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Consider a sample with data values of 10, 20, 12,17, and 16. Compute the range and interquartile range.

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:

\[IQR=8\]

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