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.

Table 1. Apartments in Wroclaw - prices in PLN by number of rooms.

	1 room	2 rooms	3 rooms	4 rooms
Min	359769	590286	632770	736669
Max	657146	888634	965829	1277691
Q1	479684.75	634757.25	769683.75	909371.5
Median	520507	677260	846303.5	964338.5
Q3	555024.75	717728.5	901078.75	1050976.75
Mean	515518.05	683567.7	833706.02	974809.96
Sd	66951.03	65072.66	86943.9	113819.21
IQR	75340	82971.25	131395	141605.25
Sx	37670	41485.62	65697.5	70802.62
Var %	0.13	0.1	0.1	0.12
IQR Var %	0.14	0.12	0.16	0.15
Skewness	-0.2	0.8	-0.42	0.33
Kurtosis	-0.38	0.48	-0.83	0.05

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")

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.

FREQUENCY TABLES

Income in kPLN

x label Freq Percent Valid Percent Cumulative Percent Valid 0-10 kUSD 2 0.2 0.2 0.2 10-20 kUSD 233 17.7 17.7 17.9 20-30 kUSD 498 37.8 37.8 55.7 30-40 kUSD 284 21.6 21.6 77.3 40-50 kUSD 134 10.2 10.2 87.5 50-60 kUSD 71 5.4 5.4 92.9 60-70 kUSD 42 3.2 3.2 96.0 70-80 kUSD 20 1.5 1.5 97.6 80-90 kUSD 10 0.8 0.8 98.3 90-100 kUSD 13 1.0 1.0 99.3 100-110 kUSD 6 0.5 0.5 99.8 110-120 kUSD 1 0.1 0.1 99.8 120-130 kUSD 1 0.1 0.1 99.9 130-140 kUSD 1 0.1 0.1 100.0 Total 1316 100.0 100.0 Missing <blank> 0 0.0 <NA> 0 0.0 Total 1316 100.0

##        # classes  Goodness of fit Tabular accuracy 
##       14.0000000        0.9740158        0.8105061

Expenditure

x label Freq Percent Valid Percent Cumulative Percent Valid 0-100 337 25.6 34.4 34.4 100-200 261 19.8 26.6 61.0 200-300 126 9.6 12.9 73.9 300-400 97 7.4 9.9 83.8 400-500 70 5.3 7.1 90.9 500-600 32 2.4 3.3 94.2 600-700 25 1.9 2.6 96.7 700-800 11 0.8 1.1 97.9 800-900 16 1.2 1.6 99.5 900-1000 5 0.4 0.5 100.0 Total 980 74.5 100.0 Missing <blank> 0 0.0 <NA> 336 25.5 Total 1316 100.0

## Warning in classIntervals(CreditCard$expenditure, n = 11, style = "fixed", :
## variable range greater than fixedBreaks

##        # classes  Goodness of fit Tabular accuracy 
##       11.0000000        0.9362416        0.8181726

Age

x label Freq Percent Valid Percent Cumulative Percent Valid 0-10 7 0.5 0.5 0.5 10-20 22 1.7 1.7 2.2 20-30 562 42.7 42.7 44.9 30-40 422 32.1 32.1 77.0 40-50 219 16.6 16.6 93.6 50-60 61 4.6 4.6 98.3 60-70 19 1.4 1.4 99.7 70-80 2 0.2 0.2 99.8 80-90 2 0.2 0.2 100.0 Total 1316 100.0 100.0 Missing <blank> 0 0.0 <NA> 0 0.0 Total 1316 100.0

##        # classes  Goodness of fit Tabular accuracy 
##        9.0000000        0.9296600        0.7164709

Ratio of monthly expenditure to yearly income

x label Freq Percent Valid Percent Cumulative Percent Valid 0-0.002 929 70.6 92.8 92.8 0.002-0.004 42 3.2 4.2 97.0 0.004-0.006 12 0.9 1.2 98.2 0.006-0.008 6 0.5 0.6 98.8 0.008-0.01 5 0.4 0.5 99.3 0.01-0.012 2 0.2 0.2 99.5 0.012-0.014 2 0.2 0.2 99.7 0.016-0.018 1 0.1 0.1 99.8 0.018-0.02 1 0.1 0.1 99.9 0.024-0.026 1 0.1 0.1 100.0 Total 1001 76.1 100.0 Missing <blank> 0 0.0 <NA> 315 23.9 Total 1316 100.0

##        # classes  Goodness of fit Tabular accuracy 
##       13.0000000        0.9393423        0.5948975

PLOTS

#income
    mean(CreditCard$income)

## [1] 3.363549

    median(CreditCard$income)

## [1] 2.9

    sd(CreditCard$income) #standard deviation

## [1] 1.692444

    var(CreditCard$income) #variance

## [1] 2.864368

    coeff_var<-sd(CreditCard$income)/mean(CreditCard$income) #coefficient of variability %
    coeff_var

## [1] 0.5031722

    IQR(CreditCard$income)# difference between quartiles =Q3-Q1 

##      75% 
## 1.753125

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

##       75% 
## 0.3022629

    min(CreditCard$income)

## [1] 0.21

    max(CreditCard$income)

## [1] 13.5

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

##        0%       10%       25%       50%       75%       95%      100% 
##  0.210000  1.800000  2.246875  2.900000  4.000000  6.891000 13.500000

#expenditure
    mean(CreditCard$expenditure)

## [1] 184.9889

    median(CreditCard$expenditure)

## [1] 101.3963

    sd(CreditCard$expenditure) #standard deviation

## [1] 272.1385

    var(CreditCard$expenditure) #variance

## [1] 74059.35

    coeff_var<-sd(CreditCard$expenditure)/mean(CreditCard$expenditure) #coefficient of variability %
    coeff_var

## [1] 1.471107

    IQR(CreditCard$expenditure)# difference between quartiles =Q3-Q1 

##      75% 
## 243.6843

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

##      75% 
## 1.201644

    min(CreditCard$expenditure)

## [1] 0

    max(CreditCard$expenditure)

## [1] 3099.505

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

##          0%         10%         25%         50%         75%         95% 
##    0.000000    0.000000    5.286458  101.396250  248.970800  636.451425 
##        100% 
## 3099.505000

card	boxplot	histogram	line1	line2	points1
yes
no

card	boxplot	histogram	line1	line2	points1
yes
no

card	boxplot	histogram	line1	line2	points1
yes
no

card	boxplot	histogram	line1	line2	points1
yes
no

Summary Table: Income Summary Statistics by Card

	No	Yes
Min	0.49	0.21
Max	11.00	13.50
Q1	2.00	2.35
Median	2.59	3.00
Q3	3.68	4.00
Mean	3.07	3.45
Sd	1.62	1.70
IQR	1.68	1.65
Sx	0.84	0.82
Var_Percent	0.53	0.49
IQR_Var_Percent	0.65	0.55
Skewness	2.04	1.92
Kurtosis	5.20	4.92

Summary Table: Expenditure Summary Statistics By Card

	No	Yes
Min	0	0.00
Max	0	3099.50
Q1	0	66.41
Median	0	149.94
Q3	0	313.41
Mean	0	238.20
Sd	0	287.57
IQR	0	247.00
Sx	0	123.50
Var_Percent	NaN	1.21
IQR_Var_Percent	NaN	1.65
Skewness	NaN	3.58
Kurtosis	NaN	20.16

Summary Table: Age Summary Statistics By Card

	No	Yes
Min	0.75	0.17
Max	80.17	83.50
Q1	25.60	25.33
Median	32.00	31.12
Q3	38.19	39.73
Mean	33.23	33.22
Sd	9.95	10.21
IQR	12.58	14.40
Sx	6.29	7.20
Var_Percent	0.30	0.31
IQR_Var_Percent	0.39	0.46
Skewness	1.09	0.77
Kurtosis	2.33	1.21

Summary Table: Ratio of expenditure to income Summary Statistics By Card

	No	Yes
Min	0	0.00
Max	0	0.03
Q1	0	0.00
Median	0	0.00
Q3	0	0.00
Mean	0	0.00
Sd	0	0.00
IQR	0	0.00
Sx	0	0.00
Var_Percent	NaN	2.69
IQR_Var_Percent	NaN	3.26
Skewness	NaN	7.27
Kurtosis	NaN	73.79