#### Subject 5. Quartiles, Quintiles, Deciles, and Percentiles

The median is the value that divides a distribution in half. Quantities such as quartiles, quintiles, deciles and percentiles perform similar functions to the median in a data set.

Quartile

There are 3 quartiles in a data set. Between them, they divide the data into 4 equal parts or quarters. The first quartile is called the lower quartile and is often denoted as Q1. The second quartile is obviously just the median, as it is the middle value of the data set. The third quartile is called the upper quartile and is often denoted as Q3.

You should note that Q1 effectively splits the data set into the lower 25% of values and the upper 75% of values whereas Q3 splits the data into the lower 75% of values and the upper 25% of values.

The distance between Q1 and Q3, namely Q3 - Q1, is called the inter-quartile range; it gives an indication of the spread of the middle 50% of the data set.

Quintile

There are 4 quintiles in a data set. Between them, they divide the data into 5 equal parts or fifths. Quintiles are not very commonly used.

You should note that the first quintile effectively splits the data set into the lower 20% of values and the upper 80% of values; the second quintile splits the data set into the lower 40% of values and the upper 60% of values, and so on.

Decile

There are 9 deciles in the data set. Between them, they divide the data into 10 equal parts, or tenths.

Obviously, the fifth decile is the median, as it is the middle value in the data set. The third decile splits the data set into the lower 30% of values and the upper 70% of values, and so on.

Percentile

There are 99 percentiles in the data set. Between them, they divide the data into 100 equal parts, or hundredths.

The fiftieth percentile is the median, as it is the middle value in the data set. The sixty-third percentile divides the data set into the lower 63% of values and the upper 37% of values, and so on.

Note that the 75th percentile is also the same value as Q3, for example.

These types of values are used to rank investment performance, such as the performance of mutual funds.

In calculating these values, it is important to first order the data set, as we did with the median. Once this is done, it is necessary to find the position of the value that you are calculating, and then the value itself (the procedure is exactly the same as that for calculating the median).