### [A, SfS] Chapter 1: Sampling, Descriptive Statistics, Intr: 1.8: Quartiles

### Quartiles

Quartiles

In this lesson you will learn:

- What
*quartiles*are and how to calculate them. - What the
*inter-quartile range*is and how to calculate it. - What the
*five-number summary*is. - What a
*boxplot*is and how to construct it.

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Another common method to get a better understanding of how a data set is distributed is to divide the data into a number of equal-sized parts. If the data set is divided up into four parts, the resulting cut-off points are called *quartiles*.

Quartiles

**Quartiles** divide a *sorted* data set into four parts, such that each part contains #1/4# of all the elements in the data set.

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 data set and #Q# is a value between 1 and 3.

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

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

\[1,\,\,\,2,\,\,\,3,\,\,\,3,\,\,\,4,\,\,\,5,\,\,\,6,\,\,\,6,\,\,\,7,\,\,\,8,\,\,\,8,\,\,\,10,\,\,\,13,\,\,\,13,\,\,\,14,\,\,\,16,\,\,\,19,\,\,\,19,\,\,\,20,\,\,\,22,\,\,\,24\]

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:

\[Q_{2}=x_{11} = 8\]

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One particularly useful measure that can be derived from the quartiles of a distribution is the *inter-quartile range*.

Inter-Quartile Range

The **inter-quartile range** #(IQR)# is the difference between the first quartile and the third quartile of a distribution.

\[IQR = Q_3 - Q_1\]

The inter-quartile range thus measures how spread out the middle #50\%# of the data are.

Classifying Outliers

The #IQR# can be used as a way to classify measurements as outliers. Under this convention, we say that the measurement is an outlier if:

- It is less than #Q1 - 1.5*IQR#, or
- It is more than #Q3 + 1.5*IQR#.

Some statisticians distinguish *moderate* outliers, identified using the criteria above, from *extreme* outliers by replacing the #1.5# with #3#.

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Quartiles can also be used to construct a so-called *five-number summary*.

Five-Number Summary

The *minimum* value of a data set is thought of as the null quartile, denoted #Q_0#, and the *maximum* value is thought of as the fourth quartile, denoted #Q_4#.

The quartiles #Q_0#, #Q_1#, #Q_2#, #Q_3# and #Q_4# together are called the **five-number summary** for measurement on a quantitative variable.

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Note that all quartiles are special cases of *quantiles*.

Quantiles

For quantitative data, the #\boldsymbol{q}#**th** **quantile** is any number such that #100q# percent of the data fall at or below that number.

The first quartile is the #0.25# quantile, because #25\%# of the data fall at or below the first quartile. The median is the #0.5# quantile, and the third quartile is the #0.75# quantile. The minimum and maximum can be thought of as the #0.0# quantile and #1.0# quantile, respectively.

Meanwhile, the #0.43# quantile is a number such that #43\%# of the data fall at or below it, and the #0.87# quantile is a number such that #87\%# of the data fall at or below it.

#\text{}#

If #p = 100q# then we can refer to the #q#th quantile instead as the #p#*th **percentile.*

Percentiles

**Percentiles** divide a *sorted* data set into one hundred parts, such that each part contains #1/100# of all the elements in the data set.

The #0.64# quantile is the #64#th percentile, and the #0.29# quantile is the #29#th percentile.

If in a national exam your score is at the #95th# percentile it means #95\%# of the people who took the exam scored at or below your score (i.e., you did very well!).

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We discussed previously that a histogram is a useful way to visualize the distribution of quantitative variable. Using the quartiles, there is an alternative to the histogram which is also commonly used: the *boxplot*.

Boxplot

A **boxplot **uses just one axis (usually vertical) to represent the numerical scale of the variable. The boxplot consists of a rectangle of arbitrary width with one side at #Q_1# and the other side at #Q_3#, and a thick line inside the box at #Q_2# (the median).

Then from the bottom of the plot, we extend a line to the minimum value of our data set, excluding any outliers, while from the top of the box we extend a line to the maximum value of our data set, excluding any outliers. These lines are often called *whiskers*. Any outliers are represented by separate points.

This boxplot shows that the measurements have a *symmetric* distribution with no outliers.

This boxplot shows that the data are *skewed to the right* (because the top whisker is longer than the bottom whisker, and the median is closer to #Q1# to #Q3#), with #4# outliers at the high end.

This boxplot shows that the data are *skewed to the left* with many outliers.

These are side-by-side boxplots. We can compare the same variable for two different groups (#A# and #B#).

We can see that the median of the variable in group #B# is greater than the median of that variable in group #A#, and we can also see that the spread of the measurements in group #A# is much smaller than the spread in group #B#. But both distributions appear symmetric, with one outlier in group #A#.

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Using R

**Five-Number Summary**

Suppose you have measurements on a quantitative variable stored in a numeric vector named #\mathtt{Salary}# in your #\mathrm{R}# workspace.

To find the minimum, first quartile, third quartile and maximum, respectively, the #\mathrm{R}# functions are:

> min(Salary)

> quantile(Salary,0.25)

> quantile(Salary,0.75)

> max(Salary)

#\text{}#

However, you can obtain all of these quantities, and the mean and median, in one go, using the #\mathtt{summary()}# function:

`> summary(Salary)`

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**Inter-Quartile Range**

You can find the inter-quartile range either by subtracting the first quartile from the third quartile, or more efficiently by using the #\mathtt{IQR()}# function:

`> IQR(Salary)`

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**Quantiles**

If you want the #q# quantile of the #\mathtt{Salary}# data, i.e., the #p = 100q# percentile, use the #\mathtt{quantile()}# function:

`> quantile(Salary,q)`

#\text{}#

**Percentiles**

If you want the percentiles from #10\%# to #90\%# incremented by tens, you could use:

`> quantile(Salary,seq(0.1,0.9,by=0.1))`

#\text{}#

**Boxplot**

If you want to make a boxplot of this variable, use the #\mathtt{boxplot()}# function:

`> boxplot(Salary)`

#\text{}#

**Side-by-Side Boxplots**

If you want side-by-side boxplots, with a separate boxplot for each of two or more groups situated next to each other, you need a second vector in #\mathrm{R}# which indicates to which group each measurement belongs. This vector can be a character vector or an integer vector. Suppose the vector is named #\mathtt{Group}#. Then use the boxplot function in this manner:

`> boxplot(Salary ~ Group)`

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