Measures of Location I: Quantiles

1. Descriptive Statistics: Frequency Distributions

Measures of Location I: Quantiles

Besides describing the characteristics of a distribution as a whole, descriptive statistics can also be used to provide more information about individual scores. One particularly useful piece of information is the location of a score relative to all other scores within the distribution.

Knowing a score's location relative to the other scores can, for instance, help you judge whether a particular score should be considered high, low, or average. Raw scores are, by themselves, not very informative in this regard.

One way to express a score's location within a distribution is to calculate its percentile rank.
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Percentile Rank

The percentile rank of a score is the percentage of scores in the distribution that are equal to or lower than it.

A student scored $70$ points at an exam and would like to know how well she did compared to her classmates. The scores of the entire class are as follows:

$34\,\,\,\,42\,\,\,\,53\,\,\,\,56\,\,\,\,57\,\,\,\,60\,\,\,\,62\,\,\,\,64\,\,\,\,64\,\,\,\,67\,\,\,\,70\,\,\,\,70\,\,\,\,72\,\,\,\,78\,\,\,\,84\,\,\,\,89$

To calculate the percentile rank of $X=70$ , first count the number of scores that are equal to or lower than $70$ , which in this case is $12$ .

Next, divide that number by the total number of scores, which in this case is $16$ , and multiply by $100\%$ .

$\cfrac{12}{16}\cdot 100\% = 75\%$

So the percentile rank of $X=70$ is $75$ .

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When a score is identified by its percentile rank, the score is called a percentile.
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Percentiles

Definition

Percentiles are the values that divide a distribution of scores into one hundred equal parts.

The $P^{th}$ percentile of distribution is the value such that $P$ percent of the scores are equal to or below it.

Formula

The of the $P^{th}$ percentile of a distribution is:

$i = \dfrac{P}{100}(n-1)+1$

where $n$ is the number of scores and $P$ is a value between 1 and 99.

Percentile Calculation

The calculation of the $P^{th}$ percentile starts by ordering the scores in the distribution from smallest to largest. Next, to find the index $i$ of the $P^{th}$ percentile, use the following formula:

$i = \dfrac{P}{100}(n-1)+1$

where $n$ the total number of scores in the distribution.

It is important to note that formula above is used to determine the location of the $P^{th}$ percentile and not the value associated with it.

If $i$ is an integer, then the $P^{th}$ percentile is the score located at the $i^{th}$ position of the ordered data.

Whenever $i$ is not an integer, linear interpolation is used to calculate the percentile:

Find the two integers closest to $i$ by rounding $i$ up and down. These indices are denoted by $i_{above}$ and $i_{below}$ , respectively.
Determine the values located at these positions. These values are denoted by $X_{above}$ and $X_{below}$ , respectively.
Calculate the $P^{th}$ percentile with the following formula: $P^{th}\text{ percentile}=X_{below} + (i - i_{below}) \cdot (X_{above} - X_{below})$

Excel Calculation

To calculate a percentile of a dataset in Excel, make use of the following function:

PERCENTILE(array, k)

array: The array or cell range of numeric values for which you want the percentile value.

k: The percentile value in the range $[0, 1]$ , inclusive.

R Calculation

To calculate a percentile of a dataset in R, make use of the following function:

quantile(x, probs)

x: The numeric vector whose sample quantiles are wanted.

probs: The numeric vector of probabilities with values in the range $[0, 1]$ .

Consider the following sample of $21$ scores:

$19,\,\,\,22,\,\,\,14,\,\,\,6,\,\,\,11,\,\,\,12,\,\,\,22,\,\,\,14,\,\,\,13,\,\,\,18,\,\,\,5,\,\,\,23,\,\,\,20,\,\,\,12,\,\,\,14,\,\,\,2,\,\,\,12,\,\,\,15,\,\,\,6,\,\,\,8,\,\,\,16$

Calculate the $30^{th}$ percentile.

$P_{30}= 12$

There are a number of different ways we can calculate the $30^{th}$ percentile. Click on one of the panels to toggle a specific solution.

Manual Calculation

To manually calculate the $30^{th}$ percentile, first sort the $n=21$ values in ascending order:

$2,\,\,\,5,\,\,\,6,\,\,\,6,\,\,\,8,\,\,\,11,\,\,\,12,\,\,\,12,\,\,\,12,\,\,\,13,\,\,\,14,\,\,\,14,\,\,\,14,\,\,\,15,\,\,\,16,\,\,\,18,\,\,\,19,\,\,\,20,\,\,\,22,\,\,\,22,\,\,\,23$

Next, to find the index $i$ of the $30^{th}$ percentile ( $P=30$ ), use the following formula:

$\begin{array}{rcl} i &=& \cfrac{P}{100}(n-1)+1\\ &=& \cfrac{30}{100}(21 - 1) + 1=7 \end{array}$
Since $i=7$ is an integer, the $30^{th}$ percentile is the score located at the $7^{th}$ position of the ordered data:

$P_{30}= X_{7} = 12$

Excel Calculation

To calculate the $30^{th}$ percentile in Excel, make use of the following function:

PERCENTILE(array, k)

array: The array or cell range of numeric values for which you want the percentile value.

k: The percentile value in the range $[0, 1]$ , inclusive.

Assuming the sample scores are located in cells A1 through A21, the Excel command to calculate the $30^{th}$ percentile is:

$= \text{PERCENTILE(A1:A21, 0.3)}$
This gives:

$P_{30} = 12$

R Calculation

To calculate the $30^{th}$ percentile in R, make use of the following function:

quantile(x, probs)

x: The numeric vector whose sample quantiles are wanted.

probs: The numeric vector of probabilities with values in the range $[0, 1]$ .

Thus, to calculate the $30^{th}$ percentile, run the following command:

$quantile(x = c(19,22,14,6,11,12,22,14,13,18,5,23,20,12,14,2,12,15,6,8,16), probs = 0.3)$
This gives:

$P_{30}= 12$

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Percentiles divide a distribution into $100$ equal parts. It is possible, however, to divide a distribution into any arbitrary number of equal parts. When dividing a distribution of scores into equal parts, the dividing values are called quantiles.
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Quantiles

If you divide the data set into $k$ equal parts, you call the dividing values $k$ -quantiles and there are always $k-1$ quantiles.

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If you divide a distribution of scores into four equal parts, the dividing values are referred to as quartiles.
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Quartiles

Definition

Quartiles are the values that divide a distribution of scores into four equal parts.

The first ( $Q_1$ ), second ( $Q_2$ ), and third ( $Q_3$ ) quartiles are equal to the 25th, 50th, and 75th percentile, respectively.

The second quartile is also called the median.

Formula

The of the $Q^{th}$ quartile of a distribution is:

$i=\dfrac{Q}{4}(n-1)+1$

where $n$ is the number of scores and $Q$ is a value between 1 and 3.

Quartile Calculation

The calculation of quartiles starts by ordering the scores in the distribution from smallest to largest. Next, to find the index $i$ of the $Q^{th}$ quartile, use the following formula:

$i=\dfrac{Q}{4}(n-1)+1$

where $n$ is the total number of scores in the dataset.

It is important to note that formula above is used to determine the location of the $Q^{th}$ quartile and not the value associated with it.

If $i$ is an integer, then the $Q^{th}$ quartile is the score located at the $i^{th}$ position of the ordered data.

Whenever $i$ is not an integer, linear interpolation is used to calculate the quartile:

Find the two integers closest to $i$ by rounding $i$ up and down. These indices are denoted by $i_{above}$ and $i_{below}$ , respectively.
Determine the values located at these positions. These values are denoted by $X_{above}$ and $X_{below}$ , respectively.
Calculate the $Q^{th}$ quartile with the following formula: $Q^{th}\text{ quartile}=X_{below} + (i - i_{below}) \cdot (X_{above} - X_{below})$

Excel Calculation

To calculate a quartile of a dataset in Excel, make use of the following function:

QUARTILE(array, quart)

array: The array or cell range of numeric values for which you want the quartile value.

quart: Indicates which quartile to return.

R Calculation

To calculate a quartile of a dataset in R, make use of the following function:

quantile(x)

x: The numeric vector whose sample quantiles are wanted.

In the output generated by R, the first, second, and third quartiles can be found under $25\%$ , $50\%$ , and $75\%$ , respectively.

Consider the following sample of $21$ scores:

$8,\,\,\,16,\,\,\,14,\,\,\,12,\,\,\,7,\,\,\,21,\,\,\,12,\,\,\,10,\,\,\,12,\,\,\,13,\,\,\,15,\,\,\,7,\,\,\,19,\,\,\,5,\,\,\,14,\,\,\,11,\,\,\,13,\,\,\,20,\,\,\,5,\,\,\,1,\,\,\,18$

Calculate the $2^{nd}$ quartile.

$Q_{2}= 12$

There are a number of different ways we can calculate the $2^{nd}$ quartile. Click on one of the panels to toggle a specific solution.

Manual Calculation

To calculate the $2^{nd}$ quartile, first sort the $n=21$ values in ascending order:

$1,\,\,\,5,\,\,\,5,\,\,\,7,\,\,\,7,\,\,\,8,\,\,\,10,\,\,\,11,\,\,\,12,\,\,\,12,\,\,\,12,\,\,\,13,\,\,\,13,\,\,\,14,\,\,\,14,\,\,\,15,\,\,\,16,\,\,\,18,\,\,\,19,\,\,\,20,\,\,\,21$

Next, to find the index $i$ of the $2^{nd}$ quartile ( $Q=2$ ), use the following formula:

$\begin{array}{rcl} i &=& \cfrac{Q}{4}(n-1)+1\\ &=& \cfrac{2}{4}(21 - 1) + 1=11 \end{array}$
Since $i=11$ is an integer, the $2^{nd}$ quartile is the score located at the $11^{th}$ position of the ordered data:

$Q_{2}=X_{11} = 12$

Excel Calculation

To calculate the $2^{nd}$ quartile in Excel, make use of the following function:

QUARTILE(array, quart)

array: The array or cell range of numeric values for which you want the quartile value.

quart: Indicates which quartile to return.

Assuming the sample scores are located in cells A1 through A21, the Excel command to calculate the $2^{nd}$ quartile is:

$= \text{QUARTILE(A1:A21, 2)}$
This gives:

$Q_{2} = 12$

R Calculation

To calculate the $2^{nd}$ quartile in R, make use of the following function:

quantile(x)

x: The numeric vector whose sample quantiles are wanted.

Thus, to calculate the $2^{nd}$ quartile, run the following command:

$quantile(x = c(8,16,14,12,7,21,12,10,12,13,15,7,19,5,14,11,13,20,5,1,18))$
Looking at the output generated by R, under $50\%$ we find:

$Q_{2}= 12$

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