Quartiles, quintiles, deciles, and percentiles

CFA level I / Quantitative Methods: Basic Concepts / Statistical Concepts and Market Returns / Quartiles, quintiles, deciles, and percentiles

The median divides a distribution in half. Quartiles divide the distribution into quarters, quintiles into fifths, deciles into tenths, and percentiles into hundredths.

Calculation of percentile: A percentile tells that how much percentage of observations are below a number. A 50th percentile means that 50 percent of the observations are below it. A 90th percentile means that 90 percent of the observations are below it. A 100 percentile is not possible as all the observations cannot be below a sample number because that number cannot be below itself. To calculate the percentile, we need to first arrange the observations in the ascending order. Then, we need to apply the below formula to calculate the percentile location.

L_y = (n+1)(y/100)

Where n is the total number of observations in the sample/population and y is the percentile number. L_y is the location of the percentile. It may or may not be a whole number. If it is not a whole number, then we need to make some arithmetic adjustment by linear interpolation to get the percentile number. The larger is the sample size; the more accurate is the percentile location.

Example 7: Calculating percentile

The returns of the 12 stocks of an index are given below (in percentages):

12.50,11.20,1.35,-0.58, -3.40, 2.46, -11.25, 8.02, 18.40, 6.70, -4.50, 5.50

Compute the 80th percentile.

Solution:

Arranging the returns in ascending order:
-11.25,-4.50,-3.40,-0.58,1.35,2.46,5.50,6.70,8.02,11.20,12.50,18.40

Location of 80th percentile = (1+12)*0.80 = 10.4.

10th position = 11.20
11th position = 12.50
10.4th position = 11.20 + (12.50-11.20)*0.4 = 11.20 + 0.52 = 11.72 percent.

The 80th percentile return is 11.72 percent.

Calculation of quartiles, quintiles, and deciles is now straightforward after knowing how to calculate a percentile. First, second, and third quartiles refer to the 25th, 50th, and 75th percentile respectively. Similarly, first, second, third, and fourth quintiles refer to 20th, 40th, 60th, and 80th percentile respectively. The 5th, 7th, and 9th deciles refer to the 50th, 70th, and 90th percentile respectively.

Example 8: Calculating quartile

Mark has invested money in a hedge fund whose return has been 12.40 percent in the recent year. He wants to check where does his hedge fund rank among the hedge funds. He would like to change the hedge fund if the fund returns fail to lie in the top quartile. The returns of hedge funds are given below (in percentage):

-34.50,11.60, 24.60, 12.40, -12.92, 8.50. 9.18, -14.48, 38.41, 7.45, 6.24, 3.06, -1.40, 12.20, 4.86

Solution:

Arranging the funds' returns in ascending order:
-34.50,-14.48,-12.92,-1.40,3.06,4.86,6.24,7.45,8.50,9.18,11.60,12.20,12.40,24.60,38.41

The top quartile lies above the cutoff mark of 3rd quartile i.e. 75th percentile.

Location of 3rd quartile = Location of 75th percentile = (1+15)*0.75 = 16*0.75= 12th.

3rd quartile return = 12.20 percent.

Any return above or equal to 12.20 percent will lie in the top quartile. The hedge fund of Mark has provided a return of 12.40 percent which is higher than 12.20 percent. So, his fund lies in the top quartile. Hence, he will not change his hedge fund.

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