# Descriptive statistics in R

- Introduction
- Data
- Minimum and maximum
- Range
- Mean
- Median
- First and third quartile
- Interquartile range
- Standard deviation and variance
- Summary
- Coefficient of variation
- Mode
- Correlation
- Contingency table
- Barplot
- Histogram
- Boxplot
- Dotplot
- Scatterplot
- Line plot
- QQ-plot
- Density plot
- Correlation plot
- Advanced descriptive statistics
- Conclusion

# Introduction

This article explains how to compute the main descriptive statistics in R and how to present them graphically. To learn more about the reasoning behind each descriptive statistics, how to compute them by hand and how to interpret them, read the article “Descriptive statistics by hand”.

To briefly recap what have been said in that article, descriptive statistics (in the broad sense of the term) is a branch of statistics aiming at summarizing, describing and presenting a series of values or a dataset. Descriptive statistics is often the first step and an important part in any statistical analysis. It allows to check the quality of the data and it helps to “understand” the data by having a clear overview of it. If well presented, descriptive statistics is already a good starting point for further analyses. There exists many measures to summarize a dataset. They are divided into two types:

- location measures and
- dispersion measures

Location measures give an understanding about the central tendency of the data, whereas dispersion measures give an understanding about the spread of the data. In this article, we focus only on the implementation in R of the most common descriptive statistics and their visualizations (when deemed appropriate). See online or in the above mentioned article for more information about the purpose and usage of each measure.

# Data

We use the dataset `iris`

throughout the article. This dataset is imported by default in R, you only need to load it by running `iris`

:

`dat <- iris # load the iris dataset and renamed it dat`

Below a preview of this dataset and its structure:

`head(dat) # first 6 observations`

```
## Sepal.Length Sepal.Width Petal.Length Petal.Width Species
## 1 5.1 3.5 1.4 0.2 setosa
## 2 4.9 3.0 1.4 0.2 setosa
## 3 4.7 3.2 1.3 0.2 setosa
## 4 4.6 3.1 1.5 0.2 setosa
## 5 5.0 3.6 1.4 0.2 setosa
## 6 5.4 3.9 1.7 0.4 setosa
```

`str(dat) # structure of dataset`

```
## 'data.frame': 150 obs. of 5 variables:
## $ Sepal.Length: num 5.1 4.9 4.7 4.6 5 5.4 4.6 5 4.4 4.9 ...
## $ Sepal.Width : num 3.5 3 3.2 3.1 3.6 3.9 3.4 3.4 2.9 3.1 ...
## $ Petal.Length: num 1.4 1.4 1.3 1.5 1.4 1.7 1.4 1.5 1.4 1.5 ...
## $ Petal.Width : num 0.2 0.2 0.2 0.2 0.2 0.4 0.3 0.2 0.2 0.1 ...
## $ Species : Factor w/ 3 levels "setosa","versicolor",..: 1 1 1 1 1 1 1 1 1 1 ...
```

The dataset contains 150 observations and 5 variables, representing the length and width of the sepal and petal and the species of 150 flowers. Length and width of the sepal and petal are numeric variables and the species is a factor with 3 levels (indicated by `num`

and `Factor w/ 3 levels`

after the name of the variables). See the different variables types in R if you need a refresh.

Regarding plots, we present the default graphs and the graphs from the well-known `{ggplot2}`

package. Graphs from the `{ggplot2}`

package usually have a better look but it requires more advanced coding skills (see the article “Graphics in R with ggplot2” to learn more). If you need to publish or share your graphs, I suggest using `{ggplot2}`

if you can, otherwise the default graphics will do the job.

*Tip:* I recently discovered the ggplot2 builder from the `{esquisse}`

addins. See how you can easily draw graphs from the `{ggplot2}`

package without having to code it yourself.

All plots displayed in this article can be customized. For instance, it is possible to edit the title, x and y-axis labels, color, etc. However, customizing plots is beyond the scope of this article so all plots are presented without any customization. Interested readers will find numerous resources online.

# Minimum and maximum

Minimum and maximum can be found thanks to the `min()`

and `max()`

functions:

`min(dat$Sepal.Length)`

`## [1] 4.3`

`max(dat$Sepal.Length)`

`## [1] 7.9`

Alternatively the `range()`

function:

```
rng <- range(dat$Sepal.Length)
rng
```

`## [1] 4.3 7.9`

gives you the minimum and maximum directly. Note that the output of the `range()`

function is actually an object containing the minimum and maximum (in that order). This means you can actually access the minimum with:

`rng[1] # rng = name of the object specified above`

`## [1] 4.3`

and the maximum with:

`rng[2]`

`## [1] 7.9`

This reminds us that, in R, there are often several ways to arrive at the same result. The method that uses the shortest piece of code is usually preferred as a shorter piece of code is less prone to coding errors and more readable.

# Range

The range can then be easily computed, as you have guessed, by subtracting the minimum from the maximum:

`max(dat$Sepal.Length) - min(dat$Sepal.Length)`

`## [1] 3.6`

To my knowledge, there is no default function to compute the range. However, if you are familiar with writing functions in R , you can create your own function to compute the range:

```
range2 <- function(x) {
range <- max(x) - min(x)
return(range)
}
range2(dat$Sepal.Length)
```

`## [1] 3.6`

which is equivalent than \(max - min\) presented above.

# Mean

The mean can be computed with the `mean()`

function:

`mean(dat$Sepal.Length)`

`## [1] 5.843333`

*Tips:*

- if there is at least one missing value in your dataset, use
`mean(dat$Sepal.Length, na.rm = TRUE)`

to compute the mean with the NA excluded. This argument can be used for most functions presented in this article, not only the mean - for a truncated mean, use
`mean(dat$Sepal.Length, trim = 0.10)`

and change the`trim`

argument to your needs

# Median

The median can be computed thanks to the `median()`

function:

`median(dat$Sepal.Length)`

`## [1] 5.8`

or with the `quantile()`

function:

`quantile(dat$Sepal.Length, 0.5)`

```
## 50%
## 5.8
```

since the quantile of order 0.5 (\(q_{0.5}\)) corresponds to the median.

# First and third quartile

As the median, the first and third quartiles can be computed thanks to the `quantile()`

function and by setting the second argument to 0.25 or 0.75:

`quantile(dat$Sepal.Length, 0.25) # first quartile`

```
## 25%
## 5.1
```

`quantile(dat$Sepal.Length, 0.75) # third quartile`

```
## 75%
## 6.4
```

You may have seen that the results above are slightly different than the results you would have found if you compute the first and third quartiles by hand. It is normal, there are many methods to compute them (R actually has 7 methods to compute the quantiles!). However, the methods presented here and in the article “descriptive statistics by hand” are the easiest and most “standard” ones. Furthermore, results do not dramatically change between the two methods.

## Other quantiles

As you have guessed, any quantile can also be computed with the `quantile()`

function. For instance, the \(4^{th}\) decile or the \(98^{th}\) percentile:

`quantile(dat$Sepal.Length, 0.4) # 4th decile`

```
## 40%
## 5.6
```

`quantile(dat$Sepal.Length, 0.98) # 98th percentile`

```
## 98%
## 7.7
```

# Interquartile range

The interquartile range (i.e., the difference between the first and third quartile) can be computed with the `IQR()`

function:

`IQR(dat$Sepal.Length)`

`## [1] 1.3`

or alternatively with the `quantile()`

function again:

`quantile(dat$Sepal.Length, 0.75) - quantile(dat$Sepal.Length, 0.25)`

```
## 75%
## 1.3
```

As mentioned earlier, when possible it is usually recommended to use the shortest piece of code to arrive at the result. For this reason, the `IQR()`

function is preferred to compute the interquartile range.

# Standard deviation and variance

The standard deviation and the variance is computed with the `sd()`

and `var()`

functions:

`sd(dat$Sepal.Length) # standard deviation`

`## [1] 0.8280661`

`var(dat$Sepal.Length) # variance`

`## [1] 0.6856935`

Remember from the article descriptive statistics by hand that the standard deviation and the variance are different whether we compute it for a sample or a population (see the difference between sample and population). In R, the standard deviation and the variance are computed as if the data represent a sample (so the denominator is \(n - 1\), where \(n\) is the number of observations). To my knowledge, there is no function by default in R that computes the standard deviation or variance for a population.

*Tip:* to compute the standard deviation (or variance) of multiple variables at the same time, use `lapply()`

with the appropriate statistics as second argument:

`lapply(dat[, 1:4], sd)`

```
## $Sepal.Length
## [1] 0.8280661
##
## $Sepal.Width
## [1] 0.4358663
##
## $Petal.Length
## [1] 1.765298
##
## $Petal.Width
## [1] 0.7622377
```

The command `dat[, 1:4]`

selects the variables 1 to 4 as the fifth variable is a qualitative variable and the standard deviation cannot be computed on such type of variable. See a recap of the different data types in R if needed.

# Summary

You can compute the minimum, \(1^{st}\) quartile, median, mean, \(3^{rd}\) quartile and the maximum for all numeric variables of a dataset at once using `summary()`

:

`summary(dat)`

```
## Sepal.Length Sepal.Width Petal.Length Petal.Width
## Min. :4.300 Min. :2.000 Min. :1.000 Min. :0.100
## 1st Qu.:5.100 1st Qu.:2.800 1st Qu.:1.600 1st Qu.:0.300
## Median :5.800 Median :3.000 Median :4.350 Median :1.300
## Mean :5.843 Mean :3.057 Mean :3.758 Mean :1.199
## 3rd Qu.:6.400 3rd Qu.:3.300 3rd Qu.:5.100 3rd Qu.:1.800
## Max. :7.900 Max. :4.400 Max. :6.900 Max. :2.500
## Species
## setosa :50
## versicolor:50
## virginica :50
##
##
##
```

*Tip:* if you need these descriptive statistics by group use the `by()`

function:

`by(dat, dat$Species, summary)`

```
## dat$Species: setosa
## Sepal.Length Sepal.Width Petal.Length Petal.Width
## Min. :4.300 Min. :2.300 Min. :1.000 Min. :0.100
## 1st Qu.:4.800 1st Qu.:3.200 1st Qu.:1.400 1st Qu.:0.200
## Median :5.000 Median :3.400 Median :1.500 Median :0.200
## Mean :5.006 Mean :3.428 Mean :1.462 Mean :0.246
## 3rd Qu.:5.200 3rd Qu.:3.675 3rd Qu.:1.575 3rd Qu.:0.300
## Max. :5.800 Max. :4.400 Max. :1.900 Max. :0.600
## Species
## setosa :50
## versicolor: 0
## virginica : 0
##
##
##
## ------------------------------------------------------------
## dat$Species: versicolor
## Sepal.Length Sepal.Width Petal.Length Petal.Width Species
## Min. :4.900 Min. :2.000 Min. :3.00 Min. :1.000 setosa : 0
## 1st Qu.:5.600 1st Qu.:2.525 1st Qu.:4.00 1st Qu.:1.200 versicolor:50
## Median :5.900 Median :2.800 Median :4.35 Median :1.300 virginica : 0
## Mean :5.936 Mean :2.770 Mean :4.26 Mean :1.326
## 3rd Qu.:6.300 3rd Qu.:3.000 3rd Qu.:4.60 3rd Qu.:1.500
## Max. :7.000 Max. :3.400 Max. :5.10 Max. :1.800
## ------------------------------------------------------------
## dat$Species: virginica
## Sepal.Length Sepal.Width Petal.Length Petal.Width
## Min. :4.900 Min. :2.200 Min. :4.500 Min. :1.400
## 1st Qu.:6.225 1st Qu.:2.800 1st Qu.:5.100 1st Qu.:1.800
## Median :6.500 Median :3.000 Median :5.550 Median :2.000
## Mean :6.588 Mean :2.974 Mean :5.552 Mean :2.026
## 3rd Qu.:6.900 3rd Qu.:3.175 3rd Qu.:5.875 3rd Qu.:2.300
## Max. :7.900 Max. :3.800 Max. :6.900 Max. :2.500
## Species
## setosa : 0
## versicolor: 0
## virginica :50
##
##
##
```

where the arguments are the name of the dataset, the grouping variable and the summary function. Follow this order, or specify the name of the arguments if you do not follow this order.

If you need more descriptive statistics, use `stat.desc()`

from the package `{pastecs}`

:

```
library(pastecs)
stat.desc(dat)
```

```
## Sepal.Length Sepal.Width Petal.Length Petal.Width Species
## nbr.val 150.00000000 150.00000000 150.0000000 150.00000000 NA
## nbr.null 0.00000000 0.00000000 0.0000000 0.00000000 NA
## nbr.na 0.00000000 0.00000000 0.0000000 0.00000000 NA
## min 4.30000000 2.00000000 1.0000000 0.10000000 NA
## max 7.90000000 4.40000000 6.9000000 2.50000000 NA
## range 3.60000000 2.40000000 5.9000000 2.40000000 NA
## sum 876.50000000 458.60000000 563.7000000 179.90000000 NA
## median 5.80000000 3.00000000 4.3500000 1.30000000 NA
## mean 5.84333333 3.05733333 3.7580000 1.19933333 NA
## SE.mean 0.06761132 0.03558833 0.1441360 0.06223645 NA
## CI.mean.0.95 0.13360085 0.07032302 0.2848146 0.12298004 NA
## var 0.68569351 0.18997942 3.1162779 0.58100626 NA
## std.dev 0.82806613 0.43586628 1.7652982 0.76223767 NA
## coef.var 0.14171126 0.14256420 0.4697441 0.63555114 NA
```

You can have even more statistics (i.e., skewness, kurtosis and normality test) by adding the argument `norm = TRUE`

in the previous function. Note that the variable `Species`

is not numeric, so descriptive statistics cannot be computed for this variable and NA are displayed.

# Coefficient of variation

The coefficient of variation can be found with `stat.desc()`

(see the line `coef.var`

in the table above) or by computing manually (remember that the coefficient of variation is the standard deviation divided by the mean):

`sd(dat$Sepal.Length) / mean(dat$Sepal.Length)`

`## [1] 0.1417113`

# Mode

To my knowledge there is no function to find the mode of a variable. However, we can easily find it thanks to the functions `table()`

and `sort()`

:

```
tab <- table(dat$Sepal.Length) # number of occurrences for each unique value
sort(tab, decreasing = TRUE) # sort highest to lowest
```

```
##
## 5 5.1 6.3 5.7 6.7 5.5 5.8 6.4 4.9 5.4 5.6 6 6.1 4.8 6.5 4.6 5.2 6.2 6.9 7.7
## 10 9 9 8 8 7 7 7 6 6 6 6 6 5 5 4 4 4 4 4
## 4.4 5.9 6.8 7.2 4.7 6.6 4.3 4.5 5.3 7 7.1 7.3 7.4 7.6 7.9
## 3 3 3 3 2 2 1 1 1 1 1 1 1 1 1
```

`table()`

gives the number of occurrences for each unique value, then `sort()`

with the argument `decreasing = TRUE`

displays the number of occurrences from highest to lowest. The mode of the variable `Sepal.Length`

is thus 5. This code to find the mode can also be applied to qualitative variables such as `Species`

:

`sort(table(dat$Species), decreasing = TRUE)`

```
##
## setosa versicolor virginica
## 50 50 50
```

or:

`summary(dat$Species)`

```
## setosa versicolor virginica
## 50 50 50
```

# Correlation

Another descriptive statistics is the correlation coefficient.

The correlation measures the *linear* relationship between two variables, and it can be computed with the `cor()`

function:

`cor(dat$Sepal.Length, dat$Sepal.Width)`

`## [1] -0.1175698`

Computing correlation in R and interpreting the results deserve a detailed explanation, so I wrote an article covering correlation and correlation test.

# Contingency table

`table()`

introduced above can also be used on two qualitative variables to create a contingency table. The dataset `iris`

has only one qualitative variable so we create a new qualitative variable just for this example. We create the variable `size`

which corresponds to `small`

if the length of the petal is smaller than the median of all flowers, `big`

otherwise:

```
dat$size <- ifelse(dat$Sepal.Length < median(dat$Sepal.Length),
"small", "big"
)
```

Here is a recap of the occurrences by size:

`table(dat$size)`

```
##
## big small
## 77 73
```

We now create a contingency table of the two variables `Species`

and `size`

with the `table()`

function:

`table(dat$Species, dat$size)`

```
##
## big small
## setosa 1 49
## versicolor 29 21
## virginica 47 3
```

or with the `xtabs()`

function:

`xtabs(~ dat$Species + dat$size)`

```
## dat$size
## dat$Species big small
## setosa 1 49
## versicolor 29 21
## virginica 47 3
```

The contingency table gives the number of cases in each subgroup. For instance, there is only one big setosa flower, while there are 49 small setosa flowers in the dataset.

To go further, we can see from the table that setosa flowers seem to be larger in size than virginica flowers. In order to check whether size is significantly associated with species, we could perform a Chi-square test of independence since both variables are categorical variables. See how to do this test by hand and in R.

Note that `Species`

are in rows and `size`

in column because we specified `Species`

and then `size`

in `table()`

. Change the order if you want to switch the two variables.

Instead of having the frequencies (i.e.. the number of cases) you can also have the relative frequencies (i.e., proportions) in each subgroup by adding the `table()`

function inside the `prop.table()`

function:

`prop.table(table(dat$Species, dat$size))`

```
##
## big small
## setosa 0.006666667 0.326666667
## versicolor 0.193333333 0.140000000
## virginica 0.313333333 0.020000000
```

Note that you can also compute the percentages by row or by column by adding a second argument to the `prop.table()`

function: `1`

for row, or `2`

for column:

```
# percentages by row:
round(prop.table(table(dat$Species, dat$size), 1), 2) # round to 2 digits with round()
```

```
##
## big small
## setosa 0.02 0.98
## versicolor 0.58 0.42
## virginica 0.94 0.06
```

```
# percentages by column:
round(prop.table(table(dat$Species, dat$size), 2), 2) # round to 2 digits with round()
```

```
##
## big small
## setosa 0.01 0.67
## versicolor 0.38 0.29
## virginica 0.61 0.04
```

See the section on advanced descriptive statistics for more advanced contingency tables.

## Mosaic plot

A mosaic plot allows to visualize a contingency table of two qualitative variables:

```
mosaicplot(table(dat$Species, dat$size),
color = TRUE,
xlab = "Species", # label for x-axis
ylab = "Size" # label for y-axis
)
```

The mosaic plot shows that, for our sample, the proportion of big and small flowers is clearly different between the three species. In particular, the virginica species is the biggest, and the setosa species is the smallest of the three species (in terms of sepal length since the variable `size`

is based on the variable `Sepal.Length`

).

For your information, a mosaic plot can also be done via the `mosaic()`

function from the `{vcd}`

package:

```
library(vcd)
mosaic(~ Species + size,
data = dat,
direction = c("v", "h")
)
```

# Barplot

Barplots can only be done on qualitative variables (see the difference with a quantitative variable here). A barplot is a tool to visualize the distribution of a qualitative variable. We draw a barplot of the qualitative variable `size`

:

`barplot(table(dat$size)) # table() is mandatory`

You can also draw a barplot of the relative frequencies instead of the frequencies by adding `prop.table()`

as we did earlier:

`barplot(prop.table(table(dat$size)))`

In `{ggplot2}`

:

```
library(ggplot2) # needed each time you open RStudio
# The package ggplot2 must be installed first
ggplot(dat) +
aes(x = size) +
geom_bar()
```

# Histogram

A histogram gives an idea about the distribution of a quantitative variable. The idea is to break the range of values into intervals and count how many observations fall into each interval. Histograms are a bit similar to barplots, but histograms are used for quantitative variables whereas barplots are used for qualitative variables. To draw a histogram in R, use `hist()`

:

`hist(dat$Sepal.Length)`

Add the arguments `breaks =`

inside the `hist()`

function if you want to change the number of bins. A rule of thumb (known as Sturges’ law) is that the number of bins should be the rounded value of the square root of the number of observations. The dataset includes 150 observations so in this case the number of bins can be set to 12.

In `{ggplot2}`

:

```
ggplot(dat) +
aes(x = Sepal.Length) +
geom_histogram()
```

By default, the number of bins is 30. You can change this value with `geom_histogram(bins = 12)`

for instance.

# Boxplot

Boxplots are really useful in descriptive statistics and are often underused (mostly because it is not well understood by the public). A boxplot graphically represents the distribution of a quantitative variable by visually displaying five common location summary (minimum, median, first/third quartiles and maximum) and any observation that was classified as a suspected outlier using the interquartile range (IQR) criterion.

The IQR criterion means that all observations above \(q_{0.75} + 1.5 \cdot IQR\) or below \(q_{0.25} - 1.5 \cdot IQR\) (where \(q_{0.25}\) and \(q_{0.75}\) correspond to first and third quartile respectively) are considered as potential outliers by R. The minimum and maximum in the boxplot are represented without these suspected outliers.

Seeing all these information on the same plot help to have a good first overview of the dispersion and the location of the data. Before drawing a boxplot of our data, see below a graph explaining the information present on a boxplot:

Now an example with our dataset:

`boxplot(dat$Sepal.Length)`

Boxplots are even more informative when presented side-by-side for comparing and contrasting distributions from two or more groups. For instance, we compare the length of the sepal across the different species:

`boxplot(dat$Sepal.Length ~ dat$Species)`

In `{ggplot2}`

:

```
ggplot(dat) +
aes(x = Species, y = Sepal.Length) +
geom_boxplot()
```

# Dotplot

A dotplot is more or less similar than a boxplot, except that:

- observations are represented as points
- it does not easily tell us about the median, first and third quartiles.

```
library(lattice)
dotplot(dat$Sepal.Length ~ dat$Species)
```

In `{ggplot2}`

:

```
ggplot(dat) +
aes(x = Species, y = Sepal.Length) +
geom_dotplot(binaxis = "y", stackdir = "center")
```

The advantage of using `{ggplot2}`

over `{lattice}`

for this plot is that we can easily see the mode.

Note that a dotplot is particularly useful when there are a limited number of observations, whereas a boxplot is more appropriate with large datasets.

# Scatterplot

Scatterplots allow to check whether there is a potential link between two quantitative variables. For this reason, scatterplots are often used to visualize a potential correlation between two variables. For instance, when drawing a scatterplot of the length of the sepal and the length of the petal:

`plot(dat$Sepal.Length, dat$Petal.Length)`

There seems to be a positive association between the two variables.

In `{ggplot2}`

:

```
ggplot(dat) +
aes(x = Sepal.Length, y = Petal.Length) +
geom_point()
```

Like boxplots, scatterplots are even more informative when differentiating the points according to a factor, in this case the species:

```
ggplot(dat) +
aes(x = Sepal.Length, y = Petal.Length, colour = Species) +
geom_point() +
scale_color_hue()
```

# Line plot

Line plots, particularly useful in time series or finance, can be created by adding the `type = "l"`

argument in the `plot()`

function:

```
plot(dat$Sepal.Length,
type = "l"
) # "l" for line
```

# QQ-plot

## For a single variable

In order to check the normality assumption of a variable (normality means that the data follow a normal distribution, also known as a Gaussian distribution), we usually use histograms and/or QQ-plots.^{1} See an article discussing about the normal distribution and how to evaluate the normality assumption in R if you need a refresh on that subject.

Histograms have been presented earlier, so here is how to draw a QQ-plot:

```
# Draw points on the qq-plot:
qqnorm(dat$Sepal.Length)
# Draw the reference line:
qqline(dat$Sepal.Length)
```

Or a QQ-plot with confidence bands with the `qqPlot()`

function from the `{car}`

package:

```
library(car) # package must be installed first
qqPlot(dat$Sepal.Length)
```

`## [1] 132 118`

If points are close to the reference line (sometimes referred as Henry’s line) and within the confidence bands, the normality assumption can be considered as met. The bigger the deviation between the points and the reference line and the more they lie outside the confidence bands, the less likely that the normality condition is met. The variable `Sepal.Length`

does not seem to follow a normal distribution because several points lie outside the confidence bands. When facing a non-normal distribution, the first step is usually to apply the logarithm transformation on the data and recheck to see whether the log-transformed data are normally distributed. Applying the logarithm transformation can be done with the `log()`

function.

In `{ggpubr}`

:

```
library(ggpubr)
ggqqplot(dat$Sepal.Length)
```

## By groups

For some statistical tests, the normality assumption is required in all groups. One solution is to draw a QQ-plot for each group by manually splitting the dataset into different groups and then draw a QQ-plot for each subset of the data (with the methods shown above). Another (easier) solution is to draw a QQ-plot for each group automatically with the argument `groups =`

in the function `qqPlot()`

from the `{car}`

package:

`qqPlot(dat$Sepal.Length, groups = dat$size)`

In `{ggplot2}`

:

```
qplot(
sample = Sepal.Length, data = dat,
col = size, shape = size
)
```

It is also possible to differentiate groups by only shape or color. For this, remove one of the argument `col`

or `shape`

in the `qplot()`

function above.

# Density plot

Density plot is a smoothed version of the histogram and is used in the same concept, that is, to represent the distribution of a numeric variable. The functions `plot()`

and `density()`

are used together to draw a density plot:

`plot(density(dat$Sepal.Length))`

In `{ggplot2}`

:

```
ggplot(dat) +
aes(x = Sepal.Length) +
geom_density()
```

# Correlation plot

The last type of descriptive plot is a correlation plot, also called a correlogram. This type of graph is more complex than the ones presented above, so it is detailed in a separate article. See how to draw a correlogram to highlight the most correlated variables in a dataset.

# Advanced descriptive statistics

We covered the main functions to compute the most common and basic descriptive statistics. There are, however, many more functions and packages to perform more advanced descriptive statistics in R. In this section, I present some of them with applications to our dataset.

`{summarytools}`

package

One package for descriptive statistics I often use for my projects in R is the `{summarytools}`

package. The package is centered around 4 functions:

`freq()`

for frequencies tables`ctable()`

for cross-tabulations`descr()`

for descriptive statistics`dfSummary()`

for dataframe summaries

A combination of these 4 functions is usually more than enough for most descriptive analyses. Moreover, the package has been built with R Markdown in mind, meaning that outputs render well in HTML reports. And for non-English speakers, built-in translations exist for French, Portuguese, Spanish, Russian and Turkish.

I illustrate each of the 4 functions in the following sections. Outputs that follow display much better in R Markdown reports, but in this article I limit myself to the raw outputs as the goal is to show how the functions work, not how to make them render well. See the setup settings in the vignette of the package if you want to print the outputs in a nice way in R Markdown.^{2}

### Frequency tables with `freq()`

The `freq()`

function produces frequency tables with frequencies, proportions, as well as missing data information.

```
library(summarytools)
freq(dat$Species)
```

```
## Frequencies
## dat$Species
## Type: Factor
##
## Freq % Valid % Valid Cum. % Total % Total Cum.
## ---------------- ------ --------- -------------- --------- --------------
## setosa 50 33.33 33.33 33.33 33.33
## versicolor 50 33.33 66.67 33.33 66.67
## virginica 50 33.33 100.00 33.33 100.00
## <NA> 0 0.00 100.00
## Total 150 100.00 100.00 100.00 100.00
```

If you do not need information about missing values, add the `report.nas = FALSE`

argument:

```
freq(dat$Species,
report.nas = FALSE # remove NA information
)
```

```
## Frequencies
## dat$Species
## Type: Factor
##
## Freq % % Cum.
## ---------------- ------ -------- --------
## setosa 50 33.33 33.33
## versicolor 50 33.33 66.67
## virginica 50 33.33 100.00
## Total 150 100.00 100.00
```

And for a minimalist output with only counts and proportions:

```
freq(dat$Species,
report.nas = FALSE, # remove NA information
totals = FALSE, # remove totals
cumul = FALSE, # remove cumuls
headings = FALSE # remove headings
)
```

```
##
## Freq %
## ---------------- ------ -------
## setosa 50 33.33
## versicolor 50 33.33
## virginica 50 33.33
```

### Cross-tabulations with `ctable()`

The `ctable()`

function produces cross-tabulations (also known as contingency tables) for pairs of categorical variables. Using the two categorical variables in our dataset:

```
ctable(
x = dat$Species,
y = dat$size
)
```

```
## Cross-Tabulation, Row Proportions
## Species * size
## Data Frame: dat
##
## ------------ ------ ------------ ------------ --------------
## size big small Total
## Species
## setosa 1 ( 2.0%) 49 (98.0%) 50 (100.0%)
## versicolor 29 (58.0%) 21 (42.0%) 50 (100.0%)
## virginica 47 (94.0%) 3 ( 6.0%) 50 (100.0%)
## Total 77 (51.3%) 73 (48.7%) 150 (100.0%)
## ------------ ------ ------------ ------------ --------------
```

Row proportions are shown by default. To display column or total proportions, add the `prop = "c"`

or `prop = "t"`

arguments, respectively:

```
ctable(
x = dat$Species,
y = dat$size,
prop = "t" # total proportions
)
```

```
## Cross-Tabulation, Total Proportions
## Species * size
## Data Frame: dat
##
## ------------ ------ ------------ ------------ --------------
## size big small Total
## Species
## setosa 1 ( 0.7%) 49 (32.7%) 50 ( 33.3%)
## versicolor 29 (19.3%) 21 (14.0%) 50 ( 33.3%)
## virginica 47 (31.3%) 3 ( 2.0%) 50 ( 33.3%)
## Total 77 (51.3%) 73 (48.7%) 150 (100.0%)
## ------------ ------ ------------ ------------ --------------
```

To remove proportions altogether, add the argument `prop = "n"`

. Furthermore, to display only the bare minimum, add the `totals = FALSE`

and `headings = FALSE`

arguments:

```
ctable(
x = dat$Species,
y = dat$size,
prop = "n", # remove proportions
totals = FALSE, # remove totals
headings = FALSE # remove headings
)
```

```
##
## ------------ ------ ----- -------
## size big small
## Species
## setosa 1 49
## versicolor 29 21
## virginica 47 3
## ------------ ------ ----- -------
```

This is equivalent than `table(dat$Species, dat$size)`

and `xtabs(~ dat$Species + dat$size)`

performed in the section on contingency tables.

To display results of the Chi-square test of independence, add the `chisq = TRUE`

argument:^{3}

```
ctable(
x = dat$Species,
y = dat$size,
chisq = TRUE, # display results of Chi-square test of independence
headings = FALSE # remove headings
)
```

```
##
## ------------ ------ ------------ ------------ --------------
## size big small Total
## Species
## setosa 1 ( 2.0%) 49 (98.0%) 50 (100.0%)
## versicolor 29 (58.0%) 21 (42.0%) 50 (100.0%)
## virginica 47 (94.0%) 3 ( 6.0%) 50 (100.0%)
## Total 77 (51.3%) 73 (48.7%) 150 (100.0%)
## ------------ ------ ------------ ------------ --------------
##
## ----------------------------
## Chi.squared df p.value
## ------------- ---- ---------
## 86.0345 2 0
## ----------------------------
```

The *p*-value is close to 0 so we reject the null hypothesis of independence between the two variables. In our context, this indicates that species and size are dependent and that there is a significant relationship between the two variables.

It is also possible to create a contingency table for each level of a third categorical variable thanks to the combination of the `stby()`

and `ctable()`

functions. There are only 2 categorical variables in our dataset, so let’s use the `tabacco`

dataset which has 4 categorical variables (i.e., gender, age group, smoker, diseased). For this example, we would like to create a contingency table of the variables `smoker`

and `diseased`

, and this for each `gender`

:

```
stby(
list(
x = tobacco$smoker, # smoker and diseased
y = tobacco$diseased
),
INDICES = tobacco$gender, # for each gender
FUN = ctable # ctable for cross-tabulation
)
```

```
## Cross-Tabulation, Row Proportions
## smoker * diseased
## Data Frame: tobacco
## Group: gender = F
##
## -------- ---------- ------------- ------------- --------------
## diseased Yes No Total
## smoker
## Yes 62 (42.2%) 85 (57.8%) 147 (100.0%)
## No 49 (14.3%) 293 (85.7%) 342 (100.0%)
## Total 111 (22.7%) 378 (77.3%) 489 (100.0%)
## -------- ---------- ------------- ------------- --------------
##
## Group: gender = M
##
## -------- ---------- ------------- ------------- --------------
## diseased Yes No Total
## smoker
## Yes 63 (44.1%) 80 (55.9%) 143 (100.0%)
## No 47 (13.6%) 299 (86.4%) 346 (100.0%)
## Total 110 (22.5%) 379 (77.5%) 489 (100.0%)
## -------- ---------- ------------- ------------- --------------
```

### Descriptive statistics with `descr()`

The `descr()`

function produces descriptive (univariate) statistics with common central tendency statistics and measures of dispersion. (See the difference between a measure of central tendency and dispersion if you need a reminder.)

A major advantage of this function is that it accepts single vectors as well as data frames. If a data frame is provided, all non-numerical columns are ignored so you do not have to remove them yourself before running the function.

The `descr()`

function allows to display:

- only a selection of descriptive statistics of your choice, with the
`stats = c("mean", "sd")`

argument for mean and standard deviation for example - the minimum, first quartile, median, third quartile and maximum with
`stats = "fivenum"`

- the most common descriptive statistics (mean, standard deviation, minimum, median, maximum, number and percentage of valid observations), with
`stats = "common"`

:

```
descr(dat,
headings = FALSE, # remove headings
stats = "common" # most common descriptive statistics
)
```

```
##
## Petal.Length Petal.Width Sepal.Length Sepal.Width
## --------------- -------------- ------------- -------------- -------------
## Mean 3.76 1.20 5.84 3.06
## Std.Dev 1.77 0.76 0.83 0.44
## Min 1.00 0.10 4.30 2.00
## Median 4.35 1.30 5.80 3.00
## Max 6.90 2.50 7.90 4.40
## N.Valid 150.00 150.00 150.00 150.00
## Pct.Valid 100.00 100.00 100.00 100.00
```

*Tip:* if you have a large number of variables, add the `transpose = TRUE`

argument for a better display.

In order to compute these descriptive statistics by group (e.g., `Species`

in our dataset), use the `descr()`

function in combination with the `stby()`

function:

```
stby(
data = dat,
INDICES = dat$Species, # by Species
FUN = descr, # descriptive statistics
stats = "common" # most common descr. stats
)
```

```
## Descriptive Statistics
## dat
## Group: Species = setosa
## N: 50
##
## Petal.Length Petal.Width Sepal.Length Sepal.Width
## --------------- -------------- ------------- -------------- -------------
## Mean 1.46 0.25 5.01 3.43
## Std.Dev 0.17 0.11 0.35 0.38
## Min 1.00 0.10 4.30 2.30
## Median 1.50 0.20 5.00 3.40
## Max 1.90 0.60 5.80 4.40
## N.Valid 50.00 50.00 50.00 50.00
## Pct.Valid 100.00 100.00 100.00 100.00
##
## Group: Species = versicolor
## N: 50
##
## Petal.Length Petal.Width Sepal.Length Sepal.Width
## --------------- -------------- ------------- -------------- -------------
## Mean 4.26 1.33 5.94 2.77
## Std.Dev 0.47 0.20 0.52 0.31
## Min 3.00 1.00 4.90 2.00
## Median 4.35 1.30 5.90 2.80
## Max 5.10 1.80 7.00 3.40
## N.Valid 50.00 50.00 50.00 50.00
## Pct.Valid 100.00 100.00 100.00 100.00
##
## Group: Species = virginica
## N: 50
##
## Petal.Length Petal.Width Sepal.Length Sepal.Width
## --------------- -------------- ------------- -------------- -------------
## Mean 5.55 2.03 6.59 2.97
## Std.Dev 0.55 0.27 0.64 0.32
## Min 4.50 1.40 4.90 2.20
## Median 5.55 2.00 6.50 3.00
## Max 6.90 2.50 7.90 3.80
## N.Valid 50.00 50.00 50.00 50.00
## Pct.Valid 100.00 100.00 100.00 100.00
```

### Data frame summaries with `dfSummary()`

The `dfSummary()`

function generates a summary table with statistics, frequencies and graphs for all variables in a dataset. The information shown depends on the type of the variables (character, factor, numeric, date) and also varies according to the number of distinct values.

`dfSummary(dat)`

```
## Data Frame Summary
## dat
## Dimensions: 150 x 6
## Duplicates: 1
##
## -----------------------------------------------------------------------------------------------------------
## No Variable Stats / Values Freqs (% of Valid) Graph Valid Missing
## ---- -------------- ----------------------- -------------------- --------------------- ---------- ---------
## 1 Sepal.Length Mean (sd) : 5.8 (0.8) 35 distinct values . . : : 150 0
## [numeric] min < med < max: : : : : (100.0%) (0.0%)
## 4.3 < 5.8 < 7.9 : : : : :
## IQR (CV) : 1.3 (0.1) : : : : :
## : : : : : : : :
##
## 2 Sepal.Width Mean (sd) : 3.1 (0.4) 23 distinct values : 150 0
## [numeric] min < med < max: : (100.0%) (0.0%)
## 2 < 3 < 4.4 . :
## IQR (CV) : 0.5 (0.1) : : : :
## . . : : : : : :
##
## 3 Petal.Length Mean (sd) : 3.8 (1.8) 43 distinct values : 150 0
## [numeric] min < med < max: : . : (100.0%) (0.0%)
## 1 < 4.3 < 6.9 : : : .
## IQR (CV) : 3.5 (0.5) : : : : : .
## : : . : : : : : .
##
## 4 Petal.Width Mean (sd) : 1.2 (0.8) 22 distinct values : 150 0
## [numeric] min < med < max: : (100.0%) (0.0%)
## 0.1 < 1.3 < 2.5 : . . :
## IQR (CV) : 1.5 (0.6) : : : : .
## : : : : : . : : :
##
## 5 Species 1. setosa 50 (33.3%) IIIIII 150 0
## [factor] 2. versicolor 50 (33.3%) IIIIII (100.0%) (0.0%)
## 3. virginica 50 (33.3%) IIIIII
##
## 6 size 1. big 77 (51.3%) IIIIIIIIII 150 0
## [character] 2. small 73 (48.7%) IIIIIIIII (100.0%) (0.0%)
## -----------------------------------------------------------------------------------------------------------
```

`describeBy()`

from the `{psych}`

package

The `describeBy()`

function from the `{psych}`

package allows to report several summary statistics (i.e., number of valid cases, mean, standard deviation, median, trimmed mean, mad: median absolute deviation (from the median), minimum, maximum, range, skewness and kurtosis) by a grouping variable.

```
library(psych)
describeBy(
dat,
dat$Species # grouping variable
)
```

```
##
## Descriptive statistics by group
## group: setosa
## vars n mean sd median trimmed mad min max range skew kurtosis
## Sepal.Length 1 50 5.01 0.35 5.0 5.00 0.30 4.3 5.8 1.5 0.11 -0.45
## Sepal.Width 2 50 3.43 0.38 3.4 3.42 0.37 2.3 4.4 2.1 0.04 0.60
## Petal.Length 3 50 1.46 0.17 1.5 1.46 0.15 1.0 1.9 0.9 0.10 0.65
## Petal.Width 4 50 0.25 0.11 0.2 0.24 0.00 0.1 0.6 0.5 1.18 1.26
## Species* 5 50 1.00 0.00 1.0 1.00 0.00 1.0 1.0 0.0 NaN NaN
## size* 6 50 1.98 0.14 2.0 2.00 0.00 1.0 2.0 1.0 -6.65 43.12
## se
## Sepal.Length 0.05
## Sepal.Width 0.05
## Petal.Length 0.02
## Petal.Width 0.01
## Species* 0.00
## size* 0.02
## ------------------------------------------------------------
## group: versicolor
## vars n mean sd median trimmed mad min max range skew kurtosis
## Sepal.Length 1 50 5.94 0.52 5.90 5.94 0.52 4.9 7.0 2.1 0.10 -0.69
## Sepal.Width 2 50 2.77 0.31 2.80 2.78 0.30 2.0 3.4 1.4 -0.34 -0.55
## Petal.Length 3 50 4.26 0.47 4.35 4.29 0.52 3.0 5.1 2.1 -0.57 -0.19
## Petal.Width 4 50 1.33 0.20 1.30 1.32 0.22 1.0 1.8 0.8 -0.03 -0.59
## Species* 5 50 2.00 0.00 2.00 2.00 0.00 2.0 2.0 0.0 NaN NaN
## size* 6 50 1.42 0.50 1.00 1.40 0.00 1.0 2.0 1.0 0.31 -1.94
## se
## Sepal.Length 0.07
## Sepal.Width 0.04
## Petal.Length 0.07
## Petal.Width 0.03
## Species* 0.00
## size* 0.07
## ------------------------------------------------------------
## group: virginica
## vars n mean sd median trimmed mad min max range skew kurtosis
## Sepal.Length 1 50 6.59 0.64 6.50 6.57 0.59 4.9 7.9 3.0 0.11 -0.20
## Sepal.Width 2 50 2.97 0.32 3.00 2.96 0.30 2.2 3.8 1.6 0.34 0.38
## Petal.Length 3 50 5.55 0.55 5.55 5.51 0.67 4.5 6.9 2.4 0.52 -0.37
## Petal.Width 4 50 2.03 0.27 2.00 2.03 0.30 1.4 2.5 1.1 -0.12 -0.75
## Species* 5 50 3.00 0.00 3.00 3.00 0.00 3.0 3.0 0.0 NaN NaN
## size* 6 50 1.06 0.24 1.00 1.00 0.00 1.0 2.0 1.0 3.59 11.15
## se
## Sepal.Length 0.09
## Sepal.Width 0.05
## Petal.Length 0.08
## Petal.Width 0.04
## Species* 0.00
## size* 0.03
```

`aggregate()`

function

The `aggregate()`

function allows to split the data into subsets and then to compute summary statistics for each. For instance, if we want to compute the mean for the variables `Sepal.Length`

and `Sepal.Width`

by `Species`

and `Size`

:

```
aggregate(cbind(Sepal.Length, Sepal.Width) ~ Species + size,
data = dat,
mean
)
```

```
## Species size Sepal.Length Sepal.Width
## 1 setosa big 5.800000 4.000000
## 2 versicolor big 6.282759 2.868966
## 3 virginica big 6.663830 2.997872
## 4 setosa small 4.989796 3.416327
## 5 versicolor small 5.457143 2.633333
## 6 virginica small 5.400000 2.600000
```

`summaryBy()`

from `{doBy}`

An alternative is the `summaryBy()`

function from the `{doBy}`

package:

```
# summary statistics by group
library(doBy)
summaryBy(Sepal.Length + Sepal.Width ~ Species,
data = dat,
FUN = summary
)
```

```
## Species Sepal.Length.Min. Sepal.Length.1st Qu. Sepal.Length.Median
## 1 setosa 4.3 4.800 5.0
## 2 versicolor 4.9 5.600 5.9
## 3 virginica 4.9 6.225 6.5
## Sepal.Length.Mean Sepal.Length.3rd Qu. Sepal.Length.Max. Sepal.Width.Min.
## 1 5.006 5.2 5.8 2.3
## 2 5.936 6.3 7.0 2.0
## 3 6.588 6.9 7.9 2.2
## Sepal.Width.1st Qu. Sepal.Width.Median Sepal.Width.Mean Sepal.Width.3rd Qu.
## 1 3.200 3.4 3.428 3.675
## 2 2.525 2.8 2.770 3.000
## 3 2.800 3.0 2.974 3.175
## Sepal.Width.Max.
## 1 4.4
## 2 3.4
## 3 3.8
```

If you are interested in some specific descriptive statistics, you can easily specify them via the `FUN`

argument:

```
summaryBy(Sepal.Length + Sepal.Width ~ Species,
data = dat,
FUN = c(mean, var)
)
```

```
## Species Sepal.Length.mean Sepal.Width.mean Sepal.Length.var
## 1 setosa 5.006 3.428 0.1242490
## 2 versicolor 5.936 2.770 0.2664327
## 3 virginica 6.588 2.974 0.4043429
## Sepal.Width.var
## 1 0.14368980
## 2 0.09846939
## 3 0.10400408
```

`group_by()`

and `summarise()`

from `{dplyr}`

Another alternative is with the `summarise()`

and `group_by()`

functions from the `{dplyr}`

package:

```
library(dplyr)
group_by(dat, Species) %>%
summarise(
mean = mean(Sepal.Length, na.rm = TRUE),
sd = sd(Sepal.Length, na.rm = TRUE)
)
```

```
## # A tibble: 3 × 3
## Species mean sd
## <fct> <dbl> <dbl>
## 1 setosa 5.01 0.352
## 2 versicolor 5.94 0.516
## 3 virginica 6.59 0.636
```

# Conclusion

Thanks for reading.

I hope this article helped you to do descriptive statistics in R. If you would like to do the same by hand or understand what these statistics represent, I invite you to read the article “Descriptive statistics by hand”.

As always, if you have a question or a suggestion related to the topic covered in this article, please add it as a comment so other readers can benefit from the discussion.

*(Note that this article is available for download on my Gumroad page.)*

Normality tests such as Shapiro-Wilk or Kolmogorov-Smirnov tests can also be used to test whether the data follow a normal distribution or not. However, in practice, normality tests are often considered as too conservative in the sense that for large sample size, a small deviation from the normality may cause the normality condition to be violated. For this reason, it is often the case that the normality condition is verified based on a combination of visual inspections (with histograms and QQ-plots) and formal test (Shapiro-Wilk test for instance).↩︎

Note that the

`plain.ascii`

and`style`

arguments are needed for this package. In our examples, these arguments are added in the settings of each chunk so they are not visible.↩︎Note that it is also possible to compute odds ratio and risk ratio. See the vignette of the package for more information on this matter as these ratios are beyond the scope of this article.↩︎

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