Quartile

In descriptive statistics, a quartile is one of four equal groups, representing a fourth of the distributed sampled population. It is a type of quantile.

In epidemiology, the four ranges defined by the three values discussed here.

Definitions

• first quartile (designated Q₁) = lower quartile = cuts off lowest 25% of data = 25th percentile
• second quartile (designated Q₂) = median = cuts data set in half = 50th percentile
• third quartile (designated Q₃) = upper quartile = cuts off highest 25% of data, or lowest 75% = 75th percentile

The difference between the upper and lower quartiles is called the interquartile range.

Computing methods

There is no universal agreement on choosing the quartile values.

The formula for locating the position of the observation at a given percentile, y, with n data points sorted in ascending order is:

$L_{y}=n\cdot {\cfrac {y}{100}}.$

• Case 1: If L is a whole number, then the value will be found halfway between positions L and L+1.
• Case 2: If L is a decimal, round to the nearest whole number. (for example, L = 1.2 becomes 1).

Example

Method 1

Example 4. Boxplot (with quartiles and an interquartile range) and a probability density function (pdf) of a normal N(0,1σ²) population

1. Use the median to divide the ordered data set into two halves. Do not include the median into the halves.
2. The lower quartile value is the median of the lower half of the data. The upper quartile value is the median of the upper half of the data.

This rule is employed by the TI-83 calculator boxplot and 1-Var Stats functions.

Method 2

1. Use the median to divide the ordered data set into two halves. Include the median into both halves.
2. The lower quartile value is the median of the lower half of the data. The upper quartile value is the median of the upper half of the data.

Example 1

Data Set: 6, 47, 49, 15, 42, 41, 7, 39, 43, 40, 36
Ordered Data Set: 6, 7, 15, 36, 39, 40, 41, 42, 43, 47, 49

Method 1	Method 2
${\begin{cases}Q_{1}=15\\Q_{2}=40\\Q_{3}=43\end{cases}}$	${\begin{cases}Q_{1}=25.5\\Q_{2}=40\\Q_{3}=42.5\end{cases}}$

Example 2

Ordered Data Set: 7, 15, 36, 39, 40, 41

Method 1	Method 2
${\begin{cases}Q_{1}=15\\Q_{2}=37.5\\Q_{3}=40\end{cases}}$	${\begin{cases}Q_{1}=25.5\\Q_{2}=37.5\\Q_{3}=39.5\end{cases}}$

Example 3

Ordered Data Set: 1, 2, 3, 4

Method 1	Method 2
${\begin{cases}Q_{1}=1.5\\Q_{2}=2.5\\Q_{3}=3.5\end{cases}}$	${\begin{cases}Q_{1}=2\\Q_{2}=2.5\\Q_{3}=3\end{cases}}$

First Quartlie can be calculated by the follwing formula if (n+1)/4 the value is not an integer. Let us consider the case that we might have 12 observations i.e n=12 then Q1=(12+1)/4 the value i.e Q1=3.25th value. To find the 3.25 the value we can use the formula Q1= 3rd value + 0.25 [4th value - 3rd value] same procedure can be adopted for any fractional value of Q1 and Q3.

See also

• Range
• Quarter
• Box plot
• Summary statistics

External links

• Quartile - from MathWorld Includes references and compares various methods to compute quartiles
• Quartiles - From MathForum.org
• Quartiles - An example how to calculate it