Quantiles and Related Visualizations

Quantiles

Quartiles, quintiles, deciles, and percentiles are values or cut points that partition a finite number of observations into nearly equal-sized subsets. The number of partitions depends on the type of cut point involved.

Quartiles

They divide data into four parts. The first quartile, Q₁, is referred to as the lower quartile, and the last quartile, Q_4, is known as the upper quartile. Q₁ splits data into the lower 25% of the values and the upper 75%. Similarly, the upper quartile subdivides data into the lower 75% of the values and the upper 25%. The difference between the upper and lower quartile is known as the interquartile range, which indicates the spread of the middle 50% of the data under review.

Quintiles

Though rarely used in practice, quintiles split a set of data into five equal parts, i.e., fifths. Therefore, the second quintile splits data into the lower 40% of the values and the upper 60%.

Deciles

The deciles subdivide data into ten equal parts. There are 10 deciles in any data set. For example, the fourth decile splits data into the lower 40% of the values and the upper 60%.

Percentiles

Percentiles split data into 100 equal parts, i.e., hundredths. So, for instance, the 77^th percentile splits the data into the lower 77% of the values and the upper 23%.

Financial analysts commonly use the four types of subdivisions to rank investment performance. You should note that quartiles, quintiles, and deciles can all be expressed as percentiles. For instance, the first quartile is just the 25^th percentile. Similarly, the fourth decile is simply the 40^th percentile. This enables the application of the formula below:

$$ \text{Position of percentile} =\cfrac {(n + 1) y}{100} $$

Where \(n\) is the number of observations and \(y\) is the percentile.

Note: You must always order the data set, usually in ascending order, before calculating any of these values.

Example: Quartiles

Given the following distribution of returns, calculate the lower quartile.

{10%   23%   12%   21%   14%   17%   16%   11%   15%   19%}

Solution

First, we have to arrange the values in ascending order:

{10%   11%   12%   14%   15%   16%   17%   19%    21%   23%}

Next, we establish the position of the first quartile. This is simply the 25^th percentile. Therefore:

$$ \begin{align*} P_{25} & =\cfrac {(10 + 1)25}{100} \\ & = 2.75^{\text{th}} \text{ value}\\ \end{align*} $$

Since the value is not straightforward, we have to extrapolate between the 2^nd and the 3^rd data points. The 25^th percentile is three-fourths (0.75) of the way from the 2^nd data point (11%) to the 3^rd data point (12%):

$$ \begin{align*} & 11\% + 0.75 * (12 – 11) \\ & = 11.75\% \end{align*} $$

Box and Whisker Plot

Box and whisker plot is used to display the dispersion of data across quartiles. A box and whisker plot consists of a “box” with “whiskers” connected to the box. It shows the following five-number summary of a set of data.

Example: Box and Whisker Plot

Consider the following box and whisker plot:

1. Which of the following is most likely the median?

1. 1. 10.
   2. 10.5.
   3. 11.

2. Which of the following is most likely the interquartile range?

1. 1. 4.5.
   2. 6.5.
   3. 11.

3. Which of the following is most likely the 3^rd Quartile?

1. 1. 4.
   2. 15.
   3. 18.

Solutions

1. B is correct. Median or Quartile 2 (Q2) = \(\frac{(10+11)}{2} = 10.5\)
2. C is correct. Interquartile Range is \(Q_3 − Q_1 = 15 − 4 = 11\)
3. B is correct. Quartile 3 (Q3) is \(\frac{3\times (n + 1)}{ 4} = \frac{3(7+1)}{ 4 }\)= 6^th term which is 15.

Quantiles in Investment Practice

• Quantiles are used to rank performance.
• Quantiles can be used in investment research for comparison purposes. For example, companies can be clustered into deciles to compare the performance of small companies with the large ones. In this case, the first decile will contain the portfolio of companies with the smallest market values, while the tenth decile will contain the companies with the largest market values.

Question

A mutual fund achieved the following rates of growth over an 11-month period:

{3%   2%   7%   8%   2%   4%   3%   7.5%   7.2%   2.7%   2.09%}

The 5^th decile from the data is closest to:

1. 2%.
2. 3%.
3. 4%.

Solution

The correct answer is B.

First, you should re-arrange the data in ascending order:

{2%   2%  2.09%  2.7%    3%   3%   4%   7%   7.2%   7.5%   8%}

Secondly, you should establish the 5^th decile. This is simply the 50^th percentile and is actually the median:

$$ \begin{align*} P_{50} & =\frac {(1 + 11) 50}{100} \\ & = 12 \times 0.5 \\ & = 6 \text{ i}.\text{e}. \text{ the }6^{\text{th}} \text{ data point}. \\ \end{align*} $$

Therefore,

$$ \text{the } 5^{\text{th}} \text{ decile} = 50^{\text{th}} \text { percentile} = \text{median} = 3\% $$