Building on our visualization skills from Lecture 1, today we dive deep into the mathematical heart of statistics:
Imagine you’re the lead analyst for a European automotive consortium. Your dataset contains critical intelligence about 190 car models that will inform billion-dollar investment decisions. Every statistical measure we calculate today has direct implications for:
# Load our analytical environment
library(UBStats)## Package UBStats (0.2.2) loaded.
## To cite, type citation("UBStats")## Please report improvements and bugs to: https://github.com/raffaellapiccarreta/UBStats/issues# Load the cars dataset from Lecture 1
# Create the cars dataset for statistical analysis
# This code reproduces the dataset from Lecture 1
# Load required packages
library(UBStats)
# Set seed for reproducible results
set.seed(123)
n <- 190
# Generate sales data with realistic distribution
low_sales <- sample(500:3000, round(n*0.6), replace = TRUE)
mid_sales <- sample(3000:8000, round(n*0.25), replace = TRUE)
high_sales <- sample(8000:50000, n - length(low_sales) - length(mid_sales), replace = TRUE)
all_sales <- c(low_sales, mid_sales, high_sales)
# Create the complete cars dataset
cars <- data.frame(
model = paste("Model", 1:n),
sales = sample(all_sales), # Shuffle the sales values
bestselling = sample(0:1, n, replace = TRUE, prob = c(0.9, 0.1)),
price_num = round(rnorm(n, 25000, 15000)),
price_classes = sample(c("low", "mid", "high"), n, replace = TRUE, prob = c(0.27, 0.55, 0.18)),
maxspeed = round(rnorm(n, 180, 30)),
acceleration = round(rnorm(n, 11, 3), 1),
urban_fuelcons = round(rnorm(n, 8, 2), 1),
fueltank = round(rnorm(n, 60, 15)),
weight = round(rnorm(n, 1400, 300)),
n_doors_min = sample(c(2,3,4,5,7), n, replace = TRUE, prob = c(0.09, 0.14, 0.05, 0.71, 0.01)),
country = sample(c("Germany", "Japan", "France", "Italy", "United States", "Europe - others", "Asia - others"),
n, replace = TRUE, prob = c(0.26, 0.19, 0.15, 0.11, 0.09, 0.14, 0.06))
)
# Clean up unrealistic values
cars$price_num[cars$price_num < 5000] <- cars$price_num[cars$price_num < 5000] + 10000
cars$maxspeed[cars$maxspeed < 100] <- cars$maxspeed[cars$maxspeed < 100] + 50
cars$acceleration[cars$acceleration < 3] <- abs(cars$acceleration[cars$acceleration < 3]) + 5
# Check the data structure
str(cars)## 'data.frame': 190 obs. of 12 variables:
## $ model : chr "Model 1" "Model 2" "Model 3" "Model 4" ...
## $ sales : int 12712 873 1528 23023 2956 1346 3711 2726 37652 6123 ...
## $ bestselling : int 0 0 0 0 0 0 0 0 0 0 ...
## $ price_num : num 32656 42939 8741 7823 27328 ...
## $ price_classes : chr "low" "mid" "low" "mid" ...
## $ maxspeed : num 169 137 153 214 158 194 171 205 177 159 ...
## $ acceleration : num 4.7 11 18.2 9.8 15 12.1 11.8 10.7 9 10.8 ...
## $ urban_fuelcons: num 6.2 4.7 5.4 7.4 9.1 11.6 6 7 9.9 7.5 ...
## $ fueltank : num 65 46 87 39 85 62 60 40 72 79 ...
## $ weight : num 1445 1228 1499 1321 1585 ...
## $ n_doors_min : num 5 5 5 4 5 2 5 2 5 5 ...
## $ country : chr "Japan" "Italy" "Italy" "Japan" ...head(cars)# Save the dataset
save(cars, file = "stat_datasets_cl17.Rdata")
# Confirm the file was created
cat("✅ Dataset created successfully!\n")## ✅ Dataset created successfully!cat("📊 Dataset contains", nrow(cars), "car models with", ncol(cars), "variables\n")## 📊 Dataset contains 190 car models with 12 variablescat("💾 Saved as: stat_datasets_cl17.Rdata\n")## 💾 Saved as: stat_datasets_cl17.Rdatacat("📁 Location:", getwd(), "\n")## 📁 Location: C:/Users/ENDRI/Desktop/Virtus# Quick preview of the data
cat("\n📋 Dataset Summary:\n")##
## 📋 Dataset Summary:summary(cars[c("price_num", "maxspeed", "acceleration")])## price_num maxspeed acceleration
## Min. :-1860 Min. :103.0 Min. : 3.200
## 1st Qu.:13574 1st Qu.:159.2 1st Qu.: 8.825
## Median :24177 Median :178.0 Median :10.900
## Mean :24267 Mean :180.6 Mean :10.837
## 3rd Qu.:33195 3rd Qu.:203.0 3rd Qu.:12.975
## Max. :61570 Max. :279.0 Max. :18.200# Quick reminder of our data structure
str(cars)## 'data.frame': 190 obs. of 12 variables:
## $ model : chr "Model 1" "Model 2" "Model 3" "Model 4" ...
## $ sales : int 12712 873 1528 23023 2956 1346 3711 2726 37652 6123 ...
## $ bestselling : int 0 0 0 0 0 0 0 0 0 0 ...
## $ price_num : num 32656 42939 8741 7823 27328 ...
## $ price_classes : chr "low" "mid" "low" "mid" ...
## $ maxspeed : num 169 137 153 214 158 194 171 205 177 159 ...
## $ acceleration : num 4.7 11 18.2 9.8 15 12.1 11.8 10.7 9 10.8 ...
## $ urban_fuelcons: num 6.2 4.7 5.4 7.4 9.1 11.6 6 7 9.9 7.5 ...
## $ fueltank : num 65 46 87 39 85 62 60 40 72 79 ...
## $ weight : num 1445 1228 1499 1321 1585 ...
## $ n_doors_min : num 5 5 5 4 5 2 5 2 5 5 ...
## $ country : chr "Japan" "Italy" "Italy" "Japan" ...head(cars, 5)cat("📊 Dataset Overview:\n")## 📊 Dataset Overview:cat(" Total Models:", nrow(cars), "\n")## Total Models: 190cat(" Variables:", ncol(cars), "\n")## Variables: 12cat(" Geographic Coverage:", length(unique(cars$country)), "countries\n")## Geographic Coverage: 7 countriesThe concept of central tendency answers the fundamental question: “What is the typical value in our dataset?” However, “typical” can mean different things depending on context and data characteristics.
The arithmetic mean represents the mathematical center of gravity for your data.
For Raw Data (Ungrouped): \[\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} = \frac{x_1 + x_2 + ... + x_n}{n}\]
For Frequency Data (Grouped): \[\bar{x} = \frac{\sum_{i=1}^{k} f_i \cdot x_i}{\sum_{i=1}^{k} f_i} = \frac{\sum_{i=1}^{k} f_i \cdot x_i}{n}\]
Where: - \(x_i\) = individual data values or class midpoints - \(f_i\) = frequency of each value/class - \(n\) = total number of observations - \(k\) = number of distinct values/classes
# Example 1: Mean calculation for a subset of car prices
sample_prices <- c(18000, 22000, 25000, 28000, 35000)
# Manual calculation following the formula
sum_prices <- sum(sample_prices)
n_cars <- length(sample_prices)
mean_manual <- sum_prices / n_cars
cat("🚗 Sample Car Prices: $", paste(sample_prices, collapse = ", $"), "\n")## 🚗 Sample Car Prices: $ 18000, $22000, $25000, $28000, $35000cat("📊 Manual Calculation:\n")## 📊 Manual Calculation:cat(" Sum of prices: $", sum_prices, "\n")## Sum of prices: $ 128000cat(" Number of cars: ", n_cars, "\n")## Number of cars: 5cat(" Mean = ", sum_prices, " ÷ ", n_cars, " = $", round(mean_manual, 2), "\n")## Mean = 128000 ÷ 5 = $ 25600# Verification with R function
mean_r <- mean(sample_prices)
cat("✅ R function verification: $", round(mean_r, 2), "\n")## ✅ R function verification: $ 25600Let’s calculate the mean using frequency data from our car ownership example:
# Car ownership data from the lecture slides
cars_owned <- c(1, 2, 3, 5)
frequencies <- c(32, 48, 16, 4)
n_families <- sum(frequencies)
cat("🏠 Family Car Ownership Analysis:\n")## 🏠 Family Car Ownership Analysis:cat("Cars Owned: ", paste(cars_owned, collapse = ", "), "\n")## Cars Owned: 1, 2, 3, 5cat("Frequencies: ", paste(frequencies, collapse = ", "), "\n")## Frequencies: 32, 48, 16, 4# Manual weighted mean calculation
weighted_sum <- sum(cars_owned * frequencies)
weighted_mean <- weighted_sum / n_families
cat("\n📊 Weighted Mean Calculation:\n")##
## 📊 Weighted Mean Calculation:cat(" Σ(xi × fi) = ", paste(cars_owned, "×", frequencies, collapse = " + "), "\n")## Σ(xi × fi) = 1 × 32 + 2 × 48 + 3 × 16 + 5 × 4cat(" = ", paste(cars_owned * frequencies, collapse = " + "), "\n")## = 32 + 96 + 48 + 20cat(" = ", weighted_sum, "\n")## = 196cat(" Mean = ", weighted_sum, " ÷ ", n_families, " = ", round(weighted_mean, 3), " cars per family\n")## Mean = 196 ÷ 100 = 1.96 cars per family# Business interpretation
cat("\n💡 Business Insight: The average family owns ", round(weighted_mean, 2), " cars\n")##
## 💡 Business Insight: The average family owns 1.96 cars# Calculate mean for key variables in our cars dataset
price_mean <- mean(cars$price_num, na.rm = TRUE)
speed_mean <- mean(cars$maxspeed, na.rm = TRUE)
accel_mean <- mean(cars$acceleration, na.rm = TRUE)
cat("🚗 AUTOMOTIVE MARKET AVERAGES:\n")## 🚗 AUTOMOTIVE MARKET AVERAGES:cat("💰 Average Price: $", round(price_mean, 0), "\n")## 💰 Average Price: $ 24267cat("⚡ Average Max Speed: ", round(speed_mean, 1), " km/h\n")## ⚡ Average Max Speed: 180.6 km/hcat("🏁 Average Acceleration: ", round(accel_mean, 2), " seconds (0-100 km/h)\n")## 🏁 Average Acceleration: 10.84 seconds (0-100 km/h)# Strategic implications
cat("\n🎯 STRATEGIC IMPLICATIONS:\n")##
## 🎯 STRATEGIC IMPLICATIONS:cat(" • New models should target price point around $", round(price_mean, 0), "\n")## • New models should target price point around $ 24267cat(" • Performance benchmark: ", round(speed_mean, 0), " km/h max speed\n")## • Performance benchmark: 181 km/h max speedcat(" • Acceleration target: Under ", round(accel_mean, 1), " seconds for competitiveness\n")## • Acceleration target: Under 10.8 seconds for competitivenessThe median represents the middle position when data is arranged in order. It divides the dataset into two equal halves.
For Odd n: \(Me = x_{(\frac{n+1}{2})}\)
For Even n: \(Me = \frac{x_{(\frac{n}{2})} + x_{(\frac{n}{2}+1)}}{2}\)
For Grouped Data: \(Me = L + \frac{\frac{n}{2} - CF_{before}}{f_{median}} \times h\)
Where: - \(L\) = lower boundary of median class - \(CF_{before}\) = cumulative frequency before median class - \(f_{median}\) = frequency of median class - \(h\) = class width
# Example with airline ticket prices
ticket_prices <- c(80, 120, 150, 90)
cat("✈️ Airline Ticket Prices to London:\n")## ✈️ Airline Ticket Prices to London:cat("Original data: $", paste(ticket_prices, collapse = ", $"), "\n")## Original data: $ 80, $120, $150, $90# Step 1: Sort the data
sorted_prices <- sort(ticket_prices)
cat("Sorted data: $", paste(sorted_prices, collapse = ", $"), "\n")## Sorted data: $ 80, $90, $120, $150# Step 2: Find the median position
n <- length(sorted_prices)
cat("n =", n, "(even number)\n")## n = 4 (even number)# Step 3: Calculate median
if (n %% 2 == 0) {
# Even number of observations
pos1 <- n / 2
pos2 <- (n / 2) + 1
median_manual <- (sorted_prices[pos1] + sorted_prices[pos2]) / 2
cat("Median position: (", pos1, " + ", pos2, ") ÷ 2\n")
cat("Median = ($", sorted_prices[pos1], " + $", sorted_prices[pos2], ") ÷ 2 = $", median_manual, "\n")
} else {
# Odd number of observations
pos <- (n + 1) / 2
median_manual <- sorted_prices[pos]
cat("Median position:", pos, "\n")
cat("Median = $", median_manual, "\n")
}## Median position: ( 2 + 3 ) ÷ 2
## Median = ($ 90 + $ 120 ) ÷ 2 = $ 105# Verification
median_r <- median(ticket_prices)
cat("✅ R verification: $", median_r, "\n")## ✅ R verification: $ 105# Using the speed distribution from lecture slides
speed_intervals <- c("[0,30)", "[30,50)", "[50,100)")
frequencies_speed <- c(4, 12, 8)
n_total <- sum(frequencies_speed)
# Calculate cumulative frequencies
cumulative_freq <- cumsum(frequencies_speed)
cat("🏎️ Car Speed Distribution Analysis:\n")## 🏎️ Car Speed Distribution Analysis:cat("Intervals: ", paste(speed_intervals, collapse = ", "), "\n")## Intervals: [0,30), [30,50), [50,100)cat("Frequencies: ", paste(frequencies_speed, collapse = ", "), "\n")## Frequencies: 4, 12, 8cat("Cumulative frequencies: ", paste(cumulative_freq, collapse = ", "), "\n")## Cumulative frequencies: 4, 16, 24# Find median class
median_position <- n_total / 2
cat("\nMedian position: n/2 =", n_total, "÷ 2 =", median_position, "\n")##
## Median position: n/2 = 24 ÷ 2 = 12# Identify median class
median_class_index <- which(cumulative_freq >= median_position)[1]
cat("Median class:", speed_intervals[median_class_index], "\n")## Median class: [30,50)# For this example, median class is [30,50)
# Using the median formula for grouped data
L <- 30 # Lower boundary of median class
CF_before <- 4 # Cumulative frequency before median class
f_median <- 12 # Frequency of median class
h <- 20 # Class width
median_grouped <- L + ((median_position - CF_before) / f_median) * h
cat("\n📊 Median Calculation for Grouped Data:\n")##
## 📊 Median Calculation for Grouped Data:cat(" L (lower boundary) =", L, "\n")## L (lower boundary) = 30cat(" n/2 =", median_position, "\n")## n/2 = 12cat(" CF_before =", CF_before, "\n")## CF_before = 4cat(" f_median =", f_median, "\n")## f_median = 12cat(" h (class width) =", h, "\n")## h (class width) = 20cat(" Median = ", L, " + ((", median_position, " - ", CF_before, ") ÷ ", f_median, ") × ", h, "\n")## Median = 30 + (( 12 - 4 ) ÷ 12 ) × 20cat(" = ", L, " + (", (median_position - CF_before), " ÷ ", f_median, ") × ", h, "\n")## = 30 + ( 8 ÷ 12 ) × 20cat(" = ", L, " + ", round((median_position - CF_before) / f_median, 3), " × ", h, "\n")## = 30 + 0.667 × 20cat(" = ", round(median_grouped, 2), " km/h\n")## = 43.33 km/h# Calculate medians for our car dataset
price_median <- median(cars$price_num, na.rm = TRUE)
speed_median <- median(cars$maxspeed, na.rm = TRUE)
accel_median <- median(cars$acceleration, na.rm = TRUE)
cat("🚗 AUTOMOTIVE MARKET MEDIANS:\n")## 🚗 AUTOMOTIVE MARKET MEDIANS:cat("💰 Median Price: $", round(price_median, 0), "\n")## 💰 Median Price: $ 24177cat("⚡ Median Max Speed: ", round(speed_median, 1), " km/h\n")## ⚡ Median Max Speed: 178 km/hcat("🏁 Median Acceleration: ", round(accel_median, 2), " seconds\n")## 🏁 Median Acceleration: 10.9 seconds# Compare with means calculated earlier
cat("\n📊 MEAN vs MEDIAN COMPARISON:\n")##
## 📊 MEAN vs MEDIAN COMPARISON:cat("Price: Mean $", round(price_mean, 0), " vs Median $", round(price_median, 0), "\n")## Price: Mean $ 24267 vs Median $ 24177cat("Speed: Mean ", round(speed_mean, 1), " vs Median ", round(speed_median, 1), " km/h\n")## Speed: Mean 180.6 vs Median 178 km/hcat("Acceleration: Mean ", round(accel_mean, 2), " vs Median ", round(accel_median, 2), " seconds\n")## Acceleration: Mean 10.84 vs Median 10.9 seconds# Skewness interpretation
if (price_mean > price_median) {
cat("\n🔍 Price distribution: RIGHT-SKEWED (mean > median)\n")
cat(" Interpretation: Some very expensive luxury cars pull the average up\n")
} else if (price_mean < price_median) {
cat("\n🔍 Price distribution: LEFT-SKEWED (mean < median)\n")
} else {
cat("\n🔍 Price distribution: SYMMETRIC (mean ≈ median)\n")
}##
## 🔍 Price distribution: RIGHT-SKEWED (mean > median)
## Interpretation: Some very expensive luxury cars pull the average upThe mode is the value that appears most frequently in the dataset.
For Discrete Data: The value(s) with highest frequency
For Grouped Data (Modal Class Method): \[Mo = L + \frac{f_1}{f_1 + f_2} \times h\]
Where: - \(L\) = lower boundary of modal class - \(f_1\) = difference between modal class frequency and previous class frequency - \(f_2\) = difference between modal class frequency and next class frequency - \(h\) = class width
# Example 1: Simple mode identification
car_doors <- c(2, 3, 4, 5, 5, 5, 5, 4, 3, 2, 5, 5)
cat("🚪 Car Door Configuration Analysis:\n")## 🚪 Car Door Configuration Analysis:cat("Data: ", paste(car_doors, collapse = ", "), "\n")## Data: 2, 3, 4, 5, 5, 5, 5, 4, 3, 2, 5, 5# Create frequency table
door_freq <- table(car_doors)
print(door_freq)## car_doors
## 2 3 4 5
## 2 2 2 6# Find mode
mode_value <- as.numeric(names(door_freq)[which.max(door_freq)])
mode_frequency <- max(door_freq)
cat("Mode: ", mode_value, " doors (appears ", mode_frequency, " times)\n")## Mode: 5 doors (appears 6 times)cat("💡 Business insight: ", mode_value, "-door configuration is most popular\n")## 💡 Business insight: 5 -door configuration is most popular# Example 2: Mode in our real dataset
country_freq <- table(cars$country)
modal_country <- names(country_freq)[which.max(country_freq)]
modal_freq <- max(country_freq)
cat("\n🌍 Manufacturing Country Mode:\n")##
## 🌍 Manufacturing Country Mode:cat("Modal country: ", modal_country, " (", modal_freq, " models)\n")## Modal country: Germany ( 40 models)cat("💡 Strategic insight: ", modal_country, " dominates car production\n")## 💡 Strategic insight: Germany dominates car production# Using our speed data example
cat("🏎️ Modal Speed Class Calculation:\n")## 🏎️ Modal Speed Class Calculation:cat("Speed intervals and frequencies:\n")## Speed intervals and frequencies:speed_data <- data.frame(
Interval = c("[0,30)", "[30,50)", "[50,100)"),
Frequency = c(4, 12, 8),
Lower_Bound = c(0, 30, 50),
Upper_Bound = c(30, 50, 100),
Class_Width = c(30, 20, 50)
)
print(speed_data)## Interval Frequency Lower_Bound Upper_Bound Class_Width
## 1 [0,30) 4 0 30 30
## 2 [30,50) 12 30 50 20
## 3 [50,100) 8 50 100 50# Identify modal class
modal_class_index <- which.max(speed_data$Frequency)
modal_class <- speed_data$Interval[modal_class_index]
modal_freq <- speed_data$Frequency[modal_class_index]
cat("\nModal class: ", modal_class, " with frequency ", modal_freq, "\n")##
## Modal class: [30,50) with frequency 12# Calculate mode using the formula
L <- speed_data$Lower_Bound[modal_class_index]
f1 <- modal_freq - speed_data$Frequency[modal_class_index - 1]
f2 <- modal_freq - speed_data$Frequency[modal_class_index + 1]
h <- speed_data$Class_Width[modal_class_index]
mode_grouped <- L + (f1 / (f1 + f2)) * h
cat("\n📊 Mode Calculation:\n")##
## 📊 Mode Calculation:cat(" L (lower boundary) =", L, "\n")## L (lower boundary) = 30cat(" f1 (", modal_freq, " - ", speed_data$Frequency[modal_class_index - 1], ") =", f1, "\n")## f1 ( 12 - 4 ) = 8cat(" f2 (", modal_freq, " - ", speed_data$Frequency[modal_class_index + 1], ") =", f2, "\n")## f2 ( 12 - 8 ) = 4cat(" h (class width) =", h, "\n")## h (class width) = 20cat(" Mode = ", L, " + (", f1, " ÷ (", f1, " + ", f2, ")) × ", h, "\n")## Mode = 30 + ( 8 ÷ ( 8 + 4 )) × 20cat(" = ", L, " + ", round(f1/(f1+f2), 3), " × ", h, "\n")## = 30 + 0.667 × 20cat(" = ", round(mode_grouped, 2), " km/h\n")## = 43.33 km/hUnderstanding when to use mean, median, or mode is crucial for accurate analysis:
cat("📋 CENTRAL TENDENCY DECISION FRAMEWORK:\n")## 📋 CENTRAL TENDENCY DECISION FRAMEWORK:cat("=" , rep("=", 50), "\n", sep="")## ===================================================cat("\n🎯 USE MEAN when:\n")##
## 🎯 USE MEAN when:cat(" ✓ Data is approximately symmetric\n")## ✓ Data is approximately symmetriccat(" ✓ No extreme outliers present\n")## ✓ No extreme outliers presentcat(" ✓ Working with interval/ratio data\n")## ✓ Working with interval/ratio datacat(" ✓ Need mathematical precision\n")## ✓ Need mathematical precisioncat(" ✓ Planning to use in further calculations\n")## ✓ Planning to use in further calculationscat("\n🎯 USE MEDIAN when:\n")##
## 🎯 USE MEDIAN when:cat(" ✓ Data is skewed (left or right)\n")## ✓ Data is skewed (left or right)cat(" ✓ Outliers are present\n")## ✓ Outliers are presentcat(" ✓ Working with ordinal data\n")## ✓ Working with ordinal datacat(" ✓ Need robust measure (resistant to extremes)\n")## ✓ Need robust measure (resistant to extremes)cat(" ✓ Income, house prices, or similar economic data\n")## ✓ Income, house prices, or similar economic datacat("\n🎯 USE MODE when:\n")##
## 🎯 USE MODE when:cat(" ✓ Working with nominal (categorical) data\n")## ✓ Working with nominal (categorical) datacat(" ✓ Need the most frequent category\n")## ✓ Need the most frequent categorycat(" ✓ Business decisions based on popularity\n")## ✓ Business decisions based on popularitycat(" ✓ Quality control (most common defect)\n")## ✓ Quality control (most common defect)cat(" ✓ Market research (most preferred option)\n")## ✓ Market research (most preferred option)# Example 1: Symmetric data - use mean
fuel_efficiency <- c(7.2, 7.8, 8.1, 8.3, 8.5, 8.7, 9.1, 9.3)
cat("⛽ Fuel Efficiency Data (L/100km): ", paste(fuel_efficiency, collapse = ", "), "\n")## ⛽ Fuel Efficiency Data (L/100km): 7.2, 7.8, 8.1, 8.3, 8.5, 8.7, 9.1, 9.3cat(" Distribution: Approximately symmetric\n")## Distribution: Approximately symmetriccat(" Best measure: MEAN = ", round(mean(fuel_efficiency), 2), " L/100km\n")## Best measure: MEAN = 8.38 L/100km# Example 2: Skewed data - use median
executive_salaries <- c(45000, 48000, 52000, 55000, 58000, 62000, 350000)
cat("\n💼 Executive Salaries: $", paste(executive_salaries, collapse = ", $"), "\n")##
## 💼 Executive Salaries: $ 45000, $48000, $52000, $55000, $58000, $62000, $350000cat(" Distribution: Right-skewed (one very high salary)\n")## Distribution: Right-skewed (one very high salary)cat(" Mean: $", round(mean(executive_salaries), 0), " (pulled up by outlier)\n")## Mean: $ 95714 (pulled up by outlier)cat(" Median: $", round(median(executive_salaries), 0), " (more representative)\n")## Median: $ 55000 (more representative)cat(" Best measure: MEDIAN\n")## Best measure: MEDIAN# Example 3: Categorical data - use mode
preferred_colors <- c("Blue", "Red", "Blue", "Green", "Blue", "Red", "Blue", "White")
color_freq <- table(preferred_colors)
modal_color <- names(color_freq)[which.max(color_freq)]
cat("\n🎨 Preferred Car Colors: ", paste(preferred_colors, collapse = ", "), "\n")##
## 🎨 Preferred Car Colors: Blue, Red, Blue, Green, Blue, Red, Blue, Whitecat(" Best measure: MODE = ", modal_color, " (most frequent choice)\n")## Best measure: MODE = Blue (most frequent choice)cat("🔧 PROFESSIONAL STATISTICAL ANALYSIS USING UBStats:\n")## 🔧 PROFESSIONAL STATISTICAL ANALYSIS USING UBStats:cat("=" , rep("=", 60), "\n", sep="")## =============================================================# Central tendency analysis for price
cat("\n💰 PRICE ANALYSIS:\n")##
## 💰 PRICE ANALYSIS:price_central <- distr.summary.x(cars$price_num, stats="central")## n n.a mode n.modes mode% median mean
## 190 0 26809 1 0.0105 24177 24267.47print(price_central)## $`Central tendency measures`
## n n.a mode n.modes mode% median mean
## 1 190 0 26809 1 0.01052632 24177 24267.47# Central tendency analysis for performance
cat("\n🏁 ACCELERATION ANALYSIS:\n")##
## 🏁 ACCELERATION ANALYSIS:accel_central <- distr.summary.x(cars$acceleration, stats="central")## n n.a mode n.modes mode% median mean
## 190 0 12.3 2 0.0263 10.9 10.84print(accel_central)## $`Central tendency measures`
## n n.a mode n.modes mode% median mean
## 1 190 0 12.3 2 0.02631579 10.9 10.83684# Central tendency analysis for speed
cat("\n⚡ MAX SPEED ANALYSIS:\n")##
## ⚡ MAX SPEED ANALYSIS:speed_central <- distr.summary.x(cars$maxspeed, stats="central")## n n.a mode n.modes mode% median mean
## 190 0 172 1 0.0368 178 180.63print(speed_central)## $`Central tendency measures`
## n n.a mode n.modes mode% median mean
## 1 190 0 172 1 0.03684211 178 180.6316cat("\n📊 AUTOMOTIVE MARKET INTELLIGENCE DASHBOARD:\n")##
## 📊 AUTOMOTIVE MARKET INTELLIGENCE DASHBOARD:cat("=" , rep("=", 55), "\n", sep="")## ========================================================# Price intelligence
cat("\n💰 PRICE INTELLIGENCE:\n")##
## 💰 PRICE INTELLIGENCE:cat(" Mean Price: $", round(mean(cars$price_num, na.rm = TRUE), 0), "\n")## Mean Price: $ 24267cat(" Median Price: $", round(median(cars$price_num, na.rm = TRUE), 0), "\n")## Median Price: $ 24177cat(" Price Range: $", round(min(cars$price_num, na.rm = TRUE), 0),
" - $", round(max(cars$price_num, na.rm = TRUE), 0), "\n")## Price Range: $ -1860 - $ 61570# Performance benchmarks
cat("\n🏎️ PERFORMANCE BENCHMARKS:\n")##
## 🏎️ PERFORMANCE BENCHMARKS:cat(" Average Top Speed: ", round(mean(cars$maxspeed, na.rm = TRUE), 1), " km/h\n")## Average Top Speed: 180.6 km/hcat(" Median Acceleration: ", round(median(cars$acceleration, na.rm = TRUE), 2), " seconds\n")## Median Acceleration: 10.9 seconds# Market segmentation insights
cat("\n🎯 MARKET SEGMENTATION INSIGHTS:\n")##
## 🎯 MARKET SEGMENTATION INSIGHTS:price_q1 <- quantile(cars$price_num, 0.25, na.rm = TRUE)
price_q3 <- quantile(cars$price_num, 0.75, na.rm = TRUE)
cat(" Budget Segment (bottom 25%): Under $", round(price_q1, 0), "\n")## Budget Segment (bottom 25%): Under $ 13574cat(" Mid-Market (25%-75%): $", round(price_q1, 0), " - $", round(price_q3, 0), "\n")## Mid-Market (25%-75%): $ 13574 - $ 33195cat(" Premium Segment (top 25%): Above $", round(price_q3, 0), "\n")## Premium Segment (top 25%): Above $ 33195# Country analysis
country_mode_freq <- table(cars$country)
top_country <- names(country_mode_freq)[which.max(country_mode_freq)]
cat("\n🌍 GEOGRAPHIC INTELLIGENCE:\n")##
## 🌍 GEOGRAPHIC INTELLIGENCE:cat(" Top Manufacturing Country: ", top_country, "\n")## Top Manufacturing Country: Germanycat(" Market Share: ", round(max(country_mode_freq)/nrow(cars)*100, 1), "%\n")## Market Share: 21.1 %The relationship between these three measures reveals crucial information about data distribution shape:
cat("📈 DISTRIBUTION SHAPE ANALYSIS:\n")## 📈 DISTRIBUTION SHAPE ANALYSIS:cat("=" , rep("=", 45), "\n", sep="")## ==============================================# Calculate measures for different variables
variables <- c("price_num", "maxspeed", "acceleration", "weight")
variable_names <- c("Price", "Max Speed", "Acceleration", "Weight")
for (i in 1:length(variables)) {
var_data <- cars[[variables[i]]]
var_mean <- mean(var_data, na.rm = TRUE)
var_median <- median(var_data, na.rm = TRUE)
cat("\n", variable_names[i], ":\n")
cat(" Mean: ", round(var_mean, 2), "\n")
cat(" Median: ", round(var_median, 2), "\n")
cat(" Difference (Mean - Median): ", round(var_mean - var_median, 2), "\n")
if (abs(var_mean - var_median) < 0.01 * var_median) {
cat(" Shape: SYMMETRIC (mean ≈ median)\n")
} else if (var_mean > var_median) {
cat(" Shape: RIGHT-SKEWED (mean > median)\n")
cat(" Interpretation: Tail extends toward higher values\n")
} else {
cat(" Shape: LEFT-SKEWED (mean < median)\n")
cat(" Interpretation: Tail extends toward lower values\n")
}
}##
## Price :
## Mean: 24267.47
## Median: 24177
## Difference (Mean - Median): 90.47
## Shape: SYMMETRIC (mean ≈ median)
##
## Max Speed :
## Mean: 180.63
## Median: 178
## Difference (Mean - Median): 2.63
## Shape: RIGHT-SKEWED (mean > median)
## Interpretation: Tail extends toward higher values
##
## Acceleration :
## Mean: 10.84
## Median: 10.9
## Difference (Mean - Median): -0.06
## Shape: SYMMETRIC (mean ≈ median)
##
## Weight :
## Mean: 1373.85
## Median: 1369.5
## Difference (Mean - Median): 4.35
## Shape: SYMMETRIC (mean ≈ median)The coefficient of skewness provides a numerical measure of distribution asymmetry:
\[SK = \frac{3(\bar{x} - Me)}{s}\]
Where \(s\) is the standard deviation.
cat("\n📐 COEFFICIENT OF SKEWNESS ANALYSIS:\n")##
## 📐 COEFFICIENT OF SKEWNESS ANALYSIS:cat("=" , rep("=", 45), "\n", sep="")## ==============================================# Calculate skewness coefficient for price
price_mean <- mean(cars$price_num, na.rm = TRUE)
price_median <- median(cars$price_num, na.rm = TRUE)
price_sd <- sd(cars$price_num, na.rm = TRUE)
skewness_coeff <- 3 * (price_mean - price_median) / price_sd
cat("Price Distribution Skewness:\n")## Price Distribution Skewness:cat(" Mean: $", round(price_mean, 0), "\n")## Mean: $ 24267cat(" Median: $", round(price_median, 0), "\n")## Median: $ 24177cat(" Standard Deviation: $", round(price_sd, 0), "\n")## Standard Deviation: $ 12473cat(" Skewness Coefficient: ", round(skewness_coeff, 3), "\n")## Skewness Coefficient: 0.022if (abs(skewness_coeff) < 0.5) {
cat(" Interpretation: APPROXIMATELY SYMMETRIC\n")
} else if (skewness_coeff > 0.5) {
cat(" Interpretation: MODERATELY RIGHT-SKEWED\n")
} else {
cat(" Interpretation: MODERATELY LEFT-SKEWED\n")
}## Interpretation: APPROXIMATELY SYMMETRICcat("\n📚 Skewness Coefficient Scale:\n")##
## 📚 Skewness Coefficient Scale:cat(" |SK| < 0.5: Approximately symmetric\n")## |SK| < 0.5: Approximately symmetriccat(" 0.5 ≤ |SK| < 1: Moderately skewed\n")## 0.5 ≤ |SK| < 1: Moderately skewedcat(" |SK| ≥ 1: Highly skewed\n")## |SK| ≥ 1: Highly skewed# STRATEGIC ANALYSIS: NEW ELECTRIC CAR DEVELOPMENT
cat("🚗 STRATEGIC ANALYSIS: NEW ELECTRIC CAR DEVELOPMENT\n")## 🚗 STRATEGIC ANALYSIS: NEW ELECTRIC CAR DEVELOPMENTcat("=============================================================\n")## =============================================================cat("SCENARIO: Your company is developing a new electric car. Use statistical analysis\n")## SCENARIO: Your company is developing a new electric car. Use statistical analysiscat("to determine optimal specifications that will be competitive in the market.\n\n")## to determine optimal specifications that will be competitive in the market.# Check if required dataset and columns exist
if (!exists("cars") || !all(c("price_num", "maxspeed", "acceleration", "country") %in% colnames(cars))) {
stop("Error: 'cars' dataset is missing or does not contain required columns (price_num, maxspeed, acceleration, country).")
}
# Problem 1: Optimal Price Positioning
cat("📊 PROBLEM 1: OPTIMAL PRICE POSITIONING\n")## 📊 PROBLEM 1: OPTIMAL PRICE POSITIONINGcat("----------------------------------------------\n")## ----------------------------------------------# Calculate central tendency and quartiles
price_stats <- summary(cars$price_num)
price_quartiles <- quantile(cars$price_num, probs = c(0.25, 0.5, 0.75), na.rm = TRUE)
cat("Central Tendency Analysis:\n")## Central Tendency Analysis:print(price_stats)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -1860 13574 24177 24267 33195 61570cat("\nQuartile Analysis:\n")##
## Quartile Analysis:cat(" 25th Percentile: $", round(price_quartiles[1], 0), "\n")## 25th Percentile: $ 13574cat(" Median: $", round(price_quartiles[2], 0), "\n")## Median: $ 24177cat(" 75th Percentile: $", round(price_quartiles[3], 0), "\n")## 75th Percentile: $ 33195# Strategic recommendation
price_25 <- price_quartiles[1]
price_75 <- price_quartiles[3]
target_price <- price_quartiles[2]
cat("\n🎯 STRATEGIC RECOMMENDATION:\n")##
## 🎯 STRATEGIC RECOMMENDATION:cat(" Target Price Range: $", round(price_25, 0), " - $", round(price_75, 0), "\n")## Target Price Range: $ 13574 - $ 33195cat(" Optimal Price Point: $", round(target_price, 0), " (median)\n")## Optimal Price Point: $ 24177 (median)cat(" Rationale: Median-based pricing captures the mainstream market.\n")## Rationale: Median-based pricing captures the mainstream market.# Problem 2: Performance Benchmarking
cat("\n\n🏁 PROBLEM 2: PERFORMANCE BENCHMARKING\n")##
##
## 🏁 PROBLEM 2: PERFORMANCE BENCHMARKINGcat("----------------------------------------------\n")## ----------------------------------------------# Top 10% performance thresholds
speed_p90 <- quantile(cars$maxspeed, 0.9, na.rm = TRUE)
accel_p10 <- quantile(cars$acceleration, 0.1, na.rm = TRUE) # Lower is better for acceleration
cat("Performance Targets for Top 10% Market:\n")## Performance Targets for Top 10% Market:cat(" Minimum Speed: ", round(speed_p90, 0), " km/h\n")## Minimum Speed: 218 km/hcat(" Maximum Acceleration: ", round(accel_p10, 2), " seconds (0-100 km/h)\n")## Maximum Acceleration: 7.28 seconds (0-100 km/h)# Problem 3: Geographic Market Analysis
cat("\n\n🌍 PROBLEM 3: GEOGRAPHIC MARKET ANALYSIS\n")##
##
## 🌍 PROBLEM 3: GEOGRAPHIC MARKET ANALYSIScat("----------------------------------------------\n")## ----------------------------------------------# Country frequency table
country_analysis <- data.frame(
Counts = table(cars$country),
Proportion = prop.table(table(cars$country)),
Percentage = prop.table(table(cars$country)) * 100
)
cat("Country Distribution:\n")## Country Distribution:print(country_analysis)## Counts.Var1 Counts.Freq Proportion.Var1 Proportion.Freq Percentage.Var1
## 1 Asia - others 14 Asia - others 0.07368421 Asia - others
## 2 Europe - others 38 Europe - others 0.20000000 Europe - others
## 3 France 33 France 0.17368421 France
## 4 Germany 40 Germany 0.21052632 Germany
## 5 Italy 19 Italy 0.10000000 Italy
## 6 Japan 33 Japan 0.17368421 Japan
## 7 United States 13 United States 0.06842105 United States
## Percentage.Freq
## 1 7.368421
## 2 20.000000
## 3 17.368421
## 4 21.052632
## 5 10.000000
## 6 17.368421
## 7 6.842105# Find modal country and European share
modal_country <- names(table(cars$country))[which.max(table(cars$country))]
european_countries <- c("Germany", "France", "Italy", "Europe - others")
european_share <- sum(cars$country %in% european_countries, na.rm = TRUE) / nrow(cars) * 100
cat("\n📈 Geographic Insights:\n")##
## 📈 Geographic Insights:cat(" Modal Manufacturing Country: ", modal_country, "\n")## Modal Manufacturing Country: Germanycat(" European Market Share: ", round(european_share, 1), "%\n")## European Market Share: 68.4 %cat(" Strategic Implication: Focus on European manufacturing partnerships.\n")## Strategic Implication: Focus on European manufacturing partnerships.cat("\n🎓 STUDENT PRACTICE EXERCISES\n")##
## 🎓 STUDENT PRACTICE EXERCISEScat("=" , rep("=", 40), "\n", sep="")## =========================================cat("Exercise 1: Manual Calculation Challenge\n")## Exercise 1: Manual Calculation Challengecat("Given airline ticket prices: $80, $120, $150, $90\n")## Given airline ticket prices: $80, $120, $150, $90cat("Tasks:\n")## Tasks:cat("a) Calculate mean, median, and mode manually\n")## a) Calculate mean, median, and mode manuallycat("b) Arrange data in ascending order\n")## b) Arrange data in ascending ordercat("c) Determine distribution shape\n")## c) Determine distribution shape# Solution for verification
tickets <- c(80, 120, 150, 90)
cat("\n✅ SOLUTION:\n")##
## ✅ SOLUTION:cat("Sorted data: $", paste(sort(tickets), collapse = ", $"), "\n")## Sorted data: $ 80, $90, $120, $150cat("Mean: $", round(mean(tickets), 2), "\n")## Mean: $ 110cat("Median: $", median(tickets), "\n")## Median: $ 105cat("Mode: No mode (all values appear once)\n")## Mode: No mode (all values appear once)cat("Shape: Mean (", round(mean(tickets), 2), ") > Median (", median(tickets), ") → RIGHT-SKEWED\n")## Shape: Mean ( 110 ) > Median ( 105 ) → RIGHT-SKEWEDcat("\n" , rep("-", 50), "\n", sep="")##
## --------------------------------------------------cat("Exercise 2: Grouped Data Challenge\n")## Exercise 2: Grouped Data Challengecat("Car ownership frequency table:\n")## Car ownership frequency table:cat("Cars Owned: 1, 2, 3, 5\n")## Cars Owned: 1, 2, 3, 5cat("Frequencies: 32, 48, 16, 4\n")## Frequencies: 32, 48, 16, 4cat("Task: Calculate weighted mean\n")## Task: Calculate weighted mean# Solution
cars_owned_ex <- c(1, 2, 3, 5)
freq_ex <- c(32, 48, 16, 4)
weighted_mean_ex <- sum(cars_owned_ex * freq_ex) / sum(freq_ex)
cat("\n✅ SOLUTION:\n")##
## ✅ SOLUTION:cat("Weighted Mean = Σ(xi × fi) / Σfi\n")## Weighted Mean = Σ(xi × fi) / Σficat(" = (1×32 + 2×48 + 3×16 + 5×4) / (32+48+16+4)\n")## = (1×32 + 2×48 + 3×16 + 5×4) / (32+48+16+4)cat(" = (32 + 96 + 48 + 20) / 100\n")## = (32 + 96 + 48 + 20) / 100cat(" = 196 / 100 = ", weighted_mean_ex, " cars per family\n")## = 196 / 100 = 1.96 cars per familycat("\n📊 EXECUTIVE SUMMARY: AUTOMOTIVE MARKET ANALYSIS\n")##
## 📊 EXECUTIVE SUMMARY: AUTOMOTIVE MARKET ANALYSIScat("=" , rep("=", 60), "\n", sep="")## =============================================================# Calculate all key statistics
price_summary <- list(
mean = mean(cars$price_num, na.rm = TRUE),
median = median(cars$price_num, na.rm = TRUE),
q1 = quantile(cars$price_num, 0.25, na.rm = TRUE),
q3 = quantile(cars$price_num, 0.75, na.rm = TRUE)
)
performance_summary <- list(
speed_mean = mean(cars$maxspeed, na.rm = TRUE),
speed_median = median(cars$maxspeed, na.rm = TRUE),
accel_mean = mean(cars$acceleration, na.rm = TRUE),
accel_median = median(cars$acceleration, na.rm = TRUE)
)
cat("\n🎯 KEY FINDINGS:\n")##
## 🎯 KEY FINDINGS:cat("\n1. PRICE POSITIONING:\n")##
## 1. PRICE POSITIONING:cat(" • Average market price: $", format(round(price_summary$mean, 0), big.mark = ","), "\n")## • Average market price: $ 24,267cat(" • Median market price: $", format(round(price_summary$median, 0), big.mark = ","), "\n")## • Median market price: $ 24,177cat(" • Price distribution: RIGHT-SKEWED (luxury segment drives average up)\n")## • Price distribution: RIGHT-SKEWED (luxury segment drives average up)cat(" • Recommended target: $", format(round(price_summary$median, 0), big.mark = ","), " (median-based pricing)\n")## • Recommended target: $ 24,177 (median-based pricing)cat("\n2. PERFORMANCE BENCHMARKS:\n")##
## 2. PERFORMANCE BENCHMARKS:cat(" • Average top speed: ", round(performance_summary$speed_mean, 0), " km/h\n")## • Average top speed: 181 km/hcat(" • Median acceleration: ", round(performance_summary$accel_median, 2), " seconds (0-100 km/h)\n")## • Median acceleration: 10.9 seconds (0-100 km/h)cat(" • Competitive threshold: ", round(performance_summary$speed_median, 0), " km/h minimum\n")## • Competitive threshold: 178 km/h minimumcat("\n3. MARKET SEGMENTATION:\n")##
## 3. MARKET SEGMENTATION:cat(" • Budget segment (Q1): Under $", format(round(price_summary$q1, 0), big.mark = ","), "\n")## • Budget segment (Q1): Under $ 13,574cat(" • Premium segment (Q3): Above $", format(round(price_summary$q3, 0), big.mark = ","), "\n")## • Premium segment (Q3): Above $ 33,195cat(" • Target segment: Mid-market ($", format(round(price_summary$q1, 0), big.mark = ","),
" - $", format(round(price_summary$q3, 0), big.mark = ","), ")\n")## • Target segment: Mid-market ($ 13,574 - $ 33,195 )# Geographic analysis
top_countries <- names(sort(table(cars$country), decreasing = TRUE))[1:3]
cat("\n4. GEOGRAPHIC OPPORTUNITIES:\n")##
## 4. GEOGRAPHIC OPPORTUNITIES:cat(" • Leading manufacturers: ", paste(top_countries, collapse = ", "), "\n")## • Leading manufacturers: Germany, Europe - others, Francecat(" • European dominance: ", round(european_share, 1), "% market share\n")## • European dominance: 68.4 % market sharecat(" • Strategic focus: European partnerships and manufacturing\n")## • Strategic focus: European partnerships and manufacturingcat("\n\n📋 TECHNICAL METHODOLOGY REPORT\n")##
##
## 📋 TECHNICAL METHODOLOGY REPORTcat("=" , rep("=", 50), "\n", sep="")## ===================================================cat("\n🔬 STATISTICAL METHODS EMPLOYED:\n")##
## 🔬 STATISTICAL METHODS EMPLOYED:cat("\n1. MEASURES OF CENTRAL TENDENCY:\n")##
## 1. MEASURES OF CENTRAL TENDENCY:cat(" • Arithmetic Mean: Σxi/n for symmetric distributions\n")## • Arithmetic Mean: Σxi/n for symmetric distributionscat(" • Median: Middle value for skewed distributions\n")## • Median: Middle value for skewed distributionscat(" • Mode: Most frequent value for categorical analysis\n")## • Mode: Most frequent value for categorical analysiscat("\n2. DATA QUALITY ASSESSMENT:\n")##
## 2. DATA QUALITY ASSESSMENT:cat(" • Sample size: ", nrow(cars), " car models\n")## • Sample size: 190 car modelscat(" • Geographic coverage: ", length(unique(cars$country)), " countries/regions\n")## • Geographic coverage: 7 countries/regionscat(" • Missing values: Handled using na.rm = TRUE\n")## • Missing values: Handled using na.rm = TRUEcat(" • Outlier detection: Visual inspection via boxplots\n")## • Outlier detection: Visual inspection via boxplotscat("\n3. DISTRIBUTION ANALYSIS:\n")##
## 3. DISTRIBUTION ANALYSIS:cat(" • Skewness assessment: Mean vs. Median comparison\n")## • Skewness assessment: Mean vs. Median comparisoncat(" • Shape determination: Visual and numerical methods\n")## • Shape determination: Visual and numerical methodscat(" • Quartile analysis: Market segmentation insights\n")## • Quartile analysis: Market segmentation insightscat("\n4. BUSINESS APPLICATIONS:\n")##
## 4. BUSINESS APPLICATIONS:cat(" • Price strategy: Median-based positioning\n")## • Price strategy: Median-based positioningcat(" • Performance targets: Percentile benchmarking\n")## • Performance targets: Percentile benchmarkingcat(" • Market analysis: Frequency-based insights\n")## • Market analysis: Frequency-based insightscat("\n🔍 COMPARATIVE ANALYSIS: GERMAN vs JAPANESE MANUFACTURERS\n")##
## 🔍 COMPARATIVE ANALYSIS: GERMAN vs JAPANESE MANUFACTURERScat("=" , rep("=", 65), "\n", sep="")## ==================================================================# Filter data by country
german_cars <- cars[cars$country == "Germany", ]
japanese_cars <- cars[cars$country == "Japan", ]
cat("Sample sizes:\n")## Sample sizes:cat(" German manufacturers: ", nrow(german_cars), " models\n")## German manufacturers: 40 modelscat(" Japanese manufacturers: ", nrow(japanese_cars), " models\n")## Japanese manufacturers: 33 models# Price comparison
cat("\n💰 PRICE COMPARISON:\n")##
## 💰 PRICE COMPARISON:german_price_stats <- c(
mean = mean(german_cars$price_num, na.rm = TRUE),
median = median(german_cars$price_num, na.rm = TRUE)
)
japanese_price_stats <- c(
mean = mean(japanese_cars$price_num, na.rm = TRUE),
median = median(japanese_cars$price_num, na.rm = TRUE)
)
cat("German cars:\n")## German cars:cat(" Mean: $", round(german_price_stats["mean"], 0), "\n")## Mean: $ 24729cat(" Median: $", round(german_price_stats["median"], 0), "\n")## Median: $ 26708cat("Japanese cars:\n")## Japanese cars:cat(" Mean: $", round(japanese_price_stats["mean"], 0), "\n")## Mean: $ 23138cat(" Median: $", round(japanese_price_stats["median"], 0), "\n")## Median: $ 21206# Performance comparison
cat("\n🏁 PERFORMANCE COMPARISON:\n")##
## 🏁 PERFORMANCE COMPARISON:german_speed <- mean(german_cars$maxspeed, na.rm = TRUE)
japanese_speed <- mean(japanese_cars$maxspeed, na.rm = TRUE)
german_accel <- mean(german_cars$acceleration, na.rm = TRUE)
japanese_accel <- mean(japanese_cars$acceleration, na.rm = TRUE)
cat("Average Top Speed:\n")## Average Top Speed:cat(" German: ", round(german_speed, 1), " km/h\n")## German: 185.4 km/hcat(" Japanese: ", round(japanese_speed, 1), " km/h\n")## Japanese: 184.3 km/hcat("Average Acceleration:\n")## Average Acceleration:cat(" German: ", round(german_accel, 2), " seconds\n")## German: 10.57 secondscat(" Japanese: ", round(japanese_accel, 2), " seconds\n")## Japanese: 10.22 seconds# Strategic insights
cat("\n🎯 STRATEGIC INSIGHTS:\n")##
## 🎯 STRATEGIC INSIGHTS:if (german_price_stats["median"] > japanese_price_stats["median"]) {
cat(" • German cars positioned as premium (higher median price)\n")
} else {
cat(" • Japanese cars positioned as premium (higher median price)\n")
}## • German cars positioned as premium (higher median price)if (german_speed > japanese_speed) {
cat(" • German manufacturers focus on performance (higher average speed)\n")
} else {
cat(" • Japanese manufacturers focus on performance (higher average speed)\n")
}## • German manufacturers focus on performance (higher average speed)cat("\n\n📊 ADVANCED MARKET SEGMENTATION ANALYSIS\n")##
##
## 📊 ADVANCED MARKET SEGMENTATION ANALYSIScat("=" , rep("=", 55), "\n", sep="")## ========================================================# Create price segments based on quartiles
price_q1 <- quantile(cars$price_num, 0.25, na.rm = TRUE)
price_q2 <- quantile(cars$price_num, 0.50, na.rm = TRUE)
price_q3 <- quantile(cars$price_num, 0.75, na.rm = TRUE)
# Segment the market
cars$price_segment <- cut(cars$price_num,
breaks = c(0, price_q1, price_q2, price_q3, Inf),
labels = c("Budget", "Economy", "Mid-Market", "Premium"),
include.lowest = TRUE)
# Analyze each segment
segment_analysis <- table(cars$price_segment)
segment_props <- prop.table(segment_analysis) * 100
cat("Market Segmentation by Price Quartiles:\n")## Market Segmentation by Price Quartiles:for(i in 1:length(segment_analysis)) {
segment_name <- names(segment_analysis)[i]
count <- segment_analysis[i]
percentage <- segment_props[i]
cat(" ", segment_name, ": ", count, " models (", round(percentage, 1), "%)\n")
}## Budget : 46 models ( 24.5 %)
## Economy : 47 models ( 25 %)
## Mid-Market : 47 models ( 25 %)
## Premium : 48 models ( 25.5 %)# Performance characteristics by segment
cat("\n🏎️ PERFORMANCE BY SEGMENT:\n")##
## 🏎️ PERFORMANCE BY SEGMENT:for(segment in names(segment_analysis)) {
segment_cars <- cars[cars$price_segment == segment & !is.na(cars$price_segment), ]
avg_speed <- mean(segment_cars$maxspeed, na.rm = TRUE)
avg_accel <- mean(segment_cars$acceleration, na.rm = TRUE)
cat(" ", segment, " segment:\n")
cat(" Average speed: ", round(avg_speed, 1), " km/h\n")
cat(" Average acceleration: ", round(avg_accel, 2), " seconds\n")
}## Budget segment:
## Average speed: 180.3 km/h
## Average acceleration: 11.72 seconds
## Economy segment:
## Average speed: 181.3 km/h
## Average acceleration: 10.96 seconds
## Mid-Market segment:
## Average speed: 178.9 km/h
## Average acceleration: 10.31 seconds
## Premium segment:
## Average speed: 182.5 km/h
## Average acceleration: 10.36 seconds# CASE STUDY: ELECTRIC VEHICLE MARKET ENTRY STRATEGY
cat("\n🔋 CASE STUDY: ELECTRIC VEHICLE MARKET ENTRY STRATEGY\n")##
## 🔋 CASE STUDY: ELECTRIC VEHICLE MARKET ENTRY STRATEGYcat("============================================================\n")## ============================================================cat("SCENARIO: A new electric vehicle startup needs to position their first model\n")## SCENARIO: A new electric vehicle startup needs to position their first modelcat("in the European market. Use statistical analysis to recommend specifications.\n\n")## in the European market. Use statistical analysis to recommend specifications.# Check if required dataset and columns exist
if (!exists("cars") || !all(c("price_num", "maxspeed", "acceleration") %in% colnames(cars))) {
stop("Error: 'cars' dataset is missing or does not contain required columns (price_num, maxspeed, acceleration).")
}
# Step 1: Market Positioning Analysis
cat("STEP 1: MARKET POSITIONING ANALYSIS\n")## STEP 1: MARKET POSITIONING ANALYSIScat("----------------------------------------\n")## ----------------------------------------# Calculate key statistics for decision making
price_stats_complete <- summary(cars$price_num)
speed_percentiles <- quantile(cars$maxspeed, c(0.25, 0.5, 0.75, 0.9), na.rm = TRUE)
accel_percentiles <- quantile(cars$acceleration, c(0.1, 0.25, 0.5, 0.75), na.rm = TRUE)
cat("Current Market Statistics:\n")## Current Market Statistics:print(price_stats_complete)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -1860 13574 24177 24267 33195 61570cat("\nPerformance Benchmarks:\n")##
## Performance Benchmarks:cat("Speed Percentiles (km/h):\n")## Speed Percentiles (km/h):cat(" 25th percentile: ", round(speed_percentiles[1], 0), "\n")## 25th percentile: 159cat(" 50th percentile: ", round(speed_percentiles[2], 0), "\n")## 50th percentile: 178cat(" 75th percentile: ", round(speed_percentiles[3], 0), "\n")## 75th percentile: 203cat(" 90th percentile: ", round(speed_percentiles[4], 0), "\n")## 90th percentile: 218cat("\nAcceleration Percentiles (seconds):\n")##
## Acceleration Percentiles (seconds):cat(" 10th percentile: ", round(accel_percentiles[1], 2), " (top 10% performance)\n")## 10th percentile: 7.28 (top 10% performance)cat(" 25th percentile: ", round(accel_percentiles[2], 2), "\n")## 25th percentile: 8.83cat(" 50th percentile: ", round(accel_percentiles[3], 2), "\n")## 50th percentile: 10.9cat(" 75th percentile: ", round(accel_percentiles[4], 2), "\n")## 75th percentile: 12.97# Step 2: Competitive Positioning Recommendations
cat("\n\nSTEP 2: STRATEGIC RECOMMENDATIONS\n")##
##
## STEP 2: STRATEGIC RECOMMENDATIONScat("----------------------------------------\n")## ----------------------------------------target_price <- median(cars$price_num, na.rm = TRUE)
target_speed <- speed_percentiles[3] # 75th percentile for competitive advantage
target_accel <- accel_percentiles[2] # 25th percentile (lower is better)
cat("🎯 RECOMMENDED SPECIFICATIONS:\n")## 🎯 RECOMMENDED SPECIFICATIONS:cat(" Target Price: $", format(round(target_price, 0), big.mark = ","), "\n")## Target Price: $ 24,177cat(" Rationale: Median pricing captures mainstream market\n\n")## Rationale: Median pricing captures mainstream marketcat(" Target Max Speed: ", round(target_speed, 0), " km/h\n")## Target Max Speed: 203 km/hcat(" Rationale: 75th percentile ensures competitive performance\n\n")## Rationale: 75th percentile ensures competitive performancecat(" Target Acceleration: ", round(target_accel, 2), " seconds (0-100 km/h)\n")## Target Acceleration: 8.83 seconds (0-100 km/h)cat(" Rationale: 25th percentile provides above-average performance\n")## Rationale: 25th percentile provides above-average performance# Step 3: Market Opportunity Analysis
cat("\n\nSTEP 3: MARKET OPPORTUNITY ANALYSIS\n")##
##
## STEP 3: MARKET OPPORTUNITY ANALYSIScat("----------------------------------------\n")## ----------------------------------------# Identify underserved segments
affordable_performance <- sum(cars$price_num <= target_price &
cars$maxspeed >= target_speed &
cars$acceleration <= target_accel, na.rm = TRUE)
total_models <- nrow(cars)
opportunity_gap <- (total_models - affordable_performance) / total_models * 100
cat("Market Gap Analysis:\n")## Market Gap Analysis:cat(" Models meeting all criteria: ", affordable_performance, " out of ", total_models, "\n")## Models meeting all criteria: 3 out of 190cat(" Market opportunity: ", round(opportunity_gap, 1), "% of market underserved\n")## Market opportunity: 98.4 % of market underservedcat(" Strategic implication: Significant opportunity for well-positioned EV\n")## Strategic implication: Significant opportunity for well-positioned EVcat("\n\n🎓 STUDENT ASSESSMENT PROBLEMS\n")##
##
## 🎓 STUDENT ASSESSMENT PROBLEMScat("=" , rep("=", 45), "\n", sep="")## ==============================================cat("PROBLEM SET A: CALCULATION MASTERY\n")## PROBLEM SET A: CALCULATION MASTERYcat("-" , rep("-", 35), "\n", sep="")## ------------------------------------cat("Problem A1: Given the following car prices (in thousands):\n")## Problem A1: Given the following car prices (in thousands):cat("$22, $18, $35, $28, $25, $45, $30, $22\n")## $22, $18, $35, $28, $25, $45, $30, $22cat("Tasks:\n")## Tasks:cat("a) Calculate the arithmetic mean\n")## a) Calculate the arithmetic meancat("b) Find the median\n")## b) Find the mediancat("c) Identify the mode\n")## c) Identify the modecat("d) Determine if the distribution is skewed\n")## d) Determine if the distribution is skewed# Solution for instructors
sample_prices_prob <- c(22, 18, 35, 28, 25, 45, 30, 22)
cat("\n✅ INSTRUCTOR SOLUTIONS:\n")##
## ✅ INSTRUCTOR SOLUTIONS:cat("a) Mean: $", round(mean(sample_prices_prob), 2), "k\n")## a) Mean: $ 28.12 kcat("b) Median: $", median(sample_prices_prob), "k\n")## b) Median: $ 26.5 kmode_freq <- table(sample_prices_prob)
mode_val <- as.numeric(names(mode_freq)[which.max(mode_freq)])
cat("c) Mode: $", mode_val, "k (appears ", max(mode_freq), " times)\n")## c) Mode: $ 22 k (appears 2 times)if (mean(sample_prices_prob) > median(sample_prices_prob)) {
cat("d) RIGHT-SKEWED (mean > median)\n")
} else if (mean(sample_prices_prob) < median(sample_prices_prob)) {
cat("d) LEFT-SKEWED (mean < median)\n")
} else {
cat("d) SYMMETRIC (mean ≈ median)\n")
}## d) RIGHT-SKEWED (mean > median)cat("\n" , rep("-", 50), "\n", sep="")##
## --------------------------------------------------cat("PROBLEM SET B: APPLIED ANALYSIS\n")## PROBLEM SET B: APPLIED ANALYSIScat("-" , rep("-", 35), "\n", sep="")## ------------------------------------cat("Problem B1: Market Research Scenario\n")## Problem B1: Market Research Scenariocat("A car dealership surveys customer satisfaction ratings (1-10 scale):\n")## A car dealership surveys customer satisfaction ratings (1-10 scale):cat("Ratings: 8, 9, 7, 10, 8, 6, 9, 8, 7, 9, 8, 10, 7, 8, 9\n")## Ratings: 8, 9, 7, 10, 8, 6, 9, 8, 7, 9, 8, 10, 7, 8, 9cat("Questions:\n")## Questions:cat("1) What is the most appropriate measure of central tendency?\n")## 1) What is the most appropriate measure of central tendency?cat("2) Calculate that measure\n")## 2) Calculate that measurecat("3) What does this tell us about customer satisfaction?\n")## 3) What does this tell us about customer satisfaction?# Solution
satisfaction <- c(8, 9, 7, 10, 8, 6, 9, 8, 7, 9, 8, 10, 7, 8, 9)
cat("\n✅ SOLUTIONS:\n")##
## ✅ SOLUTIONS:cat("1) MODE is most appropriate (ordinal scale, customer preference)\n")## 1) MODE is most appropriate (ordinal scale, customer preference)satisfaction_mode <- as.numeric(names(table(satisfaction))[which.max(table(satisfaction))])
cat("2) Mode = ", satisfaction_mode, " (appears ", max(table(satisfaction)), " times)\n")## 2) Mode = 8 (appears 5 times)cat("3) Most customers rate satisfaction as ", satisfaction_mode, "/10 - good performance\n")## 3) Most customers rate satisfaction as 8 /10 - good performancecat(" Mean = ", round(mean(satisfaction), 2), " confirms positive satisfaction\n")## Mean = 8.2 confirms positive satisfactioncat("\n📚 LECTURE 2 CONCEPTUAL FRAMEWORK SUMMARY\n")##
## 📚 LECTURE 2 CONCEPTUAL FRAMEWORK SUMMARYcat("=" , rep("=", 55), "\n", sep="")## ========================================================cat("\n🎯 CORE CONCEPTS MASTERED:\n")##
## 🎯 CORE CONCEPTS MASTERED:cat("\n1. ARITHMETIC MEAN (x̄):\n")##
## 1. ARITHMETIC MEAN (x̄):cat(" • Formula: Σxi/n (raw data) or Σ(fi×xi)/Σfi (grouped data)\n")## • Formula: Σxi/n (raw data) or Σ(fi×xi)/Σfi (grouped data)cat(" • Use: Symmetric distributions, mathematical precision\n")## • Use: Symmetric distributions, mathematical precisioncat(" • Properties: Sensitive to outliers, algebraically manipulable\n")## • Properties: Sensitive to outliers, algebraically manipulablecat("\n2. MEDIAN (Me):\n")##
## 2. MEDIAN (Me):cat(" • Definition: Middle value in ordered dataset\n")## • Definition: Middle value in ordered datasetcat(" • Use: Skewed distributions, robust measure\n")## • Use: Skewed distributions, robust measurecat(" • Properties: Resistant to outliers, positional measure\n")## • Properties: Resistant to outliers, positional measurecat("\n3. MODE (Mo):\n")##
## 3. MODE (Mo):cat(" • Definition: Most frequently occurring value\n")## • Definition: Most frequently occurring valuecat(" • Use: Categorical data, popularity analysis\n")## • Use: Categorical data, popularity analysiscat(" • Properties: Can have multiple modes or no mode\n")## • Properties: Can have multiple modes or no modecat("\n4. DISTRIBUTION SHAPE:\n")##
## 4. DISTRIBUTION SHAPE:cat(" • Symmetric: Mean ≈ Median ≈ Mode\n")## • Symmetric: Mean ≈ Median ≈ Modecat(" • Right-skewed: Mean > Median > Mode\n")## • Right-skewed: Mean > Median > Modecat(" • Left-skewed: Mode > Median > Mean\n")## • Left-skewed: Mode > Median > Meancat("\n🔧 TECHNICAL SKILLS DEVELOPED:\n")##
## 🔧 TECHNICAL SKILLS DEVELOPED:cat(" ✓ Manual calculation of all central tendency measures\n")## ✓ Manual calculation of all central tendency measurescat(" ✓ Application of formulas to grouped and ungrouped data\n")## ✓ Application of formulas to grouped and ungrouped datacat(" ✓ UBStats package proficiency\n")## ✓ UBStats package proficiencycat(" ✓ Distribution shape analysis\n")## ✓ Distribution shape analysiscat(" ✓ Business interpretation of statistical results\n")## ✓ Business interpretation of statistical resultscat("\n\n🔮 PREVIEW: LECTURE 3 - MEASURES OF VARIABILITY\n")##
##
## 🔮 PREVIEW: LECTURE 3 - MEASURES OF VARIABILITYcat("=" , rep("=", 55), "\n", sep="")## ========================================================cat("Coming up in our next session:\n")## Coming up in our next session:cat("\n📊 MEASURES OF SPREAD:\n")##
## 📊 MEASURES OF SPREAD:cat(" • Range and Interquartile Range (IQR)\n")## • Range and Interquartile Range (IQR)cat(" • Variance and Standard Deviation\n")## • Variance and Standard Deviationcat(" • Coefficient of Variation\n")## • Coefficient of Variationcat(" • Outlier detection methods\n")## • Outlier detection methodscat("\n🎯 ADVANCED TOPICS:\n")##
## 🎯 ADVANCED TOPICS:cat(" • Risk assessment in financial contexts\n")## • Risk assessment in financial contextscat(" • Quality control applications\n")## • Quality control applicationscat(" • Comparative variability analysis\n")## • Comparative variability analysiscat(" • Five-number summary and boxplots\n")## • Five-number summary and boxplotscat("\n💼 BUSINESS APPLICATIONS:\n")##
## 💼 BUSINESS APPLICATIONS:cat(" • Investment risk analysis\n")## • Investment risk analysiscat(" • Manufacturing quality control\n")## • Manufacturing quality controlcat(" • Market volatility assessment\n")## • Market volatility assessmentcat(" • Performance consistency evaluation\n")## • Performance consistency evaluationcat("\n📋 PREPARATION TASKS:\n")##
## 📋 PREPARATION TASKS:cat(" 1. Review central tendency concepts from today\n")## 1. Review central tendency concepts from todaycat(" 2. Practice manual calculations with small datasets\n")## 2. Practice manual calculations with small datasetscat(" 3. Familiarize yourself with variance formula\n")## 3. Familiarize yourself with variance formulacat(" 4. Think about real-world examples of variability\n")## 4. Think about real-world examples of variabilitycat("\n\n🏆 CAPSTONE EXERCISE: COMPREHENSIVE ANALYSIS\n")##
##
## 🏆 CAPSTONE EXERCISE: COMPREHENSIVE ANALYSIScat("=" , rep("=", 55), "\n", sep="")## ========================================================cat("SCENARIO: You are presenting to the board of directors about market positioning\n")## SCENARIO: You are presenting to the board of directors about market positioningcat("for a new luxury car model. Prepare a complete statistical brief.\n\n")## for a new luxury car model. Prepare a complete statistical brief.# Select luxury cars (top 25% by price)
luxury_threshold <- quantile(cars$price_num, 0.75, na.rm = TRUE)
luxury_cars <- cars[cars$price_num >= luxury_threshold, ]
cat("LUXURY MARKET ANALYSIS (Top 25% by Price):\n")## LUXURY MARKET ANALYSIS (Top 25% by Price):cat("Threshold: $", format(round(luxury_threshold, 0), big.mark = ","), "\n")## Threshold: $ 33,195cat("Sample size: ", nrow(luxury_cars), " models\n")## Sample size: 48 models# Complete analysis
luxury_stats <- list(
price_mean = mean(luxury_cars$price_num, na.rm = TRUE),
price_median = median(luxury_cars$price_num, na.rm = TRUE),
speed_mean = mean(luxury_cars$maxspeed, na.rm = TRUE),
speed_median = median(luxury_cars$maxspeed, na.rm = TRUE),
accel_mean = mean(luxury_cars$acceleration, na.rm = TRUE),
accel_median = median(luxury_cars$acceleration, na.rm = TRUE)
)
cat("\n📊 LUXURY SEGMENT CHARACTERISTICS:\n")##
## 📊 LUXURY SEGMENT CHARACTERISTICS:cat("Price Statistics:\n")## Price Statistics:cat(" Mean: $", format(round(luxury_stats$price_mean, 0), big.mark = ","), "\n")## Mean: $ 40,717cat(" Median: $", format(round(luxury_stats$price_median, 0), big.mark = ","), "\n")## Median: $ 38,904cat("Performance Statistics:\n")## Performance Statistics:cat(" Average Speed: ", round(luxury_stats$speed_mean, 1), " km/h\n")## Average Speed: 182.5 km/hcat(" Median Acceleration: ", round(luxury_stats$accel_median, 2), " seconds\n")## Median Acceleration: 10.1 seconds# Country analysis for luxury segment
luxury_countries <- table(luxury_cars$country)
top_luxury_country <- names(luxury_countries)[which.max(luxury_countries)]
cat("Geographic Distribution:\n")## Geographic Distribution:cat(" Leading luxury manufacturer: ", top_luxury_country, "\n")## Leading luxury manufacturer: Germanycat(" Market share: ", round(max(luxury_countries)/nrow(luxury_cars)*100, 1), "%\n")## Market share: 27.1 %cat("\n🎯 STRATEGIC RECOMMENDATIONS:\n")##
## 🎯 STRATEGIC RECOMMENDATIONS:cat(" • Target price: $", format(round(luxury_stats$price_median, -3), big.mark = ","), "\n")## • Target price: $ 39,000cat(" • Minimum speed: ", round(luxury_stats$speed_median, 0), " km/h\n")## • Minimum speed: 185 km/hcat(" • Maximum acceleration: ", round(luxury_stats$accel_median, 1), " seconds\n")## • Maximum acceleration: 10.1 secondscat(" • Benchmark against: ", top_luxury_country, " manufacturers\n")## • Benchmark against: Germany manufacturerscat("\n📈 SUCCESS METRICS:\n")##
## 📈 SUCCESS METRICS:cat(" Price positioning within luxury segment median ±10%\n")## Price positioning within luxury segment median ±10%cat(" Performance specs matching or exceeding segment averages\n")## Performance specs matching or exceeding segment averagescat(" Quality standards aligned with ", top_luxury_country, " benchmarks\n")## Quality standards aligned with Germany benchmarksToday’s comprehensive journey through descriptive statistics has equipped you with:
Remember that statistics is not just about numbers—it’s about understanding patterns, making informed decisions, and communicating insights effectively. Every measure we calculate tells a story about our data, and every story guides important business decisions.
As we continue our statistical journey, you’ll discover that today’s foundation in central tendency naturally leads to questions about variability, relationships between variables, and ultimately, to the powerful world of inferential statistics.
Next Session: We’ll explore how spread and variability around these central measures reveal even deeper insights about market dynamics, risk assessment, and competitive positioning.
“In God we trust. All others must bring data.” - W. Edwards Deming