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.

Are the yearly incomes (in USD 10,000), credit card expenditures, age, ratio of monthly credit card expenditure to yearly income - significantly different for applicants for customers with different credit risk (“card” variable - factor)?

Prepare a professional data visualizations, descriptive statistics’ tables and interpret them.

data(CreditCard)
# your code here
CreditCard_subset <- CreditCard[, c("card", "income", "expenditure", "age", "share")]

CreditCard_subset$card<-as.factor(CreditCard_subset$card)
CreditCard_subset$income<-as.numeric(CreditCard_subset$income)
CreditCard_subset$expenditure<-as.numeric(CreditCard_subset$expenditure)
CreditCard_subset$age<-as.numeric(CreditCard_subset$age)
CreditCard_subset$share<-as.numeric(CreditCard_subset$share)

# Create a frequency table for the 'card' variable
tableL <- table(CreditCard_subset$card)
table_df <- as.data.frame(tableL)
colnames(table_df) <- c("Card", "Frequency")
kbl(table_df, caption = "Acceptance of credit card") %>%
  kable_material(c("striped", "hover"))

Acceptance of credit card

Card	Frequency
no	296
yes	1023

summary_table_yes <- summary(CreditCard_subset[CreditCard_subset$card == "yes", c("income", "expenditure", "age", "share")])
summary_table_no <- summary(CreditCard_subset[CreditCard_subset$card == "no", c("income", "expenditure", "age", "share")])

summary_df_yes <- as.data.frame(summary_table_yes)
summary_df_no <- as.data.frame(summary_table_no)

summary_kable_yes <- kable(summary_df_yes, caption = "Summary Statistics Table for card accepted", align = "c") %>%
                      kable_styling(full_width = FALSE)
summary_kable_no <- kable(summary_df_no, caption = "Summary Statistics Table for card declined", align = "c") %>%
                     kable_styling(full_width = FALSE)

summary_df_yes_wide <- pivot_wider(summary_df_yes, names_from = Var2, values_from = Freq)

## Warning: Values from `Freq` are not uniquely identified; output will contain list-cols.
## • Use `values_fn = list` to suppress this warning.
## • Use `values_fn = {summary_fun}` to summarise duplicates.
## • Use the following dplyr code to identify duplicates.
##   {data} |>
##   dplyr::summarise(n = dplyr::n(), .by = c(Var1, Var2)) |>
##   dplyr::filter(n > 1L)

summary_df_no_wide <- pivot_wider(summary_df_no, names_from = Var2, values_from = Freq)

## Warning: Values from `Freq` are not uniquely identified; output will contain list-cols.
## • Use `values_fn = list` to suppress this warning.
## • Use `values_fn = {summary_fun}` to summarise duplicates.
## • Use the following dplyr code to identify duplicates.
##   {data} |>
##   dplyr::summarise(n = dplyr::n(), .by = c(Var1, Var2)) |>
##   dplyr::filter(n > 1L)

summary_kable_yes <- kable(summary_df_yes_wide, caption = "Summary Statistics Table for card accepted", align = "c") %>%
                      kable_styling(full_width = FALSE)

summary_kable_no <- kable(summary_df_no_wide, caption = "Summary Statistics Table for card declined", align = "c") %>%
                      kable_styling(full_width = FALSE)
# Display the tables
summary_kable_yes

Summary Statistics Table for card accepted

Var1	income	expenditure	age	share
	Min. : 0.210 , 1st Qu.: 2.350 , Median : 3.000 , Mean : 3.451 , 3rd Qu.: 4.000 , Max. :13.500	Min. : 0.00 , 1st Qu.: 66.52 , Median : 150.18 , Mean : 238.60 , 3rd Qu.: 313.96 , Max. :3099.51	Min. : 0.1667 , 1st Qu.:25.3333 , Median :31.0833 , Mean :33.2160 , 3rd Qu.:39.7083 , Max. :83.5000	Min. :0.000186 , 1st Qu.:0.026635 , Median :0.060209 , Mean :0.088482 , 3rd Qu.:0.113858 , Max. :0.906320

summary_kable_no

Summary Statistics Table for card declined

Var1	income	expenditure	age	share
	Min. : 0.490 , 1st Qu.: 2.000 , Median : 2.590 , Mean : 3.069 , 3rd Qu.: 3.625 , Max. :11.000	Min. :0 , 1st Qu.:0 , Median :0 , Mean :0 , 3rd Qu.:0 , Max. :0	Min. : 0.75 , 1st Qu.:25.65 , Median :31.83 , Mean :33.20 , 3rd Qu.:38.06 , Max. :80.17	Min. :0.0001091 , 1st Qu.:0.0003311 , Median :0.0004633 , Mean :0.0004768 , 3rd Qu.:0.0006000 , Max. :0.0024490

boxplot(CreditCard$income, main = "Boxplot of Income" ,ylab = "Income")

boxplot(CreditCard$expenditure, main = "Boxplot of Expenditure", ylab = "Expenditure")

boxplot(CreditCard$age, main = "Boxplot of Age", ylab = "Age" )

#We see form boxplots, that there are outliers, that we have handle with

outliersIncome <- CreditCard$income[CreditCard$income > quantile(CreditCard$income, 0.75) + 1.5 * (quantile(CreditCard$income, 0.75) - quantile(CreditCard$income, 0.25)) | 
                                CreditCard$income < quantile(CreditCard$income, 0.25) - 1.5 * (quantile(CreditCard$income, 0.75) - quantile(CreditCard$income, 0.25))]
CreditCard_clean <- CreditCard[!CreditCard$income %in% outliersIncome, ]

outliersExpenditure <- CreditCard$expenditure[CreditCard$expenditure > quantile(CreditCard$expenditure, 0.75) + 1.5 * (quantile(CreditCard$expenditure, 0.75) - quantile(CreditCard$expenditure, 0.25)) | CreditCard$expenditure < quantile(CreditCard$expenditure, 0.25) - 1.5 * (quantile(CreditCard$expenditure, 0.75) - quantile(CreditCard$expenditure, 0.25))]
CreditCard_clean <- CreditCard[!CreditCard$expenditure %in% outliersExpenditure, ]

outliersAge <- CreditCard$age[CreditCard$age > quantile(CreditCard$age, 0.75) + 1.5 * (quantile(CreditCard$age, 0.75) - quantile(CreditCard$age, 0.25)) | 
                                CreditCard$age < quantile(CreditCard$age, 0.25) - 1.5 * (quantile(CreditCard$age, 0.75) - quantile(CreditCard$age, 0.25))]
CreditCard_clean <- CreditCard[!CreditCard$age %in% outliersAge, ]



densityIncome <- ggdensity(CreditCard_clean, x = "income", 
                       fill = "card", palette = "jco")
densityAge <- ggdensity(CreditCard_clean, x = "age", 
                       fill = "card", palette = "jco")
densityExpenditure <- ggdensity(CreditCard_clean, x = "expenditure", 
                       fill = "card", palette = "jco")
CreditCard_yes <- subset(CreditCard_clean, owner == "yes")
densityIncome <- ggplot(CreditCard_yes, aes(x = income, fill = owner)) +
                 geom_density(alpha = 0.5) +
                 scale_fill_manual(values = c("yes" = "yellow")) +  
                 labs(title = "Density Plot of Income for 'yes' Owners")


CreditCard_clean$ratio <- CreditCard_clean$expenditure / (CreditCard_clean$income) /12

# histrogram
ggplot(CreditCard_clean, aes(x = ratio)) +
  geom_histogram(binwidth = 0.1, fill = "yellow", color = "blue") +
  labs(title = "Breakdown of the ratio of monthly credit card expenses to annual income",
       x = "Ratio (monthly expenses / annual income)",
       y = "How many observations") +
      scale_y_continuous(limits = c(0, 20), breaks = seq(0, 20, by = 4)) +
  theme_minimal()

## Warning: Removed 2 rows containing missing values or values outside the scale range
## (`geom_bar()`).

yes_data <- CreditCard_clean[CreditCard_clean$card == "yes", ]
no_data <- CreditCard_clean[CreditCard_clean$card == "no", ]

mean_income_yes <- aggregate(income ~ age, data = yes_data, FUN = mean)
mean_income_no <- aggregate(income ~ age, data = no_data, FUN = mean)

overall_mean_income <- mean(CreditCard_clean$income)


ggplot(mean_income_yes, aes(x = age, y = income)) +
  geom_point() +
  geom_hline(yintercept = overall_mean_income, linetype = "dashed", color = "red", size=1.5) +
  geom_line(stat = "smooth", aes(group = 1), color = "blue", size=1.5) +
  labs(title = "Average income for customers with credit risk",
       x = "Age",
       y = "Income")

## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

## `geom_smooth()` using method = 'loess' and formula = 'y ~ x'

ggplot(mean_income_no, aes(x = age, y = income)) +
  geom_point() +
  geom_hline(yintercept = overall_mean_income, linetype = "dashed", color = "red", size=1.5) +
  geom_line(stat = "smooth", aes(group = 1), color = "blue", size=1.5) +
  labs(title = "Average income for customers without credit risk",
       x = "Age",
       y = "Income")

## `geom_smooth()` using method = 'loess' and formula = 'y ~ x'

ggplot(CreditCard_clean, aes(income, card)) + 
  geom_abline(size = 1.5) +
  geom_jitter(width = 0.1, height = 0.1, color = "blue")

ggdensity(CreditCard_clean, x = "income", 
                       fill = "card", palette = "jco")

ggdensity(CreditCard_clean, x = "age", 
                       fill = "card", palette = "jco")

From the table, we observe that the mean and median of income, expenditure, and the share of individuals with credit card risk are slightly higher than those without risk. However, the age is greater among individuals without risk.

The plots reveal that the income range is wider for individuals with risk, and income tends to increase with age. Jitter plots indicate that there are more values clustered around the mean for individuals with risk, although the values are similar in both cases. Density plots suggest that the distribution of individuals based on income is similar in both cases, as well as according to age, except for individuals aged 35-37, who are more likely to be without risk