1 Data Summary and Presentation

NoteMajor Themes of Chapter 2

  • Descriptive Statistics: Learn to compute and interpret measures of central tendency (mean, median) and variability (variance, standard deviation, range)

  • Data Visualization: Master key graphical displays including histograms, box plots, stem-and-leaf diagrams, and scatter plots

  • Distribution Analysis: Understand how to assess the shape, center, spread, and outliers in datasets

  • Time Series Patterns: Recognize trends, cycles, and other temporal patterns in engineering data

  • Multivariate Relationships: Explore relationships between multiple variables using correlation analysis and scatter plot matrices

  • Quality Tools: Apply Pareto charts and other specialized plots for engineering problem-solving

ImportantLearning Objectives

After careful study of this chapter, you should be able to do the following:

  1. Compute and interpret the sample mean, sample variance, sample standard deviation, sample median, and sample range.

  2. Explain the concepts of sample mean, sample variance, population mean, and population variance.

  3. Construct and interpret visual data displays, including the stem-and-leaf display, the histogram, and the box plot and understand how these graphical techniques are useful in uncovering and summarizing patterns in data.

  4. Explain how to use box plots and other data displays to visually compare two or more samples of data.

  5. Know how to use simple time series plots to visually display the important features of time-oriented data.

  6. Construct scatter plots and compute and interpret a sample correlation coefficient.

1.1 Data Summary and Display

1.1.1 Mean

Sample Mean

NoteSample Mean

If the n observations in a sample are denoted by X_{1},X_{2},\ldots,X_{n}, the sample mean is

\begin{aligned} \overline{X} & = \frac{X_{1} + X_{2} + \cdots + X_{n}}{n} \\ \overline{X} & = \frac{\sum_{i=1}^{n} X_{i}}{n} \end{aligned}

Population Mean

NotePopulation Mean

When there is a finite number of observations (say, N) in the population, the population mean is

\begin{aligned} \mu & = \frac{\sum_{i=1}^{N} X_{i}}{N} \end{aligned}

The sample mean, \overline{X}, is a reasonable estimate of the population mean, \mu.

Example

NoteExample: O-Ring Tensile Strength

Engineering Context: This example demonstrates how the sample mean serves as a measure of central tendency in materials testing. Understanding tensile strength is crucial for ensuring component reliability in aerospace applications.

Consider the O-ring tensile strength experiment described in Chapter 1. The data from the modified rubber compound (1048, 1059, 1047, 1066, 1040, 1070, 1037, 1073) are shown in the dot diagram (Figure 1 and Figure 2).

Manual Calculation: The sample mean strength (psi) for the eight observations is:

\begin{aligned} \overline{X} & = \frac{\sum_{i=1}^{n} X_{i}}{n} \\ \overline{X} & = \frac{1037 + 1047 + \cdots + 1073}{8} \\ \overline{X} & = \frac{8440}{8} = 1055 \text{ psi} \end{aligned}

Physical Interpretation: The sample mean \overline{X} = 1055 can be thought of as a “balance point.” If each observation represents 1 pound of mass placed at the point on the x-axis, a fulcrum located at \overline{X} would exactly balance this system of weights.

Engineering Significance: This mean value provides a single representative measure that engineers can use to compare different rubber compounds or assess whether the material meets specifications.

Figure 1: Dot diagram of O-ring tensile strength. The sample mean is shown as a balance point for a system of weights.
O-Ring Tensile Strength Statistics
n sum mean
8 8440 1055 
# Load required libraries
library(fastverse)
library(kit)
library(data.table)
library(ggplot2)
library(scales)
library(kableExtra)
library(car)
library(GGally)

# O-ring tensile strength example
tensile_data <-
  data.table(TS = c(1048, 1059, 1047, 1066, 1040, 1070, 1037, 1073))

tensile_data_summary1 <-
  tensile_data %>%
  fsummarise(
    n = fnobs(TS),
    sum = fsum(TS),
    mean = fmean(TS)
  )

tensile_data_summary1 %>%
  kbl(caption = "O-Ring Tensile Strength Statistics", digits = 2) %>%
  kable_styling(bootstrap_options = c("striped", "hover"))
Figure 2: Dot diagram of O-ring tensile strength. The sample mean is shown as a balance point for a system of weights. 
# Dot diagram showing sample mean as balance point
ggplot(data = tensile_data, mapping = aes(x = TS)) +
  geom_dotplot(binwidth = 1, stackdir = "up", dotsize = 1) +
  geom_vline(aes(xintercept = fmean(TS)), color = "red", linetype = "dashed", linewidth = 1) +
  scale_x_continuous(breaks = pretty_breaks(n = 8)) +
  labs(x = "Tensile Strength (psi)", y = NULL, title = "Sample Mean as Balance Point") +
  theme_classic() +
  theme(axis.ticks.y = element_blank(), axis.text.y = element_blank())

1.1.2 Variance and Standard Deviation

Sample Variance and Sample Standard Deviation

NoteSample Variance and Sample Standard Deviation

If the n observations in a sample are denoted by X_{1},X_{2},\ldots,X_{n}, then the sample variance is

\begin{aligned} S^2 & = \frac{\sum_{i=1}^{n}\left(X_{i}-\overline{X}\right)^{2}}{n-1} \end{aligned}

The sample standard deviation, S, is the positive square root of the sample variance.

Engineering Perspective: The units of measurement for the sample variance are the square of the original units. Thus, if X is measured in psi, the variance units are (\text{psi})^2. The standard deviation has the desirable property of measuring variability in the original units (psi), making it more interpretable for engineers.

Example

NoteExample: Variability in O-Ring Strength

Why Variability Matters: In engineering applications, understanding variability is often more critical than knowing the average. High variability can indicate process instability, quality issues, or the need for tighter manufacturing controls.

The numerator of S^2 is (See Table 1): \sum_{i=1}^{n}\left(X_{i}-\overline{X}\right)^{2} = 1348

Sample Variance Calculation:

\begin{aligned} S^2 & = \frac{\sum_{i=1}^{n}\left(X_{i}-\overline{X}\right)^{2}}{n-1} = \frac{1348}{8-1} = 192.57 \text{ psi}^2 \end{aligned}

Sample Standard Deviation:

\begin{aligned} S & = \sqrt{192.57} = 13.9 \text{ psi} \end{aligned}

Engineering Interpretation: A standard deviation of 13.9 psi means that most O-ring tensile strengths fall within about 14 psi of the mean (1055 psi). This gives engineers insight into the consistency of the manufacturing process.

Computational Note: \sum_{i=1}^{n}\left(X_{i}-\overline{X}\right)^{2}=\sum_{i=1}^{n}X_{i}^{2}-\frac{\left(\sum_{i=1}^{n}X_{i}\right)^{2}}{n} (computational formula).

O-Ring Tensile Strength Statistics
var sd
192.57 13.88 
tensile_data_summary2 <-
  tensile_data %>%
  fsummarise(
    var = fvar(TS),
    sd = fsd(TS)
  )

tensile_data_summary2 %>%
  kbl(caption = "O-Ring Tensile Strength Statistics", digits = 2) %>%
  kable_styling(bootstrap_options = c("striped", "hover"))
Table 1: Variance Calculations
Variance Calculation Terms
i Xi Xi - X̄ (Xi - X̄)²
1 1048 -7 49
2 1059 4 16
3 1047 -8 64
4 1066 11 121
5 1040 -15 225
6 1070 15 225
7 1037 -18 324
8 1073 18 324
Total 8440 0 1348

How Does the Sample Variance Measure Variability?

NoteUnderstanding Variability Measurement

Engineering Insight: To see how the sample variance measures dispersion or variability, refer to Figure 3, which shows the deviations X_{i}-\overline{X} for the O-ring tensile strength data.

Key Concepts:

  • Greater variability → larger absolute deviations from the mean

  • Deviations X_{i}-\overline{X} always sum to zero

  • Squaring eliminates negative values and emphasizes larger deviations

  • Small S^2 indicates consistent process; large S^2 suggests high variability

Process Control Implications: Engineers use variance to monitor process stability. Increasing variance over time may signal equipment wear, material inconsistencies, or environmental factors affecting production.

Figure 3: How the sample variance measures variability through the deviations X_{i}-\overline{X}.

Population Variance

NotePopulation Variance

When the population is finite and consists of N values, we may define the population variance as

\begin{aligned} \sigma^2 & = \frac{\sum_{i=1}^{N}\left(X_{i}-\mu\right)^{2}}{N} \end{aligned}

The sample variance, S^2, is a reasonable estimate of the population variance, \sigma^2.

1.2 Stem-and-Leaf Diagram

NoteStem-and-Leaf Diagram

Engineering Application: Stem-and-leaf diagrams are particularly useful for initial data exploration in engineering. They preserve actual data values while showing distribution shape, making them ideal for quality control and process analysis.

A stem-and-leaf diagram provides an informative visual display of a data set X_{1},X_{2},\ldots,X_{n}, where each number X_{i} consists of at least two digits.

TipSteps for Constructing a Stem-and-Leaf Diagram

  1. Divide each number X_{i} into two parts: a stem (leading digits) and a leaf (remaining digit)

  2. List the stem values in a vertical column

  3. Record the leaf for each observation beside its stem

  4. Write the units for stems and leaves on the display

1.2.1 Example

NoteExample: Aluminum-Lithium Alloy Compressive Strength

Engineering Context: This example analyzes compressive strength data from 80 aluminum-lithium alloy specimens. Such analysis is crucial for aerospace applications where weight reduction and strength are both critical.

Data Overview: The compressive strength measurements (in psi) from Table 2 represent quality control testing of a lightweight alloy used in aircraft components.

Table 2: Compressive Strength of 80 Aluminum-Lithium Alloy Specimens
105 221 183 186 121 181 180 143
97 154 153 174 120 168 167 141
245 228 174 199 181 158 176 110
163 131 154 115 160 208 158 133
207 180 190 193 194 133 156 123
134 178 76 167 184 135 229 146
218 157 101 171 165 172 158 169
199 151 142 163 145 171 148 158
160 175 149 87 160 237 150 135
196 201 200 176 150 170 118 149

We select stem values as the numbers 7, 8, 9, \ldots, 24 (representing tens place).


  The decimal point is 1 digit(s) to the right of the |

   7 | 6
   8 | 7
   9 | 7
  10 | 15
  11 | 058
  12 | 013
  13 | 133455
  14 | 12356899
  15 | 001344678888
  16 | 0003357789
  17 | 0112445668
  18 | 0011346
  19 | 034699
  20 | 0178
  21 | 8
  22 | 189
  23 | 7
  24 | 5
# Stem-and-leaf diagram for compressive strength data
compressive_data <- fread("./data/Exmp2.4.csv")
stem(compressive_data[, CS], scale = 2, width = 80)

Engineering Interpretation: The stem-and-leaf display reveals:

  • Most compressive strengths lie between 110 and 200 psi

  • Central value is approximately 150-160 psi

  • Distribution is approximately symmetric

  • No obvious outliers or unusual patterns

  • The alloy appears to have consistent strength properties

Quality Control Insights: This symmetric distribution suggests the manufacturing process is well-controlled. Engineers can use this information to set specification limits and monitor future production.

1.3 Histograms

1.3.1 Histogram

NoteHistogram

Engineering Definition: A histogram is a graphical representation of data distribution, showing frequency or relative frequency within specified intervals (bins). Histograms are essential tools for:

  • Process capability analysis

  • Quality control monitoring

  • Design specification validation

  • Statistical process control

Each bin represents a range of values, and bar height reflects the number (or proportion) of data points in that range.

1.3.2 Example

NoteExample: Golf Ball Distance Testing

Engineering Application: The USGA tests golf balls to ensure conformance to regulations. This standardized testing demonstrates how consistent measurement protocols are essential in engineering quality control.

Testing Protocol: Balls are tested using “Iron Byron,” a mechanical device that eliminates human variability, ensuring reproducible test conditions. This exemplifies the engineering principle of controlling sources of variation.

Table 3: Golf Ball Distance Data
291.5 274.4 290.2 276.4 272.0 268.7 281.6 281.6 276.3 285.9
269.6 266.6 283.6 269.6 277.8 287.8 267.6 292.6 273.4 284.4
270.7 274.0 285.2 275.5 272.1 261.3 274.0 279.3 281.0 293.1
277.5 278.0 272.5 271.7 280.8 265.6 260.1 272.5 281.3 263.0
279.0 267.3 283.5 271.2 268.5 277.1 266.2 266.4 271.5 280.3
267.8 272.1 269.7 278.5 277.3 280.5 270.8 267.7 255.1 276.4
283.7 281.7 282.2 274.1 264.5 281.0 273.2 274.4 281.6 273.7
271.0 271.5 289.7 271.1 256.9 274.5 286.2 273.9 268.5 262.6
261.9 258.9 293.2 267.1 255.0 269.7 281.9 269.6 279.8 269.9
282.6 270.0 265.2 277.7 275.5 272.2 270.0 271.0 284.3 268.4

Histogram Design Principles: With 100 observations, \sqrt{100} = 10 suggests about 10 bins for optimal visualization. The analysis reveals important distribution characteristics for engineering decision-making.

Engineering Insights from Histograms:

  • Distribution Shape: The reasonably symmetric, bell-shaped distribution suggests the manufacturing process is well-controlled

  • Process Capability: The spread indicates natural process variation

  • Specification Conformance: Engineers can assess what percentage of balls meet distance requirements

(a) Histogram with 9 bins for the golf ball distance data
(b) Histogram with 16 bins for the golf ball distance data
(c) A relative frequency plot of the golf ball distance data
(d) A cumulative frequency plot of the golf ball distance data
Figure 4: Histogram for the golf ball distance data 
# Golf ball distance histograms
golf_data <- fread("./data/Exmp2.6.csv")

# Basic histogram with 9 bins
p1 <-
  ggplot(golf_data, aes(x = Distance)) +
  geom_histogram(bins = 9, fill = "lightblue", color = "black") +
  scale_x_continuous(breaks = pretty_breaks(n = 8)) +
  scale_y_continuous(expand = c(0, 0), limits = c(0, NA)) +
  labs(x = "Distance (yards)", y = "Frequency", title = "Histogram with 9 bins") +
  theme_classic()

# Histogram with 16 bins
p2 <-
  ggplot(golf_data, aes(x = Distance)) +
  geom_histogram(bins = 16, fill = "lightgreen", color = "black") +
  scale_x_continuous(breaks = pretty_breaks(n = 8)) +
  scale_y_continuous(expand = c(0, 0), limits = c(0, NA)) +
  labs(x = "Distance (yards)", y = "Frequency", title = "Histogram with 16 bins") +
  theme_classic()

# Relative frequency histogram
p3 <-
  ggplot(golf_data, aes(x = Distance)) +
  geom_histogram(aes(y = after_stat(count / sum(count))),
    bins = 9,
    fill = "orange", color = "black"
  ) +
  scale_x_continuous(breaks = pretty_breaks(n = 8)) +
  scale_y_continuous(expand = c(0, 0), limits = c(0, NA), labels = percent_format()) +
  labs(x = "Distance (yards)", y = "Relative Frequency", title = "Relative Frequency") +
  theme_classic()

# Cumulative frequency histogram
p4 <-
  ggplot(golf_data, aes(x = Distance)) +
  geom_histogram(aes(y = after_stat(cumsum(count))),
    bins = 10,
    fill = "pink", color = "black"
  ) +
  scale_x_continuous(breaks = pretty_breaks(n = 8)) +
  scale_y_continuous(expand = c(0, 0), limits = c(0, NA)) +
  labs(x = "Distance (yards)", y = "Cumulative Frequency", title = "Cumulative Frequency") +
  theme_classic()

print(p1)
print(p2)
print(p3)
print(p4)

1.3.3 Pareto Chart

NotePareto Chart

Engineering Application: Pareto charts are fundamental tools in quality engineering and process improvement, embodying the “80-20 rule” or “vital few vs. trivial many” principle. They help engineers prioritize improvement efforts by identifying the most significant causes of problems.

Key Features:

  • Categories ordered by frequency (highest to lowest)

  • Focuses attention on major contributors

  • Guides resource allocation for maximum impact

  • Essential for root cause analysis

1.3.4 Example

NoteExample: Aircraft Safety Analysis

Engineering Context: This analysis of aircraft accident data demonstrates how Pareto charts help engineers identify critical safety issues and prioritize design improvements.

Data Source: Aircraft accident rates from The Wall Street Journal analysis, showing hull losses per million departures between 1959-1999 for 22 aircraft types.

Table 4: Aircraft Accident Data
Aircraft Type Actual Number of Hull Losses Hull Losses/Million Departures
MD-11 5 6.54
707/720 115 6.46
DC-8 71 5.84
F-28 32 3.94
BAC 1-11 22 2.64
DC-10 20 2.57
747-Early 21 1.9

Engineering Analysis: The top three aircraft types account for the majority of incidents on a per-million-departures basis. Notably:

  • 707/720 and DC-8: Mid-1950s designs, largely out of service

  • MD-11: 1990s design with concerning safety record (5 losses out of 198 aircraft)

Process Improvement Insights: This Pareto analysis helps aviation engineers:

  • Focus safety improvements on high-risk aircraft designs

  • Understand the relationship between design era and safety performance

  • Allocate research resources to address the “vital few” risk factors

Figure 5: Pareto chart for the aircraft accident data 
# Pareto chart for aircraft accident data
aircraft_data <- fread("./data/Exmp2.8.csv")

aircraft_data %>%
  fmutate(Type = factor(Type, levels = Type[order(-HLMD)])) %>%
  ggplot(aes(x = Type, y = HLMD)) +
  geom_col(fill = "steelblue", color = "black") +
  scale_y_continuous(expand = c(0, 0), limits = c(0, NA)) +
  scale_x_discrete(guide = guide_axis(n.dodge = 2)) +
  labs(
    x = "Aircraft Type",
    y = "Hull Losses per Million Departures",
    title = "Pareto Chart: Aircraft Accident Rates"
  ) +
  theme_classic()

1.4 Box Plot

NoteBox Plot

Engineering Applications: Box plots are powerful tools for engineering data analysis, providing comprehensive information about data distribution in a compact visual format.

What Box Plots Reveal:

  • Measure of Central Tendency (median)

  • Measure of Variability (IQR, range)

  • Measure of Symmetry (distribution shape)

  • Outliers (unusual observations)

  • Group Comparisons (process/design alternatives)

Box Whisker Plot

Statistical Definitions:

\begin{align*} \text{Lower Inner Fence} & =Q_{1}-1.5\times\text{IQR}\\ \text{Upper Inner Fence} & =Q_{3}+1.5\times\text{IQR}\\ \text{Lower Outer Fence} & =Q_{1}-3\times\text{IQR}\\ \text{Upper Outer Fence} & =Q_{3}+3\times\text{IQR} \end{align*}

Outlier Classification:

  • Mild Outlier: Beyond inner fence (but inside outer fence)

  • Extreme Outlier: Beyond outer fence

Engineering Interpretation: Whiskers extend to the most extreme values that are not outliers, helping engineers identify unusual measurements that may indicate process problems or measurement errors.

Box Plot Statistics
q1 median q3 iqr lower_fence upper_fence
144.5 161.5 181 36.5 89.75 235.75
Figure 6: Box plot for compressive strength data
Figure 7: Box plot for compressive strength data 
# Box plot for compressive strength data
ggplot(compressive_data, aes(y = CS)) +
  geom_boxplot(width = 0.3, fill = "lightblue") +
  scale_y_continuous(breaks = pretty_breaks(n = 8)) +
  labs(
    y = "Compressive Strength (psi)", x = NULL,
    title = "Box Plot: Compressive Strength Distribution"
  ) +
  theme_classic() +
  theme(axis.ticks.x = element_blank(), axis.text.x = element_blank())

box_stats <- compressive_data %>%
  fsummarise(
    q1 = fquantile(CS, 0.25),
    median = fmedian(CS),
    q3 = fquantile(CS, 0.75),
    iqr = fquantile(CS, 0.75) - fquantile(CS, 0.25),
    lower_fence = fquantile(CS, 0.25) - 1.5 * (fquantile(CS, 0.75) - fquantile(CS, 0.25)),
    upper_fence = fquantile(CS, 0.75) + 1.5 * (fquantile(CS, 0.75) - fquantile(CS, 0.25))
  )

box_stats %>%
  kbl(caption = "Box Plot Statistics", digits = 2) %>%
  kable_styling(bootstrap_options = c("striped", "hover"))
Figure 8: Comparative box plots of a quality index at three plants.

1.5 Time Series Plots

NoteTime Series Plots

Engineering Importance: Time series analysis is critical in engineering for monitoring processes, identifying trends, and predicting future behavior. Understanding temporal patterns helps engineers:

  • Detect process drift or shifts

  • Identify cyclical behaviors

  • Plan maintenance schedules

  • Optimize production timing

Key Concepts:

  • Time Series: Data recorded in sequential order over time

  • Trend: Long-term increase or decrease in values

  • Cycles: Regular patterns that repeat over time

  • Seasonality: Predictable patterns related to calendar periods

Engineering Applications:

  • Equipment performance monitoring

  • Quality control charts

  • Production scheduling

  • Predictive maintenance

(a) Company sales by year
(b) Company sales by quarter
Figure 9: Company sales
(a) Annual sales trend
(b) Quarterly sales pattern
Figure 10: Time series plots showing trends and patterns 
# Time series example (simulated company sales data)
set.seed(123)
time_data <-
  data.table(
    year = 2015:2024,
    quarter = rep(1:4, length.out = 40),
    sales = 100 + 0.02 * (1:40)^2 + 10 * sin(2 * pi * (1:40) / 4) + rnorm(40, 0, 5)
  ) %>%
  fmutate(time_period = year + (quarter - 1) / 4)

# Annual sales trend
p5 <-
  time_data %>%
  fgroup_by(year) %>%
  fsummarise(annual_sales = fmean(sales)) %>%
  ggplot(aes(x = year, y = annual_sales)) +
  geom_line(color = "blue", linewidth = 1) +
  geom_point(size = 3) +
  labs(x = "Year", y = "Average Sales", title = "Annual Sales Trend") +
  theme_classic()

# Quarterly sales pattern
p6 <-
  ggplot(time_data, aes(x = time_period, y = sales)) +
  geom_line(color = "red", linewidth = 1) +
  geom_point(size = 2) +
  labs(x = "Time Period", y = "Sales", title = "Quarterly Sales Pattern") +
  theme_classic()

print(p5)
print(p6)

fwrite(time_data, "data/time_series_example.csv")
Figure 11: A digidot plot of the compressive strength data.
Figure 12: Digital dot plot of compressive strength data 
# Digital dot plot
compressive_data %>%
  fmutate(digit = round(CS / 10) * 10, count = 1) %>%
  fgroup_by(digit) %>%
  fsummarise(frequency = fsum(count)) %>%
  ggplot(aes(x = digit, y = frequency)) +
  geom_segment(aes(xend = digit, yend = 0), linewidth = 1) +
  geom_point(size = 3, color = "red") +
  labs(
    x = "Compressive Strength (grouped by 10s)", y = "Frequency",
    title = "Digital Dot Plot"
  ) +
  theme_classic()

1.6 Multivariate Data

NoteMultivariate Data

Engineering Reality: Most engineering problems involve multiple variables simultaneously. Understanding relationships between variables is essential for:

  • Process optimization

  • Design improvement

  • Root cause analysis

  • Predictive modeling

Key Concepts:

  • Univariate Analysis: Single variable (mean, variance, distribution)

  • Multivariate Analysis: Multiple variables and their relationships

  • Correlation: Strength of linear relationship between variables

  • Causation vs. Correlation: Important distinction for engineering decisions

1.6.1 Example

NoteExample: Wire Bond Strength Analysis

Engineering Context: Wire bonding is a critical process in semiconductor manufacturing. Understanding how process parameters affect bond strength is essential for reliable electronic devices.

Process Variables:

  • Pull Strength (Y): Response variable (quality measure)

  • Wire Length (X1): Process parameter

  • Die Height (X2): Design parameter

Table 5: Wire Bond Data
Obs No Pull Strength (Y) Wire Length (X1) Die Height (X2)
1 9.95 2 50
2 24.45 8 110
3 31.75 11 120
4 35.00 10 550
5 25.02 8 295
6 16.86 4 200
7 14.38 2 375
8 9.60 2 52
9 24.35 9 100
10 27.50 8 300
11 17.08 4 412
12 37.00 11 400
13 41.95 12 500
14 11.66 2 360
15 21.65 4 205
16 17.89 4 400
17 69.00 20 600
18 10.30 1 585
19 34.93 10 540
20 46.59 15 250
21 44.88 15 290
22 54.12 16 510
23 56.63 17 590
24 22.13 6 100
25 21.15 5 400

Engineering Questions:

  1. Which parameter most strongly affects bond strength?

  2. Are the parameters independent of each other?

  3. Can we predict strength from parameter settings?

(a) Scatter plot for Pull strength versus length
(b) Scatter plot for Pull strength versus die height
(c) Scatter and box plots for Pull strength versus length
(d) Scatter and box plots for Pull strength versus die height
Figure 13: Scatter diagrams and box plots for the wire bond pull strength data 
# Wire bond data analysis
wire_data <- fread("./data/Table2.9.csv")

# Scatter plot: Pull strength vs Wire length
p7 <-
  ggplot(wire_data, aes(x = X1, y = Y)) +
  geom_point(size = 3, alpha = 0.7) +
  geom_smooth(method = "lm", se = FALSE, color = "red") +
  labs(x = "Wire Length", y = "Pull Strength", title = "Pull Strength vs Wire Length") +
  theme_classic()

# Scatter plot: Pull strength vs Die height
p8 <-
  ggplot(wire_data, aes(x = X2, y = Y)) +
  geom_point(size = 3, alpha = 0.7) +
  geom_smooth(method = "lm", se = FALSE, color = "blue") +
  labs(x = "Die Height", y = "Pull Strength", title = "Pull Strength vs Die Height") +
  theme_classic()

print(p7)
print(p8)

library(car)
scatterplot(
  formula = Y ~ X1,
  data = wire_data,
  smooth = FALSE,
  regLine = FALSE,
  xlab = "Length",
  ylab = "Strength"
)

scatterplot(
  formula = Y ~ X2,
  data = wire_data,
  smooth = FALSE,
  regLine = FALSE,
  xlab = "Die height",
  ylab = "Strength"
)

1.6.2 Sample Correlation Coefficient

NoteSample Correlation Coefficient

Engineering Definition: The correlation coefficient quantifies the strength and direction of linear relationships between variables, crucial for understanding process interactions.

Given n pairs of data \left(X_{1},Y_{1}\right),\left(X_{2},Y_{2}\right),\ldots,\left(X_{n},Y_{n}\right):

r=\frac{\sum_{i=1}^{n}\left(X_{i}-\overline{X}\right)\left(Y_{i}-\overline{Y}\right)}{\sqrt{\sum_{i=1}^{n}\left(X_{i}-\overline{X}\right)^{2}\sum_{i=1}^{n}\left(Y_{i}-\overline{Y}\right)^{2}}}

where -1\leq r\leq+1.

Engineering Interpretation:

  • r ≈ +1: Strong positive relationship (as X increases, Y increases)

  • r ≈ -1: Strong negative relationship (as X increases, Y decreases)

  • r ≈ 0: Weak linear relationship (may still have nonlinear relationship)

(a) r is near +1
(b) r is near -1
(c) r is near 0, Y and X are unrelated
(d) r is near 0, Y and X are nonlinearly related
Figure 14: Scatter diagrams for different values of the sample correlation coefficient r.

1.6.3 Example

NoteExample: Shampoo Quality Analysis

Engineering Context: This multivariate analysis demonstrates how engineers evaluate product quality across multiple sensory and performance characteristics. Understanding these relationships helps optimize formulations.

Quality Variables:

  • Foam: Lathering performance

  • Scent: Fragrance intensity

  • Color: Visual appeal

  • Residue: Cleansing effectiveness

  • Region: Market segment

  • Quality: Overall consumer rating

Table 6: Data on Shampoo
Foam Scent Color Residue Region Quality
6.3 5.3 4.8 3.1 1 91
4.4 4.9 3.5 3.9 1 87
3.9 5.3 4.8 4.7 1 82
5.1 4.2 3.1 3.6 1 83
5.6 5.1 5.5 5.1 1 83
4.6 4.7 5.1 4.1 1 84
4.8 4.8 4.8 3.3 1 90
6.5 4.5 4.3 5.2 1 84
8.7 4.3 3.9 2.9 1 97
8.3 3.9 4.7 3.9 1 93
5.1 4.3 4.5 3.6 1 82
3.3 5.4 4.3 3.6 1 84
5.9 5.7 7.2 4.1 2 87
7.7 6.6 6.7 5.6 2 80
7.1 4.4 5.8 4.1 2 84
5.5 5.6 5.6 4.4 2 84
6.3 5.4 4.8 4.6 2 82
4.3 5.5 5.5 4.1 2 79
4.6 4.1 4.3 3.1 2 81
3.4 5.0 3.4 3.4 2 83
6.4 5.4 6.6 4.8 2 81
5.5 5.3 5.3 3.8 2 84
4.7 4.1 5.0 3.7 2 83
4.1 4.0 4.1 4.0 2 80

Engineering Analysis Goals:

  1. Identify which characteristics most influence overall quality

  2. Understand relationships between formulation parameters

  3. Optimize product design for target markets

Table 7: Correlation Matrix
Shampoo Data Correlation Matrix
Foam Scent Color Residue Region Quality
Foam 1.000 0.002 0.328 0.193 -0.032 0.512
Scent 0.002 1.000 0.599 0.500 0.278 -0.252
Color 0.328 0.599 1.000 0.524 0.458 -0.194
Residue 0.193 0.500 0.524 1.000 0.165 -0.489
Region -0.032 0.278 0.458 0.165 1.000 -0.507
Quality 0.512 -0.252 -0.194 -0.489 -0.507 1.000
Shampoo Data Summary Statistics
Foam_mean Foam_sd Foam_min Foam_max Scent_mean Scent_sd Scent_min Scent_max Color_mean Color_sd Color_min Color_max Residue_mean Residue_sd Residue_min Residue_max Region_mean Region_sd Region_min Region_max Quality_mean Quality_sd Quality_min Quality_max
5.5 1.45 3.3 8.7 4.91 0.67 3.9 6.6 4.9 1.02 3.1 7.2 4.03 0.7 2.9 5.6 1.5 0.51 1 2 84.5 4.36 79 97
Table 8 
# Shampoo data correlation matrix
shampoo_data <- fread("./data/Table2.11.csv")

cor_matrix <- cor(shampoo_data)
cor_matrix %>%
  round(3) %>%
  kbl(caption = "Shampoo Data Correlation Matrix") %>%
  kable_styling(bootstrap_options = c("striped", "hover"))

shampoo_summary <-
  shampoo_data %>%
  fsummarise(
    across(
      .cols = is.numeric,
      .fns = list(
        mean = fmean,
        sd = fsd,
        min = fmin,
        max = fmax
      )
    )
  )

shampoo_summary %>%
  kbl(caption = "Shampoo Data Summary Statistics", digits = 2) %>%
  kable_styling(bootstrap_options = c("striped", "hover"))
Figure 15: Matrix of scatter plots for the shampoo data 
# Matrix of scatter plots for shampoo data
ggpairs(shampoo_data,
  title = "Shampoo Quality Data: Pairwise Relationships",
  upper = list(continuous = wrap("cor", size = 3))
) +
  theme_classic()

1.7 Key Engineering Takeaways

This chapter provided essential tools for engineering data analysis:

1.7.1 Statistical Foundations

  • Measures of Central Tendency: Sample mean as balance point for process centering

  • Measures of Variability: Standard deviation for process control and capability analysis

  • Distribution Analysis: Understanding natural process variation patterns

1.7.2 Visualization Tools

  • Histograms: Assess process capability and distribution normality

  • Box Plots: Identify outliers and compare process alternatives

  • Stem-and-Leaf: Quick distribution assessment preserving actual values

  • Pareto Charts: Prioritize improvement efforts using 80-20 principle

1.7.3 Process Analysis

  • Time Series: Monitor trends, detect shifts, identify cycles

  • Correlation Analysis: Understand variable relationships for optimization

  • Multivariate Methods: Analyze complex engineering systems

1.7.4 Engineering Applications

  • Quality control and process improvement

  • Design optimization and parameter selection

  • Root cause analysis and problem solving

  • Specification setting and capability assessment

These statistical tools form the foundation for evidence-based engineering decisions and continuous improvement initiatives.