Chapter 8
Making Sense of Data
in
Six Sigma and Lean
How to tell “story” from dataset?
Quantitative Data
• Graphical Methods
–
–
–
–
–
–
Dot Plots
Stem-and-Leaf Plots
Frequency Tables
Histograms and Performance Histograms
Run Charts
Time-Series Plots
• Numerical Methods: Descriptive Statistics
How to tell “story” from dataset?
Qualitative Data
– Pie Charts
– Bar Charts
– Pareto Analysis with Lorenz Curve
How to tell “story” from dataset?
Bivarite Data
• Graphical Methods
– Scatter Plots
• Numerical Methods: Correlation Coefficient
– Pearson Coefficient
– Spearman’s Rho ()
– Kendall’s Tau () Rank Correlation
How to tell “story” from dataset?
Multi-Vari Data
• Graphical Methods
– Multi-Vari Charts
Summarizing Quantitative Data:
Dot Plots
• Dot plot is one of the most simple types of plots
Example 8.1
Minitab
Graph
Dotplot
Simple
Summarizing Quantitative Data:
Stem-and-Leaf Plots
• Stem-and-Leaf Plots are a method for showing the
frequency with which certain classes of values occur.
i160.photobucket
.com/.../treediagr
am.png
Summarizing Quantitative Data:
Frequency Tables
• constructed by
arranging collected
data values in
ascending order of
magnitude with
their corresponding
frequencies.
• Absolute
frequencies or
relative
frequencies (%)
www.sci.sdsu.edu/.../Weeks/images/Frequency.png
Summarizing Quantitative Data:
Histogram
www.statcan.gc.ca/.../ch9/images/histo1.gif
Summarizing Quantitative Data:
Run Charts
• A line graph of data points plotted in chronological
order that helps detect special causes of variation
Minitab
Graph
Time Series Plot
Simple
Summarizing Quantitative Data:
Time-Series Plots
• A time series plot is a graph showing a set of
observations taken at different points in time and
charted in a time series.
Minitab
Graph
Time Series Plot
Simple
Summarizing Quantitative Data:
Descriptive Statistics
Measures of Center
• Sample mean
• Population mean
x

x
n
x


N
• Median: the "middle" value in the dataset
• Mode: the value that occurs most often
Summarizing Quantitative Data:
Descriptive Statistics
Measures of Variation
• Range: the difference between the largest and
the smallest values in the dataset
2
• Sample variance
(
x

x
)

2
s 
n 1
s
• Sample standard deviation
2
(
x


)
• Population variance 2 
 
• Population standard deviation
N

2
(
x

x
)

n 1
2
(
x


)

N
Summarizing Quantitative Data:
Descriptive Statistics
Measures of Variation
• Coefficient of Variation (CV)
• Interquartile Range (IQR)
s
CV 
x
IQR  Q3  Q1
Summarizing Quantitative Data:
Descriptive Statistics
• Minimum
• Maximum
• Median
• First Quartile
• Third Quartile
Minitab:
Stat
Basic Statistics
Display Descriptive..
• Boxplot
Summarizing Quantitative Data:
Descriptive Statistics
Identifying Potential Outliers
• Lower inner fence (LIF) = Q1  (1.5  IQR )
• Upper inner fence (UIF) = Q3  (1.5  IQR )
• Lower outer fence (LOF) = Q1  (3.0  IQR )
• Upper outer fence (UOF) = Q3  (3.0  IQR )
• Mild outliers: data fall between the two lower
fences and between the two upper fences
• Extreme outliers: data fall below the LOF or
above the UOF
Summarizing Quantitative Data:
Descriptive Statistics
Measures of Positions
• Percentiles
– Percentiles divide the dataset into 100 equal parts
– Percentiles measure position from the bottom
– Percentiles are most often used for determining the
relative standing of an individual in a population or the
rank position of the individual.
• z scores
– Standard normal distribution ( = 0 and  = 1)
x
xx
z
z

s
Summarizing Qualitative Data:
Graphical Displays
• Pie Chart
http://techie-teacher-wannabe.wikispaces.com/file/view/SocialPieChart.png/96606670/So
cialPieChart.png
Summarizing Qualitative Data:
Graphical Displays
• Bar Graph
www.creationfactor.net/images/graph-bar.jpg
Summarizing Qualitative Data:
Graphical Displays
• Pareto
Analysis
with
Lorenz
Curve
www.spcforexcel.com/files/images/ccpareto.gif
Summarizing Bivariate Data:
Scatterplot
Minitab:
Graph
Scatterplot
Simple
Summarizing Bivariate Data:
Correlation Coefficient
• Pearson Correlation Coefficient
r
 xy 
( x )(  y )
n
2
2




(
x
)
(
y
)
 x2  
  y2  


n  
n 

Minitab:
Stat
Regression
Regression
Summarizing Bivariate Data:
Correlation Coefficient
• Spearman’s Rho ()
– A measure of the linear relationship between two variables.
– It differs from Pearson's correlation only in that the computations
are done after the numbers are converted to ranks.
– When converting to ranks, the smallest value on X becomes a
rank of 1, etc.
– D (Difference) is calculated between the pair of ranks
rs  1 
6 D 2
n(n 2  1)
Summarizing Bivariate Data:
Correlation Coefficient
• Spearman’s Rho () Example
GPA
3.99
3.97
3.93
3.92
3.91
3.85
3.84
3.77
Salary
57.7
61.2
57.3
54.6
64.7
55.3
52.2
54.1
GPA Rank
8
7
6
5
4
3
2
1
Salary Rank
6
7
5
3
8
4
1
2
D
2
0
1
2
-4
-1
1
-1
D2
4
0
1
4
16
1
1
1
6 D 2
6(28)
rs  1 
 1
 .667
2
2
n(n  1)
8(8  1)
=28
Summarizing Bivariate Data:
Correlation Coefficient
• Kendall’s Tau ()
– A measure of the linear relationship between two variables.
– It differs from Pearson's correlation only in that the computations
are done after the numbers are converted to ranks.
– When converting to ranks, the smallest value on X becomes a
rank of 1, etc.
– P is # of pairs with both ranks higher
r 
4 P
n(n  1)
1
Summarizing Bivariate Data:
Correlation Coefficient
• Kendall’s Tau () Example
•GPAExample 3.99 3.97 3.93 3.92
3.91
3.85
3.84
3.77
Salary
57.7
61.2
57.3
54.6
64.7
55.3
52.2
54.1
GPA Rank
8
7
6
5
4
3
2
1
Salary Rank
6
7
5
3
8
4
1
2
P
0
0
2
3
0
4
6
6
4 P
4(21)
r 
1
 1  .50
n(n  1)
8(8  1)
=21
Summarizing Multi-Vari Data:
Multi-Vari Charts
• Show patterns of variation from several possible
causes on a single chart, or set of charts
• Obtains a first look at the process stability over
time. Can be constructed in various ways to get
the “best view”.
– Positional: variation within a part or process
– Cyclical: variation between consecutive parts or process steps
– Temporal: Time variability
Graphical Tool:
Multi-Vari Charts
Cus. Size
Product
Cus. Type
Satis.
1
1
2
3.54
2
1
3
3.16
Cus. Size: 1 = small
2 = large
1
2
2
2.42
Product:
2
2
2
2.70
1
1
3
3.31
2
1
2
4.12
2
2
1
3.24
2
2
2
4.47
2
1
2
3.83
1
1
1
2.94
http://www.qimacros.com/qiwizard/multivari-chart.html
1 = Consumer
2 = Manuf.
Cus. Type: 1 = Gov’t
2 = Commercial
3 = Education
Graphical Tool:
Multi-Vari Charts
Minitab:
Stat
Quality Tools
Multi Vari Chart