Descriptive Statistics in R – An Introduction

Before diving into Descriptive Statistics in R we will first look at the different sources and types of data and focus on data measurement scales. To summarise data we will study various measures of central tendency and measures of variation.

You can download the data files for this tutorial here.

There are two main sources of data collection namely Primary and Secondary source.

Primary data sources include information collected and processed directly by the researcher, such as data collected through  surveys and interviews. Secondary data sources include information retrieved through preexisting sources such as Census data being used to study the impact of education on income.

Types of Data

In general there are two types of data,  structured data and unstructured data.

Structured data is stored in a standardized format for providing information. It is usually stored in well-defined schemas such as databases. It is generally tabular with columns and rows that clearly define its attributes.

Unstructured data is not organised in a pre-defined manner. For example Emails, tweets, blogs and so on.

Types of Data - Unstructured data vs structured data

Measurement Scales

Measurement scales are used to measure variables in statistics. There are four types of measurement scales – nominal, ordinal, interval and ratio scale.

Nominal scale: It is applied to qualitative data where the objects or items are classified into various distinct groups or categories depending on the type of the characteristic under study. Even if they are coded numerically, the order of values has no meaning. Examples are Location,Gender.

Ordinal Scale: It is applied to kind of data which are rank ordered. The different types characteristic have a logical or ordered relationship. These ranks only indicate   as to which category is better. For example Ranking the features of a product on a scale of 1 to 5. Here order of values is meaniniful.

Measurement scale

An interval scale provides more a powerful measure than an ordinal scale. It allows us not only to rank order items that are measured, but also to measure and find the difference between them. For example, temperature measured in degrees Celsius.

In addition to all the properties of an interval scale , a ratio scale features an identifiable true zero point.  Examples of ratio scales are physical dimensions such as weight, height,  distance and so on.

Types of measurement scale

Let’s look at an example to make the concept clearer.

We can observe that gender and region are measured on a nominal scale, age is measured on a ratio scale as age has a true zero point, and satisfaction level is measured on an ordinal scale.

Measurement scale example

Measures of Central Tendency

Central tendency is a descriptive summary of a dataset through a single value that reflects the center of the data distribution. The three most widely used measures of central tendency are mean, median and mode.

The mean is defined as the sum of all values of the variable divided by the total number of values. The median is the middle value. If N is odd and if N is even, it is the average of the two middle values. The mode is the most frequently occurring observation in a data set.

measures of central trendancy

Calculating Mean, Median and Mode

The formulae for calculating mean, median and mode are simple, but let’s quickly revise them.

In our example, the mean of the marks of 12 students is obtained by adding all the marks and dividing it by 12. Here, the mean is 14.83

To find the median,  we first arrange the data in ascending order.

Since the number of observations equals 12and therefore an even number,  the median is average of the two middle values, 16 and 17.  Therefore the median equals 16.5

17 is the most frequently occurring value. Therefore the mode equals 17

Calculating mean example
calculating median example
mean median mode

The trimmed mean

 A trimmed mean is a method of averaging that removes a small specified percentage of the largest and smallest values before calculating the mean.  The use of a trimmed mean helps eliminate the influence of outliers. . Typically, 5% of data points at each end are excluded.  Note that the trimmed mean will give an accurate estimate if the underlying distribution is symmetric.

trimmed mean

The best measure of central tendency

When data is measured on a nominal scale one can only calculate the mode. For data measured on an ordinal scale, the best measure of central tendency is the median. The mean is  appropriate when the distribution is symmetric and the measurement scale is an interval or ratio.

For a skewed (that is, not symmetric) distribution, the mean is generally not at the center and the median is a better measure of central tendency.

best measure of central trendancy

Measures of Variation

While measures of central tendency are used to estimate the central value of a dataset, measures of dispersion are important for describing the spread of data. 

Two data sets can have an equal mean (that is, measure of central tendency) but vastly different variability.  Take our example of two cricketers, where both batsmen have the same average score, but the spread around the mean is different.

The most commonly used measures of variation are range , interquartile range and standard deviation

Range, Interquartile Range and Standard Deviation

The range is defined as the difference between the highest and lowest values in a dataset. The disadvantage of defining range as a measure of dispersion is that it does not take into account all values for calculation.

The interquartile range is defined as the difference between the third quartile denoted by 𝑸_𝟑   and the lower quartile denoted by  𝑸_𝟏 . 75% of observations lie below the third quartile and 25% of observations lie below the first quartile.

Variance is defined as the sum of squares of deviations from the mean, divided by the total number of observations. The standard deviation is the positive square root of the variance.  The standard deviation is preferred instead of variance as it has the same units as the original values.

range interquartile range and standard diviation

Calculating range, interquartile range and Standard Deviation

In our example, the data of 12 student marks in an examination, the range is the difference between the highest observation and lowest observation. The highest observation is 20 and lowest is 8. Therefore range is 12

To obtain the interquartile range, we first arrange data in ascending order.  The third quartile is  3(n/4)th value, as shown,  that is the 9th Value= 18.  The first quartile is n/4th value. That is the3rd Value= 11. Therefore the interquartile range = 7.

Variance is obtained by adding the squared deviations from the mean and dividing them by n.  For our example, n is 12.

Therefore the variance equals 15.47 and the standard deviation is the positive root of 15.97, which equals 3.93

range formular
Inter quartile range calculation
Standard deviation calculation

The Coefficient of Variation

If we want to compare the variation in two sets of data, then the coefficient of variation should be used and not variance or standard deviation.

The coefficient of variation is a relative measure of variation, whereas standard deviation is an absolute measure of variation.

The coefficient of variation is computed as standard deviation divided by the mean and then expressed as a percentage.

A higher value of coefficient of variation implies more variation in our data.

coefficient of variations

Case Study 1

Let’s consider a simple example in which runs scored by two batsmen are recorded. Our objective is to compare the performance of two batsmen using the measures of central tendency and the measure of variation.

case study - 1

We can observe that the mean for both batsmen is 70 but the coefficient of variation for batsman A is 13.97% and for batsman B it is 57.32%.

We can see that variability in performance of Batsman B is more than that of Batsman A. Hence, we can infer that Batsman A is a more consistent performer than Batsman B.

Case study - batsmen mean and variance

Case study 2

Our next objective is to describe the variables present in the data of 100 retailers in the platinum segment of an FMCG company.

The sample size is 100 and the dataset has the following variables:  Retailer, Zone, Retailer_Age, Perindex, Growth and NPS_Category

case study-2

This is a snapshot of the dataset.  Each row of data is information about one retailer with a unique Retailer ID.

Zone, Retailer Age and NPS category are categorical variables, whereas performance index and growth are numeric continuous variables. The NPS stands for net promoter score and indicates loyalty to the company.

retail data

We use the read.csv function to import the dataset. 

To obtain a summary of the variables we use the summary function.For a continuous or numeric variable, the summary function gives a summary in the form of : minimum, 1st quartile, median, mean, 3rd quartile, maximum and count of Not Applicables  (if any).

Importing Data

 retail_data <-read.csv("Retail_Data.csv" header=TRUE) 

#Checking the variable features using summary function

 summary(retail_data) 

# Output

Case study output in R

Understanding Data Through Visualization

A boxplot shows the distribution of data based on a five number summary (“minimum”, first quartile (Q1), median, third quartile (Q3), and “maximum”). It can tell you about your outliers and what their values are. It can also tell you if your data is symmetrical, how tightly your data is grouped, and if and how your data is skewed.

The boxplot function is used in R to display data.

Here we can see that the Perindex variable is distributed symmetrically whereas the Growth variable is Positively Skewed.  We’ll look at skewness in detail in the next session

 boxplot(retail_data$Perindex, data= retail_data, main =  "BoxPlot (Perindex)",ylab = "Perindex",col = "darkorange")
 
 boxplot(retail_data$Growth, data= retail_data, main =  "BoxPlot (Growth)",ylab = "Growth",col = "darkorange") 
Descriptive Statistics - Box Plot

Measures of central tendency in R

The mean function is used to obtain the mean of a variable and the median function is used to obtain the median of a variable.

If the data contains missing values, the argument na.rm = TRUE should be defined in the mean function.

Here we will prefer mean for the perindex variable and median for the growth variable as growth distribution is skewed.

So as we’ve seen, the Perindex Variable is symmetric, hence its mean value is considered, whereas for our Growth Variable, which is Positively Skewed, median would be a better measure.

# Mean for Perindex & Growth Variables

mean(retail_data$Perindex) 
[1] NA  

mean() in R, gives mean of the variable. 

mean(retail_data$Perindex,na.rm = T) 
[1] 70.49697 

Using na.rm=T excludes the missing values from the mean

 mean(retail_data$Growth,na.rm = T) 
 [1] 5.1528 

# Median for Perindex & Growth Variables

 median(retail_data$Perindex,na.rm = T) 
 [1] 71.15 

median() in R, gives median of the variable. 

median(retail_data$Growth,na.rm = T)
[1] 4.495 

The trimmed_mean  function is used to obtain the trimmed mean. We should specify the percentage of observations to be excluded on each side within the trimmed_mean function. For example, to exclude 10% of observation, 0.1 should be specified.

# Trimmed Mean

 trimmed_mean_PI <- mean(retail_data$Perindex,0.10,na.rm=T) 
 trimmed_mean_PI 
 [1] 70.5842 

Using 0.10 in the mean(), excludes 10% observations from each side of the data from the mean

 trimmed_mean_G <- mean(retail_data$Growth,0.10,na.rm = T)
 trimmed_mean_G 
 [1] 4.825 

There is no standard function in R to obtain the mode. You can use the table function to create a frequency table and then extract the observation that has the highest frequency

# Measure of Central Tendency for Categorical Variable

# Mode using Frequency Table

freq <- table(retail_data$Zone)
freq
 
 East North South  West 
 15    25     32    28  

table() in R, gives the  frequency of counts of the variable mentioned.

Measures of Dispersion in R

The range function gives the highest and lowest values of a dataset. To obtain the difference between the highest and lowest values the diff  function is used.

The IQR function is used to obtain the inter quartile range in R

# Range, Difference & Inter Quartile Range

 r_PI <- range(retail_data$Perindex,na.rm = T)
 r_PI 
 [1] 46.53 92.49 

range() in R, gives minimum and maximum values of that variable

 r_G <- range(retail_data$Growth,na.rm = T)
 r_G 
 [1]  1.47 17.50 
 diff(r_PI)
 [1] 45.96  

diff() calculates difference between all values of that vector

 diff(r_G) 
 [1] 16.03 
 IQR(retail_data$Perindex,na.rm = T)
 [1] 12.095  

IQR() in R gives the Inter-Quartile range of the variable

 IQR(retail_data$Growth,na.rm = T)
 [1] 3.2825  

To obtain standard deviation the sd function is used. The var function is used to obtain variance.

There’s no standard function in R to obtain a coefficient of variation. We obtain the standard deviation and the mean using the sd and mean functions respectively, and then the coefficient of variation is obtained in R by dividing standard deviation by mean.

# Standard Deviation/ Variance

 sd(retail_data$Perindex,na.rm = T) 
 [1] 9.569232 

sd() in R, gives standard deviation  of the variable 

 sd(retail_data$Growth) 
 [1] 2.620525 
 var(retail_data$Perindex,na.rm = T) 
 [1] 91.5702 

var() in R, gives variance of the variable 

 var(retail_data$Growth) 
 [1] 6.867152 

# Coefficient of Variation

 cv_PI <- sd(retail_data$Perindex,na.rm = T)/ mean(retail_data$Perindex,na.rm = T)
 cv_PI 
 [1] 0.1357396 

There is no standard function for CV in R. Hence we calculate it by definition.

 cv_G <- sd(retail_data$Growth)/mean(retail_data$Growth)
 cv_G 
 [1] 0.5085633 

Let’s now recap, In this tutorial we studied various measures of central tendency and dispersion and how to describe our data using these measures in R. This tutorial is based on lessons from the Data Analytics in R unit of the Digita Schools Advanced Diploma in Data Analytics and Postgraduate Diploma in Data Science

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