Fre

qu

en

cy



A 3-Parameter Weibull Models Tuesday Data


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





320240160800

Median

Mean

1401301201101009080



Wednesday Descriptive Statistics

What Distribution Models Wednesday?

3202802402001601208040

20

15

10

5

0

Shape 1.430

Scale 104.1

Thresh 24.66

N 52

Data Usage

Fre

qu

en

cy


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





400300200100

Median

Mean

1401301201101009080



Thursday Descriptive Statistics

What Distribution Models Thursday?

400300200100

25

20

15

10

5

0

Shape # 1.364

Scale # 85.54

Thresh # 40.89

N 52

Data Usage

Fre

qu

en

cy



A 3-Parameter Weibull Models Thursday Data


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





4003002001000

Median

Mean

1401301201101009080



Friday Descriptive Statistics

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

Fre

qu

en

cy



A 3-Parameter Weibull Models Friday Data


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





6004803602401200

Median

Mean

16014012010080



Saturday Descriptive Statistics

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

Fre

qu

en

cy



A 3-Parameter Weibull Models Saturday Data


THE SIMULATION

The Simulation Equation

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

ASSESSING CAPABILITY

FROM SIMULATION RESULTS

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

[email protected]

a six sigma analysis of mobile data usage...a six sigma analysis of mobile data usage minitab...

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