###### Expected Value, Variance, and Standard ...

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.

They divide data into **four** parts. The first quartile, Q_{1}, is referred to as the lower quartile, and the last quartile, Q4, is known as the upper quartile. Q_{1} splits the data into the lower 25% of the values and the upper 75%. Similarly, the upper quartile subdivides the data into the lower 75% of the values and the upper 25%. The difference between the upper quartile and the lower quartile is known as the **interquartile range**, which indicates the spread of the middle 50% of the data.

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

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 split data into **100** equal parts, i.e., hundredths. So, for instance, the 77^{th }percentile splits 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.

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

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.*

Given the following distribution of returns, determine 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-fourth (0.75) of the way from the 2^{nd} data point (11%) to the 3^{rd} data point (12%):

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

QuestionA 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%}

Determine the 5

^{th}decile from the data.A. 4%

B. 3%

C. 2%

SolutionThe correct answer is B.

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

{2% 2% 2.7% 2.09% 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 themedian:$$ \begin{align*} P_{50} & =\cfrac {(1 + 11) 50}{100} \\ & = 12 * 0.5 \\ & = 6 \text{ i}.\text{e}. \text{ the }6^{\text{th}} \text{ data point}. \\ \end{align*} $$

Therefore,

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