Quartiles, Deciles and Percentiles
Introduction:
All of us are aware of the concept of the median in Statistics, the middle value or the mean of the two middle values, of an array. We have learned that the median divides a set of data into two equal parts. In the same way, there are also certain other values which divide a set of data into four, ten or hundred equal parts. Such values are referred as quartiles, deciles, and percentiles respectively.
Collectively, the quartiles, deciles and percentiles and other values obtained by equal sub-division of the data are called Quartiles.
Quartiles:
The values which divide an array (a set of data arranged in ascending or descending order) into four equal parts are called Quartiles. The first, second and third quartiles are denoted by Q1, Q2,Q3 respectively. The first and third quartiles are also called the lower and upper quartiles respectively. The second quartile represents the median, the middle value.
Quartiles for Ungrouped Data:
Quartiles for ungrouped data are calculated by the following formula.
In order to apply formulae, we need to arrange the above data into ascending order i.e. in the form of an array.
Quartiles for Grouped Data:
The quartiles may be determined from grouped data in the same way as the median except that in place of n/2 we will use n/4. For calculating quartiles from grouped data we will form cumulative frequency column. Quartiles for grouped data will be calculated from the following formula:
Where,
l = lower class boundary of the class containing the Q1 or Q3
, i.e. the class corresponding to the cumulative frequency in which n/4 or 3n/4 lies
h = class interval size of the class containing Q1 or Q3
f = frequency of the class containing Q1 or Q3
n = number of values, or the total frequency.
C.F = cumulative frequency of the class preceding the class containing Q1 or Q3
Quartile Deviation
Definition
The quartile deviation is half the difference between the third quartile and the first quartile of a frequency distribution, or simply distribution. Mathematically, quartile deviation would be represented as follows:
Quartile deviation is also known as semi-interquartile range. Here, the difference between the third and first quartiles is called interquartile range. The interquartile range may be taken as measure of dispersion (i.e. the extent to which the values are spread out from the average).
Characteristics of Quartile Deviation
- The quartile deviation is simple to understand and easy to calculate.
- As a measure of variation, it is superior to the range because it is not affected by extreme values.
- The values of quartile deviation might be the same for two dissimilar distributions provided the quartiles are the same.
- It is not utilized in algebraic manipulation.
- It is of the use only in a case where one wants to study the dispersion of items in the middle or the main body of the series. This usually happens in a frequency distribution where distribution tends to be intense in the middle or the main body of the series and the distribution of items towards the extremes are not of much significance.
Explanation
We are aware of the fact that 50% of the values lie between Q1 and Q3. The range i.e. Median +/- Q.D, also contains approximately 50% of the values. In a symmetrical distribution, the quartile are equi-distant from the median and the quartile deviation measures the distance from the median to the lower quartile or the distance from the median to the upper quartile. Therefore, the in a symmetrical distribution, if we measure quartile deviation below and above the median than it will includes the central 50% of the values of the distribution. We always need to remember that, in a distribution where there is not complete symmetry, quartile deviation measures the average distance from the quartiles to the median.
Deciles:
The values which divide an array into ten equal parts are called deciles. The first, second,…… ninth decile d1-d2-d3
corresponds to median. The second, fourth, sixth and eighth deciles which collectively divide the data into five equal parts are called quintiles.
Deciles for Ungrouped Data:
Deciles for ungrouped data will be calculated from the following formula:
Percentiles:
The values which divide an array into one hundred equal parts are called percentiles. The first, second,……. Ninety-ninth percentile are denoted by
The 50th percentile (p50) corresponds to the median. The 25th percentile (p25)corresponds to the first quartile and the 75th percentile (p75)corresponds to the third quartile.
Percentiles for Ungrouped Data:
Percentile from ungrouped data could be calculated from the following formula:
Percentiles for Grouped Data:
Percentiles can also be calculated for grouped data which is done with the help of following formula:
