Week 4 prob
2024-09-25
Now we will make 5 samples from the original data set and name them as data frame(df_) 1,2,3,4,5. To sampling the dataset will collect the randomly from any rows or columns.These sub samples should each include both categorical and continuous (numeric) data.
# Load necessary libraries
library(dplyr)
# Set a seed for reproducibility
set.seed(123)
# Get the number of rows in the dataset
num_rows <- nrow(data)
# Sample 50% of the data 5 times with replacement
df_1 <- data[sample(1:num_rows, size = floor(0.5 * num_rows), replace = TRUE), ]
df_2 <- data[sample(1:num_rows, size = floor(0.5 * num_rows), replace = TRUE), ]
df_3 <- data[sample(1:num_rows, size = floor(0.5 * num_rows), replace = TRUE), ]
df_4 <- data[sample(1:num_rows, size = floor(0.5 * num_rows), replace = TRUE), ]
df_5 <- data[sample(1:num_rows, size = floor(0.5 * num_rows), replace = TRUE), ]We Will display the first few rows of each sample.
The sample 1
head(df_1)## idx transaction_id user_id transaction_date
## 2463 2463 6e9aaecd-6d8d-4614-9618-9249c8d7c913 USER_08108 2024-02-17 09:43
## 2511 2511 4897848c-d285-4e7d-877a-3b7e42c3937f USER_07206 2024-02-21 00:46
## 2227 2227 da2b83e0-5022-497e-87e9-dde5bd325bbc USER_05183 2024-01-31 03:49
## 526 526 3167b375-b764-4d20-8fcd-84615cbad432 USER_08995 2023-09-26 04:45
## 4291 4291 5955ca9b-0304-4008-95d5-47c45415e9fc USER_04125 2024-06-29 21:42
## 2986 2986 21419024-75d9-46ef-928c-8e3935cfbd6e USER_02349 2024-03-27 04:11
## product_category product_name merchant_name product_amount
## 2463 Streaming Service Basic Plan Netflix 755.49
## 2511 Bus Ticket Non-AC Seater MakeMyTrip Bus 2253.46
## 2227 Electricity Bill Residential Bill BSES 128.21
## 526 Gaming Credits Legendary Weapon Free Fire 2359.70
## 4291 Rent Payment 1BHK Apartment Rent Nestaway 3252.79
## 2986 Streaming Service Student Offer Hotstar 1652.35
## transaction_fee cashback loyalty_points payment_method transaction_status
## 2463 36.79 94.56 141 Wallet Balance Successful
## 2511 15.38 56.24 59 Bank Transfer Successful
## 2227 25.41 51.28 222 Wallet Balance Successful
## 526 44.21 50.01 490 UPI Pending
## 4291 35.53 22.54 844 Bank Transfer Failed
## 2986 38.07 81.17 702 Wallet Balance Successful
## merchant_id device_type location
## 2463 MERCH_0228 Android Urban
## 2511 MERCH_0332 iOS Urban
## 2227 MERCH_0916 Android Suburban
## 526 MERCH_0758 Android Urban
## 4291 MERCH_0217 iOS Urban
## 2986 MERCH_0440 Android UrbanThe sample 2
head(df_2)## idx transaction_id user_id transaction_date
## 754 754 a5c55dcf-8dd1-43d1-ab58-36602030967f USER_09609 2023-10-12 02:40
## 2813 2813 35098c21-1943-42f9-9b5c-4edcd8ab6fcf USER_09786 2024-03-14 11:38
## 4702 4702 690508d9-cd89-40b2-ba42-c386ff4201db USER_02781 2024-07-29 17:01
## 2581 2581 800df07c-3cfd-489f-9c91-b06d0ea40747 USER_00161 2024-02-25 17:25
## 2468 2468 3ea438e5-acc1-45b8-b615-abda3fd0985e USER_04602 2024-02-17 17:19
## 3927 3927 1122df72-43d1-4feb-9b03-636d6f77db16 USER_00977 2024-06-03 11:59
## product_category product_name merchant_name product_amount
## 754 Internet Bill Broadband 50Mbps Plan Hathway 4909.70
## 2813 Education Fee Python Programming Course Vedantu 5177.35
## 4702 Streaming Service Student Offer ZEE5 9805.53
## 2581 Rent Payment Co-working Space Housing.com 6479.56
## 2468 Flight Booking Bangalore to Kolkata MakeMyTrip 7504.16
## 3927 Movie Ticket 3 Idiots SPI Cinemas 542.76
## transaction_fee cashback loyalty_points payment_method transaction_status
## 754 6.72 21.52 227 Wallet Balance Successful
## 2813 31.29 79.01 129 Debit Card Successful
## 4702 45.21 72.15 51 Wallet Balance Successful
## 2581 41.07 37.74 844 Credit Card Failed
## 2468 10.10 11.18 735 Credit Card Successful
## 3927 26.01 85.77 820 Credit Card Successful
## merchant_id device_type location
## 754 MERCH_0950 Web Urban
## 2813 MERCH_0447 Android Suburban
## 4702 MERCH_0355 iOS Urban
## 2581 MERCH_0926 Android Urban
## 2468 MERCH_0678 Android Suburban
## 3927 MERCH_0498 Android UrbanThe sample 3
head(df_3)## idx transaction_id user_id transaction_date
## 3805 3805 ef72d2b6-3546-49ad-8910-e3044b6bb54a USER_00534 2024-05-25 15:37
## 4599 4599 1923ce46-f3ff-4f55-a13a-81927b0dc9c2 USER_04643 2024-07-22 18:28
## 3362 3362 f80abda3-c503-4b4f-8195-c72dce348433 USER_04861 2024-04-24 06:42
## 4065 4065 72457eb4-c2bc-472e-a4f5-b9ed98a7cc55 USER_03237 2024-06-14 05:50
## 3845 3845 7cc0e519-6d67-4aa4-ac3c-7637bd83683d USER_03871 2024-05-28 16:29
## 1873 1873 645362b8-bdc8-48a4-8ba6-80b7b6d2a8c3 USER_05353 2024-01-03 21:31
## product_category product_name merchant_name
## 3805 Education Fee IELTS Training Coursera
## 4599 Taxi Fare Airport Transfer Rapido
## 3362 Water Bill Commercial Property Hyderabad Water Board
## 4065 Taxi Fare Rental Package Uber
## 3845 Taxi Fare Rental Package Jugnoo
## 1873 Internet Bill Broadband 50Mbps Plan Jio Fiber
## product_amount transaction_fee cashback loyalty_points payment_method
## 3805 6198.02 27.58 30.02 392 UPI
## 4599 6066.91 30.37 33.47 173 Debit Card
## 3362 7026.12 10.88 99.96 328 Wallet Balance
## 4065 8214.70 4.55 76.09 96 Bank Transfer
## 3845 1422.18 12.09 63.56 67 Bank Transfer
## 1873 4846.55 0.45 71.20 895 UPI
## transaction_status merchant_id device_type location
## 3805 Successful MERCH_0120 Android Urban
## 4599 Successful MERCH_0849 Android Urban
## 3362 Successful MERCH_0598 Android Suburban
## 4065 Successful MERCH_0852 Android Urban
## 3845 Successful MERCH_0043 iOS Urban
## 1873 Successful MERCH_0323 iOS UrbanThe sample 4
head(df_4)## idx transaction_id user_id transaction_date
## 3218 3218 f2fa2caf-4f2d-4865-aff9-528f5869d42d USER_00295 2024-04-14 00:46
## 384 384 6a207895-38bc-41d9-b2b5-48674247e183 USER_08989 2023-09-17 05:55
## 1928 1928 a034e187-4c6f-4796-9a1c-68d5d23349e9 USER_03450 2024-01-08 02:20
## 627 627 93b911e2-2660-4a04-b1bb-0934529bf842 USER_05563 2023-10-03 03:11
## 4987 4987 de0930db-c7b0-4ec8-91e7-fe22331f7642 USER_06427 2024-08-17 09:28
## 4365 4365 27309781-1ce6-46b7-a688-5b363328a28d USER_04781 2024-07-04 17:27
## product_category product_name merchant_name product_amount
## 3218 Hotel Booking Suite Treebo 1383.56
## 384 Internet Bill Fiber 100Mbps Plan Hathway 1317.00
## 1928 Rent Payment Commercial Space Rent NoBroker 5885.22
## 627 Internet Bill Business Internet Plan Jio Fiber 2228.22
## 4987 Mobile Recharge Annual Plan 2999 Jio 3352.43
## 4365 Rent Payment PG Accommodation NoBroker 234.25
## transaction_fee cashback loyalty_points payment_method transaction_status
## 3218 19.75 25.64 748 UPI Successful
## 384 49.72 17.21 190 Wallet Balance Successful
## 1928 44.32 51.88 365 UPI Successful
## 627 30.48 35.30 885 Wallet Balance Successful
## 4987 21.01 18.73 316 Credit Card Successful
## 4365 22.17 79.62 873 Debit Card Successful
## merchant_id device_type location
## 3218 MERCH_0554 Android Suburban
## 384 MERCH_0022 iOS Urban
## 1928 MERCH_0176 iOS Suburban
## 627 MERCH_0266 Android Urban
## 4987 MERCH_0096 Android Rural
## 4365 MERCH_0068 Web RuralThe sample 5
head(df_5)## idx transaction_id user_id transaction_date
## 4302 4302 57baf3ed-a809-4118-b929-68de04a4d4eb USER_09534 2024-06-30 09:48
## 4290 4290 0c1337a9-500d-4702-b814-f77b111d5ae2 USER_02841 2024-06-29 21:30
## 1445 1445 bde9e0d4-a31b-43fb-afe3-8c48e6e2c963 USER_09078 2023-12-03 13:51
## 4573 4573 d6d5cb9b-fe77-46fc-b857-d2a7a184fe6a USER_06332 2024-07-20 15:29
## 770 770 2f2178ab-52a4-4144-a134-91e0eac258ea USER_07853 2023-10-13 07:22
## 3565 3565 ca5cda0d-fbed-4e90-bb9c-5662687593f2 USER_06133 2024-05-08 18:28
## product_category product_name merchant_name product_amount
## 4302 Gas Bill Piped Natural Gas Adani Gas 9385.57
## 4290 Water Bill Residential Connection Delhi Jal Board 7979.07
## 1445 Electricity Bill Agricultural Bill Tata Power 1040.50
## 4573 Internet Bill Student Special Plan BSNL Broadband 9992.47
## 770 Gift Card Rs. 5000 Luxury Card Shoppers Stop 4316.54
## 3565 Electricity Bill Commercial Bill BSES 9899.56
## transaction_fee cashback loyalty_points payment_method transaction_status
## 4302 21.39 41.20 770 Bank Transfer Successful
## 4290 3.05 31.16 726 Bank Transfer Successful
## 1445 17.70 39.88 216 Credit Card Successful
## 4573 34.61 72.06 124 Debit Card Successful
## 770 6.94 46.12 666 Credit Card Successful
## 3565 37.15 37.94 229 Debit Card Successful
## merchant_id device_type location
## 4302 MERCH_0478 Android Urban
## 4290 MERCH_0470 Android Urban
## 1445 MERCH_0290 iOS Urban
## 4573 MERCH_0678 Android Rural
## 770 MERCH_0533 Android Urban
## 3565 MERCH_0645 Android UrbanScrutinize To scrutinize the subsamples, we can compare various statistics, such as the distribution of categorical variables and summary statistics for numeric variables we use Group_by function.
Now we will Compare Categorical Data: We’ll count how often each category (e.g., product_category, transaction_status) appears in each subsample.
# Compare categorical distributions across subsamples
compare_categorical <- function(df, variable) {
df %>%
group_by(!!sym(variable)) %>%
summarise(count = n()) %>%
arrange(desc(count))
}
# Check distribution of product_category across subsamples
cat_df1 <- compare_categorical(df_1, "product_category")
cat_df2 <- compare_categorical(df_2, "product_category")
cat_df3 <- compare_categorical(df_3, "product_category")
cat_df4 <- compare_categorical(df_4, "product_category")
cat_df5 <- compare_categorical(df_5, "product_category")We will display the Compared Samples
cat_df1## # A tibble: 20 × 2
## product_category count
## <chr> <int>
## 1 Streaming Service 152
## 2 Education Fee 151
## 3 Hotel Booking 145
## 4 Movie Ticket 142
## 5 Electricity Bill 138
## 6 Rent Payment 137
## 7 Gaming Credits 130
## 8 Water Bill 127
## 9 Gift Card 125
## 10 Mobile Recharge 123
## 11 Food Delivery 120
## 12 Bus Ticket 118
## 13 Loan Repayment 118
## 14 Internet Bill 117
## 15 Flight Booking 115
## 16 Online Shopping 114
## 17 Gas Bill 111
## 18 Grocery Shopping 110
## 19 Insurance Premium 106
## 20 Taxi Fare 101cat_df2## # A tibble: 20 × 2
## product_category count
## <chr> <int>
## 1 Streaming Service 161
## 2 Education Fee 155
## 3 Gas Bill 150
## 4 Electricity Bill 141
## 5 Food Delivery 135
## 6 Movie Ticket 135
## 7 Water Bill 135
## 8 Hotel Booking 130
## 9 Taxi Fare 130
## 10 Rent Payment 128
## 11 Internet Bill 127
## 12 Mobile Recharge 123
## 13 Online Shopping 120
## 14 Bus Ticket 119
## 15 Loan Repayment 118
## 16 Grocery Shopping 111
## 17 Insurance Premium 105
## 18 Gift Card 98
## 19 Gaming Credits 93
## 20 Flight Booking 86cat_df3## # A tibble: 20 × 2
## product_category count
## <chr> <int>
## 1 Movie Ticket 154
## 2 Streaming Service 146
## 3 Water Bill 142
## 4 Electricity Bill 139
## 5 Education Fee 137
## 6 Hotel Booking 135
## 7 Gas Bill 131
## 8 Rent Payment 130
## 9 Taxi Fare 129
## 10 Grocery Shopping 125
## 11 Internet Bill 125
## 12 Food Delivery 123
## 13 Gift Card 122
## 14 Bus Ticket 118
## 15 Gaming Credits 115
## 16 Online Shopping 114
## 17 Insurance Premium 111
## 18 Mobile Recharge 110
## 19 Flight Booking 101
## 20 Loan Repayment 93cat_df4## # A tibble: 20 × 2
## product_category count
## <chr> <int>
## 1 Streaming Service 160
## 2 Hotel Booking 148
## 3 Water Bill 143
## 4 Taxi Fare 138
## 5 Electricity Bill 136
## 6 Movie Ticket 136
## 7 Rent Payment 135
## 8 Education Fee 131
## 9 Food Delivery 130
## 10 Loan Repayment 125
## 11 Gas Bill 124
## 12 Gift Card 123
## 13 Mobile Recharge 118
## 14 Bus Ticket 114
## 15 Gaming Credits 114
## 16 Online Shopping 113
## 17 Grocery Shopping 110
## 18 Insurance Premium 109
## 19 Flight Booking 100
## 20 Internet Bill 93cat_df5## # A tibble: 20 × 2
## product_category count
## <chr> <int>
## 1 Education Fee 151
## 2 Streaming Service 144
## 3 Water Bill 140
## 4 Taxi Fare 134
## 5 Grocery Shopping 133
## 6 Rent Payment 132
## 7 Movie Ticket 130
## 8 Gas Bill 129
## 9 Bus Ticket 127
## 10 Internet Bill 125
## 11 Mobile Recharge 123
## 12 Electricity Bill 122
## 13 Food Delivery 121
## 14 Hotel Booking 121
## 15 Online Shopping 117
## 16 Flight Booking 116
## 17 Insurance Premium 114
## 18 Gaming Credits 109
## 19 Loan Repayment 108
## 20 Gift Card 104Now we will Compare Numeric Data: We’ll compute summary statistics (mean, median, etc.) for numeric variables like product_amount or cashback.
# Function to calculate summary statistics for numeric columns
compare_numeric <- function(df, variable) {
df %>%
summarise(mean = mean(!!sym(variable), na.rm = TRUE),
median = median(!!sym(variable), na.rm = TRUE),
sd = sd(!!sym(variable), na.rm = TRUE),
min = min(!!sym(variable), na.rm = TRUE),
max = max(!!sym(variable), na.rm = TRUE))
}
# Compare product_amount across subsamples
num_df1 <- compare_numeric(df_1, "product_amount")
num_df2 <- compare_numeric(df_2, "product_amount")
num_df3 <- compare_numeric(df_3, "product_amount")
num_df4 <- compare_numeric(df_4, "product_amount")
num_df5 <- compare_numeric(df_5, "product_amount")We will display the summary statistics of each dataframes
num_df1## mean median sd min max
## 1 5033.087 4981.41 2878.074 11.75 9992.47num_df2## mean median sd min max
## 1 4967.581 4932.35 2880.227 13.33 9993.54num_df3## mean median sd min max
## 1 4906.752 4923.43 2876.466 10.09 9993.54num_df4## mean median sd min max
## 1 5015.371 5005.695 2881.142 10.09 9993.54num_df5## mean median sd min max
## 1 5031.011 5025.56 2908.699 10.09 9994.83Visualization of comparisons
# Load necessary libraries
library(ggplot2)
library(dplyr)
# Combine summary statistics for each subsample into one data frame
summary_stats <- data.frame(
Sample = rep(c("df_1", "df_2", "df_3", "df_4", "df_5"), each = 5),
Statistic = rep(c("Mean", "Median", "Standard Deviation", "Min", "Max"), times = 5),
Value = c(mean(df_1$product_amount, na.rm = TRUE), median(df_1$product_amount, na.rm = TRUE), sd(df_1$product_amount, na.rm = TRUE), min(df_1$product_amount, na.rm = TRUE), max(df_1$product_amount, na.rm = TRUE),
mean(df_2$product_amount, na.rm = TRUE), median(df_2$product_amount, na.rm = TRUE), sd(df_2$product_amount, na.rm = TRUE), min(df_2$product_amount, na.rm = TRUE), max(df_2$product_amount, na.rm = TRUE),
mean(df_3$product_amount, na.rm = TRUE), median(df_3$product_amount, na.rm = TRUE), sd(df_3$product_amount, na.rm = TRUE), min(df_3$product_amount, na.rm = TRUE), max(df_3$product_amount, na.rm = TRUE),
mean(df_4$product_amount, na.rm = TRUE), median(df_4$product_amount, na.rm = TRUE), sd(df_4$product_amount, na.rm = TRUE), min(df_4$product_amount, na.rm = TRUE), max(df_4$product_amount, na.rm = TRUE),
mean(df_5$product_amount, na.rm = TRUE), median(df_5$product_amount, na.rm = TRUE), sd(df_5$product_amount, na.rm = TRUE), min(df_5$product_amount, na.rm = TRUE), max(df_5$product_amount, na.rm = TRUE))
)Means for all samples
ggplot(summary_stats %>% filter(Statistic == "Mean"), aes(x = Sample, y = Value, fill = Sample)) +
geom_bar(stat = "identity") +
labs(title = "Mean Product Amount by Sample", x = "Sample", y = "Mean") +
theme_minimal()
Observation: Sample 1 has a higher mean product amount compared to others indicating that transactions in that subsample have, on average, larger values.Other subsample 3 have lower mean, indicating smaller average spending amounts.A higher mean driven by a few large purchases and high-value products electronics or luxury goods.A lower mean reflect frequent small transactions such as everyday purchases e.g., groceries, low-cost services.
Medians for all samples
ggplot(summary_stats %>% filter(Statistic == "Median"), aes(x = Sample, y = Value, fill = Sample)) +
geom_bar(stat = "identity") +
labs(title = "Median Product Amount by Sample", x = "Sample", y = "Median") +
theme_minimal()
Observation :The meadian of Sample 5 is high and the median of sample 3 is less compared to all.Median reflects the “typical” transaction amount, which is less affected by extreme values. A higher median indicate that most transactions in a subsample consist of moderately expensive purchases as electronics, while a lower median points to more routine, inexpensive transactions groceries.
Standard deviation for all samples
ggplot(summary_stats %>% filter(Statistic == "Standard Deviation"), aes(x = Sample, y = Value, fill = Sample)) +
geom_bar(stat = "identity") +
labs(title = "Standard Deviation of Product Amount by Sample", x = "Sample", y = "Standard Deviation") +
theme_minimal()
Observations: The Sample 5 and sample 3 has high standard deviation and less standard deviation respectively.A high standard deviation could reflect a mix of different types of purchases large electronics purchases alongside smaller daily purchases.A lower standard deviation indicate that transactions are uniform in size, potentially from more homogeneous purchasing behavior repetitive, small transactions.
Minimum for all samples
ggplot(summary_stats %>% filter(Statistic == "Min"), aes(x = Sample, y = Value, fill = Sample)) +
geom_bar(stat = "identity") +
labs(title = "Minimum Product Amount by Sample", x = "Sample", y = "Min") +
theme_minimal()
Observation: The minimum of all samples is high in Sample 2 and less minimum in sample 3,4,5 because the high minimum sample evn has less expenses on large products compared to all.Small or negligible transactions may come from refunds, discounts, or low-cost items, which are common across subsamples regardless of category.
Maximum for all samples
ggplot(summary_stats %>% filter(Statistic == "Max"), aes(x = Sample, y = Value, fill = Sample)) +
geom_bar(stat = "identity") +
labs(title = "Maximum Product Amount by Sample", x = "Sample", y = "Max") +
theme_minimal()
Observation: Subsample 5 show higher maximum values compared to others, indicating that those subsamples include some very large transactions.The lower maximum subsample 1 means that the most expensive transaction is still moderate in size.A higher maximum reflect high-value purchases, such as luxury goods or significant service payments.A lower maximum indicate that customers in this subsample did not make any very large transactions, sticking to lower-value purchases.
- Anomaly in One Sub-Sample
High Transaction Amounts: In a sub-sample with lower average spending ( groceries or low-cost services), a very high transaction amount ( $10,000 purchase) would be considered an anomaly. However, in a sub-sample consisting of higher spending categories (electronics or luxury products), such a transaction would be typical and not considered an anomaly.
Low Transaction Amounts: In sub-samples where high-value purchases are the norm (electronics), a very low transaction amount (under $5) might be seen as anomalous.
This could indicate an unusual product, a refund, or a data entry error. In other sub-samples, particularly for categories like groceries or services, small transactions are common, and thus this wouldn’t be considered an anomaly. Thus, what constitutes an anomaly largely depends on the context of the sub-sample and the expected range of transaction amounts within that category.
- Consistent Aspects Across All Sub-Samples:
Small Transactions: Even in sub-samples where the overall spending is higher ( electronics or travel), there may still be a small portion of low-value transactions that are consistent across all sub-samples. These could be for low-cost services, accessories, or small products.
Transaction Frequency: The number of transactions in each sub-sample might be somewhat consistent if the data was sampled with equal probability across the dataset. Even if the amounts differ, the count of transactions across categories might stay roughly stable.
Payment Method Distribution: If a certain payment method (credit card or mobile wallet) is dominant in the dataset, this could be consistent across all sub-samples, showing similar adoption of payment methods regardless of the specific product category or transaction amount. These consistent elements provide insights into the broader spending behavior and purchasing trends across all transaction types.
OPTIONAL I WORKED ON IT
Monte Carlo simulations
library(dplyr)
library(ggplot2)
set.seed(123)
samples <- list()
for (i in 1:5) {
samples[[i]] <- data %>%
sample_frac(size = 0.5, replace = TRUE)
}
summary_stats <- data.frame(
Sample = character(),
Mean_Spending = numeric(),
Median_Spending = numeric(),
SD_Spending = numeric()
)
for (i in 1:5) {
sample_mean <- mean(samples[[i]]$product_amount, na.rm = TRUE)
sample_median <- median(samples[[i]]$product_amount, na.rm = TRUE)
sample_sd <- sd(samples[[i]]$product_amount, na.rm = TRUE)
summary_stats <- rbind(summary_stats, data.frame(
Sample = paste("Sample", i),
Mean_Spending = sample_mean,
Median_Spending = sample_median,
SD_Spending = sample_sd
))
}
ggplot(summary_stats, aes(x = Sample)) +
geom_bar(aes(y = Mean_Spending, fill = "Mean"), stat = "identity", alpha = 0.6, color = "darkblue", linewidth = 1) +
geom_point(aes(y = Median_Spending, color = "Median"), size = 4, shape = 18) + # Larger points for median
geom_errorbar(aes(ymin = Mean_Spending - SD_Spending, ymax = Mean_Spending + SD_Spending),
width = 0.2, color = "black", linewidth = 1.2) + # Prominent error bars
labs(title = "Mean, Median, and Standard Deviation of Spending Across 5 Samples",
x = "Samples", y = "Spending Amount") +
scale_fill_manual(name = "Legend", values = c("Mean" = "lightblue")) + # Use distinct color for mean
scale_color_manual(name = "Legend", values = c("Median" = "red")) + # Red for median points
theme_minimal() +
theme(axis.text.x = element_text(angle = 45, hjust = 1),
plot.title = element_text(hjust = 0.5),
legend.position = "top") +
geom_text(aes(y = Mean_Spending, label = round(Mean_Spending, 2)), vjust = -1, size = 3) +
geom_text(aes(y = Median_Spending, label = round(Median_Spending, 2), color = "Median"), vjust = 1.5, size = 3) 
Observations from the Monte Carlo Simulation and Plot:
Categories like “Travel” and “Electronics” are likely to show higher variability in spending. This is because large purchases, like vacations or expensive electronic items, can skew both the average and the total spending. Categories with Consistent Spending,Categories such as “Groceries” and “Food & Beverages” typically show more consistent spending patterns across simulations, as these are everyday, frequent purchases that have less variability in price compared to luxury or high-cost items. Anomalies in Certain Subsamples:In some subsamples, anomalies may appear in categories like “Luxury” or “Electronics” if large transactions are over-represented. However, such outliers would not be present in categories with low transaction variance like “Groceries.” Monte Carlo simulations smooth out these anomalies, showing what a “normal” average should look like after accounting for randomness.