1 Derive Percentiles in R

The percentile function in R is quantile(). It takes the data and a probability value or a set of probability values as input.

Derive the 50th percentile or median:

Values = c(2, 4, 6, 8, 10, 12, 14, 16, 18)

quantile(Values, 0.5)

50% 
 10 

For other percentile values, the quantile() function also takes a type argument depending on the methodology to be used. The default is type = 7. Here we present 3 commonly used types of the available types 1 to 9.

For \(x_1, x_2, ... ,x_n\) sorted from lowest to highest as \(x_{(1)}, x_{(2)}, ..., x_{(k)}, ... ,x_{(n)}\).

The \(k\)th orders statistic or \(k\)th highest value \(x_{(k)}\) is the following percentile:

Type 4: \(\frac{k}{n}*100\)th percentile.

Type 6: \(\frac{k}{n+1}*100\)th percentile.

Type 7: \(\frac{k-1}{n-1}*100\)th percentile.

Examples using values = c(2, 4, 6, 8, 10, 12, 14, 16, 18) with \(n=9\) values:

The 33.33th percentile with type = 4; which should be 3rd highest value as \(\frac{k}{n}*100 = 33.33 \Rightarrow k = 3\) or \((\frac{3}{9}*100 = 33.33)\).

quantile(Values, 1/3, type = 4)

33.33333% 
        6 

The 40th percentile with type = 6; which should be 4th highest value as \(\frac{k}{n+1}*100 = 40 \Rightarrow k = 4\) or \((\frac{4}{10}*100 = 40)\).

quantile(Values, 0.4, type = 6)

40% 
  8 

The 75th percentile with type = 7; which should be 7th highest value as \(\frac{k-1}{n-1}*100 = 75 \Rightarrow k = 7\) or \((\frac{7-1}{9-1}*100 = 75)\).

quantile(Values, 0.75, type = 7)

75% 
 14 

2 What Percentile is a Number in R?

Using the same type definitions as above:

Values = c(2, 4, 6, 8, 10, 12, 14, 16, 18)

Type 4: \(\frac{k}{n}*100\)th percentile.

# What percentile is 6?
x = 6
length(which(Values <= x))/length(Values)*100

[1] 33.33333

Type 6: \(\frac{k}{n+1}*100\)th percentile.

# What percentile is 8?
x = 8
length(which(Values <= x))/(length(Values)+1)*100

[1] 40

Type 7: \(\frac{k-1}{n-1}*100\)th percentile.

# What percentile is 14?
x = 14
length(which(Values < x))/(length(Values)-1)*100

[1] 75

3 Derive Quantiles in R

Quantiles are the points that split the data range into intervals with equal probabilities.

\(n-\)quantile splits the data into \(n\) intervals of equal probabilities.

Values = c(2, 4, 6, 8, 10, 12, 14, 16, 18)

Derive \(2\)-quantile:

quantile(Values, 1/2)

50% 
 10 

Derive \(3\)-quantile:

quantile(Values, (1:2)/3)

Or:

quantile(Values, c(1/3, 2/3))

33.33333% 66.66667% 
 7.333333 12.666667 

Derive \(5\)-quantile:

quantile(Values, (1:4)/5)

Or:

quantile(Values, c(1/5, 2/5, 3/5, 4/5))

 20%  40%  60%  80% 
 5.2  8.4 11.6 14.8 

4 Derive Quartiles or Five Number Summary in R

Quartiles are also \(4\)-quantile as defined above. The five number summary is the \(4\)-quantile including the extremes (the minimum and maximum values).

The quartiles or \(4\)-quantile values are \(Q_1\) (greater than 25% of the data), \(Q_2\) (greater than 50% of the data), and \(Q_3\) (greater than 75% of the data).

Derive Quartiles:

Values = c(2, 4, 6, 8, 10, 12, 14, 16, 18)

quantile(Values, (1:3)/4)

Or:

quantile(Values, c(1/4, 2/4, 3/4))

25% 50% 75% 
  6  10  14 

Derive five number summary:

The five number summary are \(\text{minimum}\), \(Q_1\), \(Q_2\), \(Q_3\), and \(\text{maximum}\). This is done with the fivenum() function.

Values = c(2, 4, 6, 8, 10, 12, 14, 16, 18)

fivenum(Values)

[1]  2  6 10 14 18

5 Interquartile Range in R

The interquartile range is \(Q_3 - Q_1\) as defined above. This is done with the IQR() function, and it also takes the "type" argument as the quantile() function. The default type is type = 7.

Values = c(2, 4, 6, 8, 10, 12, 14, 16, 18)

IQR(Values)

# Or:
IQR(Values, type =7)

[1] 8