### Chapter 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\%# on an exam and wants to know how well she did compared to her classmates. The grades 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 index 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:\[P^{th}\text{ percentile}=X_{below} + (i - i_{below}) \cdot (X_{above} - X_{below})\]

There are a number of different ways we can calculate the #80^{th}# percentile. Click on one of the panels to toggle a specific solution.

\[1,\,\,\,1,\,\,\,2,\,\,\,2,\,\,\,2,\,\,\,3,\,\,\,4,\,\,\,5,\,\,\,7,\,\,\,8,\,\,\,12,\,\,\,13,\,\,\,14,\,\,\,14,\,\,\,15,\,\,\,17,\,\,\,20,\,\,\,21,\,\,\,21,\,\,\,23,\,\,\,25\]

Next, to find the index #i# of the #80^{th}# percentile (#P=80#), use the following formula:

\[\begin{array}{rcl}

i &=& \cfrac{P}{100}(n-1)+1\\

&=& \cfrac{80}{100}(21 - 1) + 1=17

\end{array}\]

Since #i=17# is an integer, the #80^{th}# percentile is the score located at the #17^{th}# position of the ordered data:

\[P_{80}= X_{17} = 20\]

Assuming the sample scores are located in cells A1 through A21, the Excel command to calculate the #80^{th}# percentile is:

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.

\[= \text{PERCENTILE(A1:A21, 0.8)}\]

This gives:

\[P_{80} = 20\]

Thus, to calculate the #80^{th}# percentile, run the following command:

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]#.

\[quantile(x = c(2,21,13,1,15,14,2,2,7,3,20,5,1,8,12,4,21,25,23,17,14), probs = 0.8)\]

This gives:

\[P_{80}= 20\]

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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 index 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 the 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:\[Q^{th}\text{ quartile}=X_{below} + (i - i_{below}) \cdot (X_{above} - X_{below})\]

There are a number of different ways we can calculate the #1^{st}# quartile. Click on one of the panels to toggle a specific solution.

\[2,\,\,\,8,\,\,\,9,\,\,\,10,\,\,\,13,\,\,\,14,\,\,\,19,\,\,\,20,\,\,\,20,\,\,\,24,\,\,\,25,\,\,\,25,\,\,\,25\]

Next, to find the index #i# of the #1^{st}# quartile (#Q=1#), use the following formula:

\[\begin{array}{rcl}

i &=& \cfrac{Q}{4}(n-1)+1\\

&=& \cfrac{1}{4}(13 - 1) + 1=4

\end{array}\]

Since #i=4# is an integer, the #1^{st}# quartile is the score located at the #4^{th}# position of the ordered data:

\[Q_{1}=X_{4} = 10\]

Assuming the sample scores are located in cells A1 through A13, the Excel command to calculate the #1^{st}# quartile is:

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.

\[= \text{QUARTILE(A1:A13, 1)}\]

This gives:

\[Q_{1} = 10\]

Thus, to calculate the #1^{st}# quartile, run the following command:

quantile(x)

x: The numeric vector whose sample quantiles are wanted.

\[quantile(x = c(2,19,9,10,25,25,14,20,20,8,13,25,24))\]

Looking at the output generated by R, under #25\%# we find:

\[Q_{1}= 10\]

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