Q55. - Algebra 1

Question

Find the range, median, lower quartile, and upper quartile for {16,19,21,24,25,31,35}.

Step-by-Step Solution

Answer

Range = 19

Median = 24

Lower quartile = 19

Upper quartile = 31

Step-1. Apply the concept of range, median, lower quartile, and upper quartile

• The range is the difference between the largest observation and the smallest observation of a data set.
• The median is the middle value when a data set is ordered from least to greatest. It is the second quartile (Q₂) and divides the data into two half.

• The lower quartile Q₁ is the median of the lower half of the data. 
• The upper quartile Q₃ is the median of the upper half of the data.

Step-2. Calculation of range.

Given:

{16,19,21,24,25,31,35}.

Arrange the data set in ascending order:

16,19,21,24,25,31,35

$\begin{array}{l}Range=L\arg est-Smallest\\ \Rightarrow Range=35-16\\ \Rightarrow Range=19\end{array}$

Step-3. Calculation of median.

Given:

{16,19,21,24,25,31,35}.

Arrange the data set in ascending order:

16,19,21,24,25,31,35

So, the median is the average of ${\left(\frac{7+1}{2}\right)}^{th}=4^{th}$ observation

Therefore, the median is $Q_2=24$.

Step-4. Calculation of lower quartile.

Given:

{16,19,21,24,25,31,35}.

Arrange the data set in ascending order:

16,19,21,24,25,31,35

From step 3, the median of the data set is 24 and the data set is odd 

Hence, the lower half of the data set is :

16,19,21

The lower quartile is the median of the lower half of the data set:

$Q_1=19$

Step-5. Calculation of upper quartile.

Given:

{16,19,21,24,25,31,35}.

Arrange the data set in ascending order:

16,19,21,24,25,31,35

From step 3, the median of the data set is 24 and the data set is odd 

Hence, the upper half of the data set is:

25,31,35

The upper quartile is the median of the upper half of the data set:

$\Rightarrow Q_3=31$