Download 2-6 Multivariate Data

2-1 Data Summary and Display
Engineering Example
Suppose that an engineer is developing a rubber compound for use in
O-rings. The O-rings are to be employed as seals in plasma etching
tools used in the semiconductor industry, so their resistance to acids
and other corrosive substances is an important characteristic. The
engineer uses the standard rubber compound to produce eight Orings in a development laboratory and measures the tensile strength
of each specimen after immersion in a nitric acid solution at 30°C for
25 minutes [refer to the American Society for Testing and Materials
(ASTM) Standard D 1414 and the associated standards for many
interesting aspects of testing rubber O-rings]. The tensile strengths (in
psi) of the eight O-rings are 1030, 1035, 1020, 1049, 1028, 1026,
1019, and 1010. Then, he decides to consider a modified formulation
of the rubber in which a Teflon additive is included. Eight O-ring
specimens are made from this modified rubber compound and
subjected to the nitric acid emersion test described earlier. The tensile
test results are 1037, 1047, 1066, 1048, 1059, 1073, 1070, and 1040.
2-1 Data Summary and Display
2-1 Data Summary and Display
Population Mean
For a finite population with N measurements, the mean is
The sample mean is a reasonable estimate of the
population mean.
2-1 Data Summary and Display
Sample Variance and Sample Standard Deviation
2-1 Data Summary and Display
The sample variance is
The sample standard deviation is
2-1 Data Summary and Display
Computational formula for s2
2-1 Data Summary and Display
Population Variance
When the population is finite and consists of N values,
we may define the population variance as
The sample variance is a reasonable estimate of the
population variance.
2-2 Stem-and-Leaf Diagram
Steps for Constructing a Stem-and-Leaf Diagram
2-2 Stem-and-Leaf Diagram
2-2 Stem-and-Leaf Diagram
2-2 Stem-and-Leaf Diagram
2-2 Stem-and-Leaf Diagram
2-2 Stem-and-Leaf Diagram
Q1 = (20+21)/2 = (143+145)/2 = 144
Q2 = (40+41)/2 = 161.5
Q3 = (60+61)/2 = 181
2-2 Stem-and-Leaf Diagram
2-3 Histograms
A histogram is a more compact summary of data than a
stem-and-leaf diagram. To construct a histogram for
continuous data, we must divide the range of the data into
intervals, which are usually called class intervals, cells, or
bins. If possible, the bins should be of equal width to
enhance the visual information in the histogram.
2-3 Histograms
2-3 Histograms
2-3 Histograms
2-3 Histograms
2-3 Histograms
An important variation of the histogram is the Pareto chart.
This chart is widely used in quality and process
improvement studies where the data usually represent
different types of defects, failure modes, or other categories
of interest to the analyst. The categories are ordered so that
the category with the largest number of frequencies is on
the left, followed by the category with the second largest
number of frequencies, and so forth.
2-3 Histograms
2-4 Box Plots
• The box plot is a graphical display that
simultaneously describes several important features of
a data set, such as center, spread, departure from
symmetry, and identification of observations that lie
unusually far from the bulk of the data.
• Whisker
• Outlier
• Extreme outlier
2-4 Box Plots
2-4 Box Plots
2nd quartile = median = 161.5
1st quartile = 143.5
IQR = Q3 – Q1 = 181 – 143.5 = 37.5
1.5 IQR = 56.25
Q1 - 1.5 IQR = 143.5 – 56.25 = 87.25
3rd quartile = 181
IQR = Q3 – Q1 = 181 – 143.5 = 37.5
1.5 IQR = 56.25
Q3 + 1.5 IQR = 237.25
2-4 Box Plots
2-5 Time Series Plots
• A time series or time sequence is a data set in
which the observations are recorded in the order in
which they occur.
• A time series plot is a graph in which the vertical
axis denotes the observed value of the variable (say x)
and the horizontal axis denotes the time (which could
be minutes, days, years, etc.).
• When measurements are plotted as a time series, we
often see
•trends,
•cycles, or
•other broad features of the data
2-5 Time Series Plots
2-5 Time Series Plots
2-5 Time Series Plots
2-6 Multivariate Data
• The dot diagram, stem-and-leaf diagram, histogram,
and box plot are descriptive displays for univariate
data; that is, they convey descriptive information
about a single variable.
•Many engineering problems involve collecting and
analyzing multivariate data, or data on several
different variables.
•In engineering studies involving multivariate
data, often the objective is to determine the
relationships among the variables or to build an
empirical model.
2-6 Multivariate Data
2-6 Multivariate Data
2-6 Multivariate Data
Sample Correlation Coefficient
The strength of a linear relationship between two variables
2-6 Multivariate Data
Strong when 0.8≤ r ≤ 1, weak 0 ≤ r ≤ 0.5, and moderate otherwise
2-6 Multivariate Data
2-6 Multivariate Data
2-6 Multivariate Data
2-6 Multivariate Data
2-6 Multivariate Data