Descriptive Statistics

Univariate Statistics

Data

In our example this week, we are going to use the fake data - about real estates in Wroclaw - prices by districts, size of apartments and many more.

Preprocessing

As you can see, not all formats of our variables are adapted. We need to prepare appropriate formats of our variables according to their measurement scale and future application.

apartments$district<-as.factor(apartments$district)
apartments$building_type<-as.factor(apartments$building_type)
apartments$rooms<-factor(apartments$rooms,ordered=TRUE)
attach(apartments)
apartments$price_PLN<-as.numeric(apartments$price_PLN)
apartments$price_EUR<-as.numeric(apartments$price_EUR)

Frequency Tables

In the first step of our analysis, we will group our data into a simple frequency table.

First, let’s look at the distribution of housing prices in our sample and verify tabular validity using the TAI measure:

Ok, it looks quite ugly, so let’s wrap it up using the ‘kable’ package:

Apartments in Wroclaw - prices in kPLN

x label Freq Percent Valid Percent Cumulative Percent Valid 350-450 kPLN 9 4.5 4.5 4.5 450-550 kPLN 21 10.5 10.5 15.0 550-650 kPLN 33 16.5 16.5 31.5 650-750 kPLN 36 18.0 18.0 49.5 750-850 kPLN 31 15.5 15.5 65.0 850-950 kPLN 36 18.0 18.0 83.0 950-1050 kPLN 21 10.5 10.5 93.5 1050-1150 kPLN 10 5.0 5.0 98.5 1150-1250 kPLN 2 1.0 1.0 99.5 1250-1350 kPLN 1 0.5 0.5 100.0 Total 200 100.0 100.0 Missing <blank> 0 0.0 <NA> 0 0.0 Total 200 100.0

##        # classes  Goodness of fit Tabular accuracy 
##       10.0000000        0.9780872        0.8508467

As we can see - the TAI index is quite high. 0.85 means that we can accept the proposed construction of the frequency table.

Basic plots

In this section, we should represent our data using basic (pre-installed in R) graphics. Select the most appropriate graphs depending on the scale of the selected variables. Explore the heterogeneity of the distribution by presenting the data by group (e.g., by neighborhood, building type, etc.). Don’t forget about main titles, labels and legends. Read more about graphical parameters here.

Note that the echo = FALSE parameter has been added to the code snippet to prevent printing the R code that generated the graph.

ggplot2 plots

Now, let’s use the ggplot2 and ggpubr libraries to plot.

Ggplot2 allows you to show the average value for each group using the stat_summary() function. You no longer need to calculate average values before creating a graph!

RainCloud Plot

Faceting

Faceting generates small multiples, each showing a different subset of the data. They are a powerful tool for exploratory data analysis: you can quickly compare patterns in different parts of the data and see if they are the same or different. Read more here.

Univariate Statistics

Before automatically reporting the full summary table of descriptive statistics, this time your goal is to measure the central tendency of the price distribution. Compare the mean, median, and mode along with positional measures - quantiles - by district and building type or number of rooms in the apartment.

    mean(price_PLN)

## [1] 760035

    median(price_PLN)

## [1] 755719.5

    sd(price_PLN) #standard deviation

## [1] 186099.8

    var(price_PLN) #variance

## [1] 34633125960

    coeff_var<-sd(price_PLN)/mean(price_PLN) #coefficient of variability %
    coeff_var

## [1] 0.2448568

    IQR(price_PLN)# difference between quartiles =Q3-Q1 

##      75% 
## 282686.5

    sx<-IQR(price_PLN)/2  #interquartile deviation
    coeff_varx<-sx/median(price_PLN) #IQR coefficient of variability %
    coeff_varx

##       75% 
## 0.1870314

    min(price_PLN)

## [1] 359769

    max(price_PLN)

## [1] 1277691

    quantile(price_PLN,probs=c(0,0.1,0.25,0.5,0.75,0.95,1),na.rm=TRUE)

##        0%       10%       25%       50%       75%       95%      100% 
##  359769.0  518806.8  619073.8  755719.5  901760.2 1054250.8 1277691.0

Ok, we have calculated all of the basic summary statistics above. Let’s wrap them up together now.

rooms	boxplot	histogram	line1	line2	points1
1
2
3
4

Summary tables

Ok, now we will finally summarize the basic measures of central tendency for prices by district/building type using the ‘kable’ package. Feel free to customize your final report. See some hints here.

gtsummary

We can calculate easily descriptive statistics also using gtsummary package:

apartments %>%
  select(price_PLN,rooms) %>%
  tbl_summary(label= price_PLN ~ "Price",digits=c(price_PLN)~2,by=rooms,type = all_continuous() ~ "continuous2", statistic = all_continuous() ~ c("{N_nonmiss}", "{median} ({p25}, {p75})", "{min}, {max}"),missing = "no")

Characteristic	1, N = 44	2, N = 50	3, N = 58	4, N = 48
Price
N	44.00	50.00	58.00	48.00
Median (IQR)	520,507.00 (479,684.75, 555,024.75)	677,260.00 (634,757.25, 717,728.50)	846,303.50 (769,683.75, 901,078.75)	964,338.50 (909,371.50, 1,050,976.75)
Range	359,769.00, 657,146.00	590,286.00, 888,634.00	632,770.00, 965,829.00	736,669.00, 1,277,691.00

dfSummary

dfSummary() creates a summary table with statistics, frequencies and graphs for all variables in a data frame. The information displayed is type-specific (character, factor, numeric, date) and also varies according to the number of distinct values.

When using dfSummary() in R Markdown documents, it is generally a good idea to exclude a column or two to avoid margin overflow. Since the Valid and Missing columns are redundant, we can drop either one of them.

dfSummary(apartments,
          plain.ascii  = FALSE, 
          style        = "grid", 
          graph.magnif = 0.75, 
          valid.col    = FALSE,
          tmp.img.dir  = "/tmp")

## temporary images written to 'C:\tmp'

Data Frame Summary

apartments

Dimensions: 200 x 6
Duplicates: 0

No	Variable	Stats / Values	Freqs (% of Valid)	Graph	Missing
1	price_PLN [numeric]	Mean (sd) : 760035 (186099.8) min < med < max: 359769 < 755719.5 < 1277691 IQR (CV) : 282686.5 (0.2)	200 distinct values		0 (0.0%)
2	price_EUR [numeric]	Mean (sd) : 175934 (43078.6) min < med < max: 83280 < 174935 < 295762 IQR (CV) : 65436.2 (0.2)	200 distinct values		0 (0.0%)
3	rooms [ordered, factor]	1. 1 2. 2 3. 3 4. 4	44 (22.0%) 50 (25.0%) 58 (29.0%) 48 (24.0%)		0 (0.0%)
4	size [numeric]	Mean (sd) : 46.2 (20.1) min < med < max: 17 < 43.7 < 87.7 IQR (CV) : 30.2 (0.4)	162 distinct values		0 (0.0%)
5	district [factor]	1. Biskupin 2. Krzyki 3. Srodmiescie	65 (32.5%) 79 (39.5%) 56 (28.0%)		0 (0.0%)
6	building_type [factor]	1. kamienica 2. niski blok 3. wiezowiec	61 (30.5%) 63 (31.5%) 76 (38.0%)		0 (0.0%)

To produce optimal results, summarytools has its own version of the base by() function. It’s called stby(), and we use it exactly as we would by():

(stats_by_rooms <- stby(data      = apartments, INDICES   = apartments$rooms, FUN       = descr, stats     = "common", transpose = TRUE))

## Non-numerical variable(s) ignored: rooms, district, building_type

Descriptive Statistics
apartments
Group: rooms = 1
N: 44

	Mean	Std.Dev	Min	Median	Max	N.Valid	Pct.Valid
price_EUR	119332.95	15497.90	83280.00	120488.00	152117.00	44.00	100.00
price_PLN	515518.05	66951.03	359769.00	520507.00	657146.00	44.00	100.00
size	19.28	1.46	17.00	19.10	21.90	44.00	100.00

Group: rooms = 2
N: 50

	Mean	Std.Dev	Min	Median	Max	N.Valid	Pct.Valid
price_EUR	158233.22	15063.13	136640.00	156773.00	205702.00	50.00	100.00
price_PLN	683567.70	65072.66	590286.00	677260.00	888634.00	50.00	100.00
size	36.80	4.46	29.60	35.95	43.70	50.00	100.00

Group: rooms = 3
N: 58

	Mean	Std.Dev	Min	Median	Max	N.Valid	Pct.Valid
price_EUR	192987.55	20125.88	146475.00	195904.00	223572.00	58.00	100.00
price_PLN	833706.02	86943.90	632770.00	846303.50	965829.00	58.00	100.00
size	53.33	7.21	41.20	53.45	65.20	58.00	100.00

Group: rooms = 4
N: 48

	Mean	Std.Dev	Min	Median	Max	N.Valid	Pct.Valid
price_EUR	225650.42	26347.03	170525.00	223226.50	295762.00	48.00	100.00
price_PLN	974809.96	113819.21	736669.00	964338.50	1277691.00	48.00	100.00
size	72.05	10.18	53.30	70.85	87.70	48.00	100.00

Tidy Tables

When generating freq() or descr() tables, it is possible to turn the results into “tidy” tables with the use of the tb() function (think of tb as a diminutive for tibble). For example:

apartments %>%
  descr(stats = "common") %>%
  tb()

## # A tibble: 3 × 8
##   variable      mean       sd    min      med       max n.valid pct.valid
##   <chr>        <dbl>    <dbl>  <dbl>    <dbl>     <dbl>   <dbl>     <dbl>
## 1 price_EUR 175934.   43079.   83280 174935    295762       200       100
## 2 price_PLN 760035.  186100.  359769 755720.  1277691       200       100
## 3 size          46.2     20.1     17     43.7      87.7     200       100

Here are some examples showing how lists created using stby() or group_by() can be transformed into tidy tibbles.

grouped_descr <- stby(data    = apartments,INDICES = apartments$rooms, FUN     = descr, stats   = "common")

grouped_descr %>% tb()

## # A tibble: 12 × 9
##    rooms variable      mean        sd      min      med    max n.valid pct.valid
##    <fct> <chr>        <dbl>     <dbl>    <dbl>    <dbl>  <dbl>   <dbl>     <dbl>
##  1 1     price_EUR 119333.   15498.    83280   120488   1.52e5      44       100
##  2 1     price_PLN 515518.   66951.   359769   520507   6.57e5      44       100
##  3 1     size          19.3      1.46     17       19.1 2.19e1      44       100
##  4 2     price_EUR 158233.   15063.   136640   156773   2.06e5      50       100
##  5 2     price_PLN 683568.   65073.   590286   677260   8.89e5      50       100
##  6 2     size          36.8      4.46     29.6     36.0 4.37e1      50       100
##  7 3     price_EUR 192988.   20126.   146475   195904   2.24e5      58       100
##  8 3     price_PLN 833706.   86944.   632770   846304.  9.66e5      58       100
##  9 3     size          53.3      7.21     41.2     53.4 6.52e1      58       100
## 10 4     price_EUR 225650.   26347.   170525   223226.  2.96e5      48       100
## 11 4     price_PLN 974810.  113819.   736669   964338.  1.28e6      48       100
## 12 4     size          72.0     10.2      53.3     70.8 8.77e1      48       100

A Bridge to Other Packages

stby(data    = apartments, 
     INDICES = apartments$rooms, 
     FUN     = descr, 
     stats   = "fivenum") %>%
  tb(order = 3) %>%
  kable(format = "html", digits = 2) %>%
  collapse_rows(columns = 1, valign = "top")

variable	rooms	min	q1	med	q3	max
price_EUR	1	83280.0	110881.0	120488.00	128568.00	152117.0
	2	136640.0	146754.0	156773.00	166259.00	205702.0
	3	146475.0	177478.0	195904.00	208599.00	223572.0
	4	170525.0	209827.5	223226.50	243300.00	295762.0
price_PLN	1	359769.0	479005.5	520507.00	555411.50	657146.0
	2	590286.0	633978.0	677260.00	718237.00	888634.0
	3	632770.0	766707.0	846303.50	901149.00	965829.0
	4	736669.0	906455.0	964338.50	1051055.50	1277691.0
size	1	17.0	18.1	19.10	20.60	21.9
	2	29.6	32.9	35.95	40.50	43.7
	3	41.2	47.9	53.45	59.70	65.2
	4	53.3	64.2	70.85	82.15	87.7

Your turn!

Your task this week is to: prepare your own descriptive analysis for the “CreditCard” dataset (AER package). It is a cross-sectional dataframe on the credit history for a sample of applicants for a type of credit card.

library(AER)
library(dplyr)
library(ggplot2)

data(CreditCard)

# analysis by credit risk
summary_by_card <- CreditCard %>%
  group_by(card) %>%
  summarise(
    mean_yearly_income = mean(income),
    median_yearly_income = median(income),
    mean_expenditure = mean(expenditure),
    median_expenditure = median(expenditure),
    mean_age = mean(age),
    median_age = median(age),
    mean_expenditure_income_ratio = mean(expenditure / income),
    median_expenditure_income_ratio = median(expenditure / income)
  )

# Histogram of yearly income by credit risk
income_histogram <- ggplot(CreditCard, aes(x = income, fill = card)) +
  geom_histogram(binwidth = 2, alpha = 0.7, position = "identity") +
  labs(title = "Relation between yearly income and credit risk",
       x = "Yearly Income (in 10k dollars)",
       y = "Frequency",
       fill = "Credit Risk")


expenditure_boxplot <- ggplot(CreditCard, aes(x = card, y = expenditure, fill = card)) +
  geom_boxplot() +
  labs(title = "Expenditure Distribution by Credit Risk",
       x = "Credit Risk",
       y = "Expenditure",
       fill = "Credit Risk")

# graphs
print(income_histogram)

print(expenditure_boxplot)

# your code here
# Load required libraries
library(AER) # for accessing the CreditCard dataset
library(dplyr) # for data manipulation
library(ggplot2)# for data visualization
library(tidyr)

# View the structure of the dataset
str(CreditCard)

## 'data.frame':    1319 obs. of  12 variables:
##  $ card       : Factor w/ 2 levels "no","yes": 2 2 2 2 2 2 2 2 2 2 ...
##  $ reports    : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ age        : num  37.7 33.2 33.7 30.5 32.2 ...
##  $ income     : num  4.52 2.42 4.5 2.54 9.79 ...
##  $ share      : num  0.03327 0.00522 0.00416 0.06521 0.06705 ...
##  $ expenditure: num  124.98 9.85 15 137.87 546.5 ...
##  $ owner      : Factor w/ 2 levels "no","yes": 2 1 2 1 2 1 1 2 2 1 ...
##  $ selfemp    : Factor w/ 2 levels "no","yes": 1 1 1 1 1 1 1 1 1 1 ...
##  $ dependents : num  3 3 4 0 2 0 2 0 0 0 ...
##  $ months     : num  54 34 58 25 64 54 7 77 97 65 ...
##  $ majorcards : num  1 1 1 1 1 1 1 1 1 1 ...
##  $ active     : num  12 13 5 7 5 1 5 3 6 18 ...

# Define a function to calculate mode
get_mode <- function(x) {
  ux <- unique(x)
  ux[which.max(tabulate(match(x, ux)))]
}
      #Data cleansing
      # Calculate the first and third quartiles
      Q1 <- quantile(CreditCard$income, 0.25)
      Q3 <- quantile(CreditCard$income, 0.75)
      
      # Calculate the IQR
      IQR <- Q3 - Q1
      
      # Define the lower and upper bounds for outliers
      lower_bound <- Q1 - 1.5 * IQR
      upper_bound <- Q3 + 1.5 * IQR
      
      # Filter the dataset to remove outliers
      CreditCard_filtered <- CreditCard %>%
        filter(income >= lower_bound & income <= upper_bound)

# Summary statistics an making one table for better visualization
summary_by_card <- CreditCard %>%
  group_by(card) %>%
  summarise(
    mean_reports = round(mean(reports,na.rm = TRUE),3),
    median_reports = round(median(reports,na.rm = TRUE),3),
    mode_reports = get_mode(reports),
    mean_age = round(mean(age,na.rm = TRUE),3),
    median_age = round(median(age,na.rm = TRUE),3),
    mode_age = get_mode(age),
    mean_income = round(mean(income,na.rm = TRUE),3),
    median_income = round(median(income,na.rm = TRUE),3),
    mode_income = get_mode(income),
    mean_share = round(mean(share,na.rm = TRUE),3),
    median_share = round(median(share,na.rm = TRUE),3),
    mode_share = get_mode(share),
    mean_expenditure = round(mean(expenditure,na.rm = TRUE),3),
    median_expenditure = round(median(expenditure,na.rm = TRUE),3),
    mode_expenditure = get_mode(expenditure)
  )%>%
  pivot_wider(names_from = card, values_from = everything(), names_glue = "{.value}-{card}")

summary_by_card_wide <- as.data.frame(summary_by_card)


# Visualization: Histogram of Age by Credit Risk
age_histogram <- ggplot(CreditCard, aes(x = age, fill = card)) +
  geom_histogram(binwidth = 2, alpha = 0.7, position = "identity") +
  labs(title = "Distribution of Age by Credit Risk",
       x = "Age",
       y = "Frequency",
       fill = "Credit Risk")

# Visualization: Boxplot of Income by Credit Risk
income_boxplot <- ggplot(CreditCard, aes(x = card, y = income, fill = card)) +
  geom_boxplot() +
  labs(title = "Income Distribution by Credit Risk",
       x = "Credit Risk",
       y = "Income",
       fill = "Credit Risk")

# Visualization: Scatterplot of Share vs. Expenditure by Credit Risk
share_expenditure_scatter <- ggplot(CreditCard, aes(x = share, y = expenditure, color = card)) +
  geom_point() +
  labs(title = "Share vs. Expenditure by Credit Risk",
       x = "Share",
       y = "Expenditure",
       color = "Credit Risk")
# Distribution of income by credit risk (card)
income_distribution <- ggplot(CreditCard, aes(x = income, fill = card)) +
  geom_density(alpha = 0.5) +
  labs(title = "Distribution of Income by Credit Risk",
       x = "Income",
       y = "Density",
       fill = "Credit Risk")

# Histogram of income by credit risk
income_credit_histogram <- ggplot(CreditCard, aes(x = income, fill = card)) +
  geom_histogram(binwidth = 1000, alpha = 0.7, position = "identity") +
  labs(title = "Histogram of Income by Credit Risk",
       x = "Income",
       y = "Frequency",
       fill = "Credit Risk")

# Boxplot of age by credit risk
age_boxplot <- ggplot(CreditCard, aes(x = card, y = age, fill = card)) +
  geom_boxplot() +
  labs(title = "Boxplot of Age by Credit Risk",
       x = "Credit Risk",
       y = "Age",
       fill = "Credit Risk")

# Frequency distribution of expenditures by credit risk
expenditure_histogram <- ggplot(CreditCard, aes(x = expenditure, fill = card)) +
  geom_histogram(binwidth = 50, alpha = 0.7, position = "identity") +
  labs(title = "Frequency Distribution of Expenditures by Credit Risk",
       x = "Expenditure",
       y = "Frequency",
       fill = "Credit Risk") +
  theme_minimal()

# Frequency distribution of shares by credit risk
share_histogram <- ggplot(CreditCard, aes(x = share, fill = card)) +
  geom_histogram(binwidth = 0.05, alpha = 0.7, position = "identity") +
  labs(title = "Frequency Distribution of Shares by Credit Risk",
       x = "Share",
       y = "Frequency",
       fill = "Credit Risk") +
  theme_minimal()

# Display the visualizations
print(income_distribution)

print(expenditure_histogram)

print(share_histogram)

print(income_credit_histogram)

print(age_boxplot)

print(age_histogram)

print(income_boxplot)

print(share_expenditure_scatter)

# Making one summary for both card types
print_summary_side_by_side <- function(summary_df) {
  cat("\n------------------------------------------------------------\n")
  cat(sprintf("%-30s %-20s %s\n", "Statistic", "Credit Risk: no", "Credit Risk: yes"))
  cat("------------------------------------------------------------\n")
  for (stat in c("mean_reports", "median_reports", "mode_reports", 
                 "mean_age", "median_age", "mode_age", 
                 "mean_income", "median_income", "mode_income", 
                 "mean_share", "median_share", "mode_share", 
                 "mean_expenditure", "median_expenditure", "mode_expenditure")) {
    no_label <- paste(stat, "no", sep = "-")
    yes_label <- paste(stat, "yes", sep = "-")
    cat(sprintf("%-30s %-20s %s\n", stat, 
                format(summary_df[[no_label]], nsmall = 3), 
                format(summary_df[[yes_label]], nsmall = 3)))
  }
  cat("------------------------------------------------------------\n")
}

CreditCard %>%
  descr(stats = "all") %>%
  tb()

## # A tibble: 9 × 16
##   variable        mean       sd      min      q1     med      q3     max     mad
##   <chr>          <dbl>    <dbl>    <dbl>   <dbl>   <dbl>   <dbl>   <dbl>   <dbl>
## 1 active        7.00     6.31   0        2   e+0 6   e+0 1.1 e+1 4.6 e+1 5.93e+0
## 2 age          33.2     10.1    0.167    2.54e+1 3.12e+1 3.94e+1 8.35e+1 9.51e+0
## 3 dependents    0.994    1.25   0        0       1   e+0 2   e+0 6   e+0 1.48e+0
## 4 expenditure 185.     272.     0        4.58e+0 1.01e+2 2.49e+2 3.10e+3 1.50e+2
## 5 income        3.37     1.69   0.21     2.24e+0 2.9 e+0 4   e+0 1.35e+1 1.19e+0
## 6 majorcards    0.817    0.387  0        1   e+0 1   e+0 1   e+0 1   e+0 0      
## 7 months       55.3     66.3    0        1.2 e+1 3   e+1 7.2 e+1 5.4 e+2 3.11e+1
## 8 reports       0.456    1.35   0        0       0       0       1.4 e+1 0      
## 9 share         0.0687   0.0947 0.000109 2.23e-3 3.88e-2 9.36e-2 9.06e-1 5.66e-2
## # ℹ 7 more variables: iqr <dbl>, cv <dbl>, skewness <dbl>, se.skewness <dbl>,
## #   kurtosis <dbl>, n.valid <dbl>, pct.valid <dbl>

print_summary_side_by_side(summary_by_card_wide)

## 
## ------------------------------------------------------------
## Statistic                      Credit Risk: no      Credit Risk: yes
## ------------------------------------------------------------
## mean_reports                   1.588                0.129
## median_reports                 1.000                0.000
## mode_reports                   0.000                0.000
## mean_age                       33.203               33.216
## median_age                     31.833               31.083
## mode_age                       25.83333             26.08333
## mean_income                    3.069                3.451
## median_income                  2.590                3.000
## mode_income                    2.500                3.000
## mean_share                     0.000                0.088
## median_share                   0.000                0.060
## mode_share                     0.00048              0.0032
## mean_expenditure               0.000                238.602
## median_expenditure             0.000                150.177
## mode_expenditure               0.000                0.000
## ------------------------------------------------------------

Based on the visualizations provided, we can see some major differences between for customers with different credit risk. The biggest difference we can see in incomes. Those with cards (card=yes), tend to have higher income.They also have bigger and way more varied expenditures. The age, however, doesn’t really differ with both groups. Monthly credit card expenditure to yearly income is distinct between the two groups, those having a credit card using a more significant proportion of their income.