The question **aims** to find a **range and quartile range**.

The** range** is the **difference between the largest and the smallest value**. In statistics, the scope of the data collection is the difference between the most **significant** and **smallest values**. The **difference** here is clear: the data set’s range is the result of high and low sample output. In **descriptive statistics**, however, the concept of scope has a complex meaning. The **scope/range** is the size of the smallest interval (statistics) that contains** all the data** and provides an indication of **statistical dispersion**—measured by the same units as the data. Relying only on two perspectives is very useful in representing the spread of small data sets.

In **descriptive statistics**, the **interquartile range** $(IQR)$ is a **measure of statistical scattering**, which is the **data spread**. $IQR$ can also be called midspread, middle $50\%$, fourth spread, or $H$ spread. It is the **difference** between $75$ and $25$ **percent of data**.

## Expert Answer

The **range is the difference between the largest and the smallest value.**

\[Range=(largest\: value-smallest\: value)\]

The **largest value** is $20$ and the **smallest value** is $10$.

\[Range=(20-10)\]

\[Range=10\]

The lower quartile, or **first quartile** $(Q1)$, is the **amount** at which $25\%$ of data points are subtracted when arranged in **increasing order**.

The **first quartile** is defined as the **median of the data values** **below the median**.

\[Q_{1}=\dfrac{10+12}{2}\]

\[Q_{1}=11\]

The upper quartile, or **third quartile** $(Q_{3})$, is the value at which $75\%$ of the** data points** are **subdivided** when arranged in **increasing order**.

The **third quartile is defined as the median of the data values above the median**.

\[Q_{3}=\dfrac{17+20}{2}\]

\[Q_{3}=18.5\]

The **interquartile range** $(IQR)$ is the **difference between the first quartile** $Q_{1}$ and the **third quartile** $Q_{3}$.

\[IQR=Q_{3}-Q_{1}\]

\[IQR=18.5-11\]

\[IQR=7.5\]

The **interquartile range** is $7.5$.

## Numerical Results

The **range** is calculated as:

\[Range=10\]

The** interquartile range** $(IQR)$ is calculated as:

\[IQR=7.5\]

## Example

**Data values of the sample are $8$, $20$, $14$, $17$, and $18$. Calculate the range and range of the interquartile.**

**Solution:**

The** range is the difference between the largest and the smallest value**.

\[Range=(largest\: value-smallest\: value)\]

The **largest value** is $20$ and the **smallest value** is $8$.

\[Range=(20-8)\]

\[Range=12\]

The lower quartile, or** first quartile** $(Q1)$, is the **amount** at which $25\%$ of data points are **subtracted** when arranged in **increasing order.**

The **first quartile** is defined as the **median of the data values below the median**.

\[Q_{1}=\dfrac{8+14}{2}\]

\[Q_{1}=11\]

The upper quartile, or **third quartile** $(Q_{3})$, is the value at which $75\%$ of the data points are **subdivided** when arranged in** increasing order**.

The **third quartile** is defined as the **median of the data values above the median.**

\[Q_{3}=\dfrac{18+20}{2}\]

\[Q_{3}=19\]

The **interquartile range** $(IQR)$ is the **difference between the first quartile **$Q_{1}$** and the third quartile** $Q_{3}$.

\[IQR=Q_{3}-Q_{1}\]

\[IQR=19-11\]

\[IQR=8\]

The **interquartile range** is $8$.

The **range** is calculated as:

\[Range=12\]

The** interquartile range** $(IQR)$ is calculated as:

\[IQR=8\]