# Quantiles in R

Considering a value \(p\), being \(0 < p < 1\) the quantile of order \(p\) is the value that leaves a proportion of the data below (\(p\)) and the rest \((1-p)\) above that value. Notice that quantiles are the generalization of the median which is the quantile for \(p = 0.5\). In R, you can make use of the `quantile`

function to calculate any quantile for any numeric vector.

## Syntax

The `quantile`

function calculates the sample quantiles of a numeric vector (`x`

). By default, this function calculates the quartiles specified inside `probs`

, but you can also input any other probabilities to compute any percentile.

```
quantile(x, # Numeric vector
probs = seq(0, 1, 0.25), # Quantiles (By default the quartiles: 0, 0.25, 0.5, 0.75, 1)
na.rm = FALSE, # If TRUE, removes missing values
names = TRUE, # If TRUE, the result keeps name attributes
type = 7, # Integer between 1 and 9 to select a quantile algorithm
digits = 7, # If names = TRUE, is the number of digits of the percentages
...) # Additional arguments if needed
```

## Quartiles

Quartiles are quantiles of order 0.25, 0.5 and 0.75 and they **divide the sample into four parts with the same frequency**. Usually, quartiles are denoted by \(Q_1\), \(Q_2\) and \(Q_3\).

```
# Sample data
set.seed(1)
x <- rnorm(100)
# Calculate the quartiles of x
quantile(x)
```

```
0% 25% 50% 75% 100%
-2.2146999 -0.4942425 0.1139092 0.6915454 2.4016178
```

Recall that the **quartile 0.5 is equal to the median**:

```
# Sample data
set.seed(1)
x <- rnorm(100)
# Calculate the median of x
median(x) # 0.1139092
```

Note that you can **remove the name attributes** from the output setting `names = FALSE`

.

```
# Sample data
set.seed(1)
x <- rnorm(100)
# Calculate the quartiles of x
quantile(x, names = FALSE)
```

`-2.2146999 -0.4942425 0.1139092 0.6915454 2.4016178`

### Remove missing values

If your numeric vector contains missing values you wonāt be able to calculate the quantiles, so you will need to set `na.rm = TRUE`

to remove the missing values before the calculation.

```
# Sample data
set.seed(1)
x <- rnorm(100)
# Missing value
x[1] <- NA
# Calculate the quartiles of x removing missing values
quantile(x, na.rm = TRUE)
```

```
0% 25% 50% 75% 100%
-2.2146999 -0.4757753 0.1532533 0.6933514 2.4016178
```

### Quantile algorithms

The calculation of the quantiles are based on **one of the nine algorithms discussed in Hyndman and Fan (1996)**. By default, the seventh algorithm is used, but you can select other passing an integer between 1 and 9 to `type`

. Read the previous reference for further information about each algorithm.

```
# Sample data
set.seed(1)
x <- rnorm(100)
# Calculate the quartiles of x using Type 8 algorithm
quantile(x, type = 8)
```

```
0% 25% 50% 75% 100%
-2.2146999 -0.5156992 0.1139092 0.6939534 2.4016178
```

### Visual representation

It is important to note that a box plot can be used to visualize quartiles, but the method used inside the `boxplot`

function is not the same as the one used inside `quartile`

, so the output may vary slightly.

```
# Sample data
set.seed(1)
x <- rnorm(100)
quartile <- quantile(x)
# Box plot
boxplot(x, col = 4, horizontal = TRUE)
text(quartile[2], 1.25, expression(Q[1]))
text(quartile[3], 1.25, expression(Q[2]))
text(quartile[4], 1.25, expression(Q[3]))
```

## Deciles

Deciles are quantiles of order 0.1, 0.2, ā¦, 0.9 and **divide the sample into 10 equal-frequency parts**. In order to calculate them you can input a sequence from 0 to 1 by 0.1 to `probs`

, as shown in the example below.

```
# Sample data
set.seed(1)
x <- rnorm(100)
# Calculate the deciles of x
quantile(x, probs = seq(0, 1, by = 0.1))
```

```
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
-2.21469989 -1.05265747 -0.61386923 -0.37534202 -0.07670313 0.11390916 0.37707993 0.58121734 0.77125360 1.18106508 2.40161776
```

## Percentiles

Percentiles are quantiles of the order 0.01, 0.02, ā¦ , 0.99 and **divide the sample into 100 equal-frequency parts**. If you want to calculate the percentiles of a numeric vector you will need to specify a sequence from 0 to 1 by 0.01 inside `probs`

.

```
# Sample data
set.seed(1)
x <- rnorm(100)
# Calculate the percentiles of x
quantile(x, probs = seq(0, 1, by = 0.01))
```

```
0% 1% 2% 3% 4% 5% 6% 7% 8%
-2.214699887 -1.991605178 -1.808646490 -1.532008555 -1.472864961 -1.381744198 -1.282620249 -1.255240517 -1.226934278
```